Background: #fff
Foreground: #000
PrimaryPale: #8cf
PrimaryLight: #18f
PrimaryMid: #04b
PrimaryDark: #014
SecondaryPale: #ffc
SecondaryLight: #fe8
SecondaryMid: #db4
SecondaryDark: #841
TertiaryPale: #eee
TertiaryLight: #ccc
TertiaryMid: #999
TertiaryDark: #666
Error: #f88
<!--{{{-->
<div class='toolbar' macro='toolbar [[ToolbarCommands::EditToolbar]]'></div>
<div class='title' macro='view title'></div>
<div class='editor' macro='edit title'></div>
<div macro='annotations'></div>
<div class='editor' macro='edit text'></div>
<div class='editor' macro='edit tags'></div><div class='editorFooter'><span macro='message views.editor.tagPrompt'></span><span macro='tagChooser excludeLists'></span></div>
<!--}}}-->
To get started with this blank [[TiddlyWiki]], you'll need to modify the following tiddlers:
* [[SiteTitle]] & [[SiteSubtitle]]: The title and subtitle of the site, as shown above (after saving, they will also appear in the browser title bar)
* [[MainMenu]]: The menu (usually on the left)
* [[DefaultTiddlers]]: Contains the names of the tiddlers that you want to appear when the TiddlyWiki is opened
You'll also need to enter your username for signing your edits: <<option txtUserName>>
<<importTiddlers>>
<!--{{{-->
<link rel='alternate' type='application/rss+xml' title='RSS' href='index.xml' />
<!--}}}-->
These [[InterfaceOptions]] for customising [[TiddlyWiki]] are saved in your browser

Your username for signing your edits. Write it as a [[WikiWord]] (eg [[JoeBloggs]])

<<option txtUserName>>
<<option chkSaveBackups>> [[SaveBackups]]
<<option chkAutoSave>> [[AutoSave]]
<<option chkRegExpSearch>> [[RegExpSearch]]
<<option chkCaseSensitiveSearch>> [[CaseSensitiveSearch]]
<<option chkAnimate>> [[EnableAnimations]]

----
Also see [[AdvancedOptions]]
<!--{{{-->
<div class='header' role='banner' macro='gradient vert [[ColorPalette::PrimaryLight]] [[ColorPalette::PrimaryMid]]'>
<div class='headerShadow'>
<span class='siteTitle' refresh='content' tiddler='SiteTitle'></span>&nbsp;
<span class='siteSubtitle' refresh='content' tiddler='SiteSubtitle'></span>
</div>
<div class='headerForeground'>
<span class='siteTitle' refresh='content' tiddler='SiteTitle'></span>&nbsp;
<span class='siteSubtitle' refresh='content' tiddler='SiteSubtitle'></span>
</div>
</div>
<div id='mainMenu' role='navigation' refresh='content' tiddler='MainMenu'></div>
<div id='sidebar'>
<div id='sidebarOptions' role='navigation' refresh='content' tiddler='SideBarOptions'></div>
<div id='sidebarTabs' role='complementary' refresh='content' force='true' tiddler='SideBarTabs'></div>
</div>
<div id='displayArea' role='main'>
<div id='messageArea'></div>
<div id='tiddlerDisplay'></div>
</div>
<!--}}}-->
/*{{{*/
body {background:[[ColorPalette::Background]]; color:[[ColorPalette::Foreground]];}

a {color:[[ColorPalette::PrimaryMid]];}
a:hover {background-color:[[ColorPalette::PrimaryMid]]; color:[[ColorPalette::Background]];}
a img {border:0;}

h1,h2,h3,h4,h5,h6 {color:[[ColorPalette::SecondaryDark]]; background:transparent;}
h1 {border-bottom:2px solid [[ColorPalette::TertiaryLight]];}
h2,h3 {border-bottom:1px solid [[ColorPalette::TertiaryLight]];}

.button {color:[[ColorPalette::PrimaryDark]]; border:1px solid [[ColorPalette::Background]];}
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.button:active {color:[[ColorPalette::Background]]; background:[[ColorPalette::SecondaryMid]]; border:1px solid [[ColorPalette::SecondaryDark]];}

.header {background:[[ColorPalette::PrimaryMid]];}
.headerShadow {color:[[ColorPalette::Foreground]];}
.headerShadow a {font-weight:normal; color:[[ColorPalette::Foreground]];}
.headerForeground {color:[[ColorPalette::Background]];}
.headerForeground a {font-weight:normal; color:[[ColorPalette::PrimaryPale]];}

.tabSelected {color:[[ColorPalette::PrimaryDark]];
	background:[[ColorPalette::TertiaryPale]];
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.tabUnselected {color:[[ColorPalette::Background]]; background:[[ColorPalette::TertiaryMid]];}
.tabContents {color:[[ColorPalette::PrimaryDark]]; background:[[ColorPalette::TertiaryPale]]; border:1px solid [[ColorPalette::TertiaryLight]];}
.tabContents .button {border:0;}

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#sidebarOptions input {border:1px solid [[ColorPalette::PrimaryMid]];}
#sidebarOptions .sliderPanel {background:[[ColorPalette::PrimaryPale]];}
#sidebarOptions .sliderPanel a {border:none;color:[[ColorPalette::PrimaryMid]];}
#sidebarOptions .sliderPanel a:hover {color:[[ColorPalette::Background]]; background:[[ColorPalette::PrimaryMid]];}
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.wizard h2 {color:[[ColorPalette::Foreground]]; border:none;}
.wizardStep {background:[[ColorPalette::Background]]; color:[[ColorPalette::Foreground]];
	border:1px solid [[ColorPalette::PrimaryMid]];}
.wizardStep.wizardStepDone {background:[[ColorPalette::TertiaryLight]];}
.wizardFooter {background:[[ColorPalette::PrimaryPale]];}
.wizardFooter .status {background:[[ColorPalette::PrimaryDark]]; color:[[ColorPalette::Background]];}
.wizard .button {color:[[ColorPalette::Foreground]]; background:[[ColorPalette::SecondaryLight]]; border: 1px solid;
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.wizard .button:active {color:[[ColorPalette::Background]]; background:[[ColorPalette::Foreground]]; border: 1px solid;
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.wizard .notChanged {background:transparent;}
.wizard .changedLocally {background:#80ff80;}
.wizard .changedServer {background:#8080ff;}
.wizard .changedBoth {background:#ff8080;}
.wizard .notFound {background:#ffff80;}
.wizard .putToServer {background:#ff80ff;}
.wizard .gotFromServer {background:#80ffff;}

#messageArea {border:1px solid [[ColorPalette::SecondaryMid]]; background:[[ColorPalette::SecondaryLight]]; color:[[ColorPalette::Foreground]];}
#messageArea .button {color:[[ColorPalette::PrimaryMid]]; background:[[ColorPalette::SecondaryPale]]; border:none;}

.popupTiddler {background:[[ColorPalette::TertiaryPale]]; border:2px solid [[ColorPalette::TertiaryMid]];}

.popup {background:[[ColorPalette::TertiaryPale]]; color:[[ColorPalette::TertiaryDark]]; border-left:1px solid [[ColorPalette::TertiaryMid]]; border-top:1px solid [[ColorPalette::TertiaryMid]]; border-right:2px solid [[ColorPalette::TertiaryDark]]; border-bottom:2px solid [[ColorPalette::TertiaryDark]];}
.popup hr {color:[[ColorPalette::PrimaryDark]]; background:[[ColorPalette::PrimaryDark]]; border-bottom:1px;}
.popup li.disabled {color:[[ColorPalette::TertiaryMid]];}
.popup li a, .popup li a:visited {color:[[ColorPalette::Foreground]]; border: none;}
.popup li a:hover {background:[[ColorPalette::SecondaryLight]]; color:[[ColorPalette::Foreground]]; border: none;}
.popup li a:active {background:[[ColorPalette::SecondaryPale]]; color:[[ColorPalette::Foreground]]; border: none;}
.popupHighlight {background:[[ColorPalette::Background]]; color:[[ColorPalette::Foreground]];}
.listBreak div {border-bottom:1px solid [[ColorPalette::TertiaryDark]];}

.tiddler .defaultCommand {font-weight:bold;}

.shadow .title {color:[[ColorPalette::TertiaryDark]];}

.title {color:[[ColorPalette::SecondaryDark]];}
.subtitle {color:[[ColorPalette::TertiaryDark]];}

.toolbar {color:[[ColorPalette::PrimaryMid]];}
.toolbar a {color:[[ColorPalette::TertiaryLight]];}
.selected .toolbar a {color:[[ColorPalette::TertiaryMid]];}
.selected .toolbar a:hover {color:[[ColorPalette::Foreground]];}

.tagging, .tagged {border:1px solid [[ColorPalette::TertiaryPale]]; background-color:[[ColorPalette::TertiaryPale]];}
.selected .tagging, .selected .tagged {background-color:[[ColorPalette::TertiaryLight]]; border:1px solid [[ColorPalette::TertiaryMid]];}
.tagging .listTitle, .tagged .listTitle {color:[[ColorPalette::PrimaryDark]];}
.tagging .button, .tagged .button {border:none;}

.footer {color:[[ColorPalette::TertiaryLight]];}
.selected .footer {color:[[ColorPalette::TertiaryMid]];}

.error, .errorButton {color:[[ColorPalette::Foreground]]; background:[[ColorPalette::Error]];}
.warning {color:[[ColorPalette::Foreground]]; background:[[ColorPalette::SecondaryPale]];}
.lowlight {background:[[ColorPalette::TertiaryLight]];}

.zoomer {background:none; color:[[ColorPalette::TertiaryMid]]; border:3px solid [[ColorPalette::TertiaryMid]];}

.imageLink, #displayArea .imageLink {background:transparent;}

.annotation {background:[[ColorPalette::SecondaryLight]]; color:[[ColorPalette::Foreground]]; border:2px solid [[ColorPalette::SecondaryMid]];}

.viewer .listTitle {list-style-type:none; margin-left:-2em;}
.viewer .button {border:1px solid [[ColorPalette::SecondaryMid]];}
.viewer blockquote {border-left:3px solid [[ColorPalette::TertiaryDark]];}

.viewer table, table.twtable {border:2px solid [[ColorPalette::TertiaryDark]];}
.viewer th, .viewer thead td, .twtable th, .twtable thead td {background:[[ColorPalette::SecondaryMid]]; border:1px solid [[ColorPalette::TertiaryDark]]; color:[[ColorPalette::Background]];}
.viewer td, .viewer tr, .twtable td, .twtable tr {border:1px solid [[ColorPalette::TertiaryDark]];}

.viewer pre {border:1px solid [[ColorPalette::SecondaryLight]]; background:[[ColorPalette::SecondaryPale]];}
.viewer code {color:[[ColorPalette::SecondaryDark]];}
.viewer hr {border:0; border-top:dashed 1px [[ColorPalette::TertiaryDark]]; color:[[ColorPalette::TertiaryDark]];}

.highlight, .marked {background:[[ColorPalette::SecondaryLight]];}

.editor input {border:1px solid [[ColorPalette::PrimaryMid]];}
.editor textarea {border:1px solid [[ColorPalette::PrimaryMid]]; width:100%;}
.editorFooter {color:[[ColorPalette::TertiaryMid]];}
.readOnly {background:[[ColorPalette::TertiaryPale]];}

#backstageArea {background:[[ColorPalette::Foreground]]; color:[[ColorPalette::TertiaryMid]];}
#backstageArea a {background:[[ColorPalette::Foreground]]; color:[[ColorPalette::Background]]; border:none;}
#backstageArea a:hover {background:[[ColorPalette::SecondaryLight]]; color:[[ColorPalette::Foreground]]; }
#backstageArea a.backstageSelTab {background:[[ColorPalette::Background]]; color:[[ColorPalette::Foreground]];}
#backstageButton a {background:none; color:[[ColorPalette::Background]]; border:none;}
#backstageButton a:hover {background:[[ColorPalette::Foreground]]; color:[[ColorPalette::Background]]; border:none;}
#backstagePanel {background:[[ColorPalette::Background]]; border-color: [[ColorPalette::Background]] [[ColorPalette::TertiaryDark]] [[ColorPalette::TertiaryDark]] [[ColorPalette::TertiaryDark]];}
.backstagePanelFooter .button {border:none; color:[[ColorPalette::Background]];}
.backstagePanelFooter .button:hover {color:[[ColorPalette::Foreground]];}
#backstageCloak {background:[[ColorPalette::Foreground]]; opacity:0.6; filter:alpha(opacity=60);}
/*}}}*/
/*{{{*/
* html .tiddler {height:1%;}

body {font-size:.75em; font-family:arial,helvetica; margin:0; padding:0;}

h1,h2,h3,h4,h5,h6 {font-weight:bold; text-decoration:none;}
h1,h2,h3 {padding-bottom:1px; margin-top:1.2em;margin-bottom:0.3em;}
h4,h5,h6 {margin-top:1em;}
h1 {font-size:1.35em;}
h2 {font-size:1.25em;}
h3 {font-size:1.1em;}
h4 {font-size:1em;}
h5 {font-size:.9em;}

hr {height:1px;}

a {text-decoration:none;}

dt {font-weight:bold;}

ol {list-style-type:decimal;}
ol ol {list-style-type:lower-alpha;}
ol ol ol {list-style-type:lower-roman;}
ol ol ol ol {list-style-type:decimal;}
ol ol ol ol ol {list-style-type:lower-alpha;}
ol ol ol ol ol ol {list-style-type:lower-roman;}
ol ol ol ol ol ol ol {list-style-type:decimal;}

.txtOptionInput {width:11em;}

#contentWrapper .chkOptionInput {border:0;}

.externalLink {text-decoration:underline;}

.indent {margin-left:3em;}
.outdent {margin-left:3em; text-indent:-3em;}
code.escaped {white-space:nowrap;}

.tiddlyLinkExisting {font-weight:bold;}
.tiddlyLinkNonExisting {font-style:italic;}

/* the 'a' is required for IE, otherwise it renders the whole tiddler in bold */
a.tiddlyLinkNonExisting.shadow {font-weight:bold;}

#mainMenu .tiddlyLinkExisting,
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	#sidebarTabs .tiddlyLinkNonExisting {font-weight:normal; font-style:normal;}
#sidebarTabs .tiddlyLinkExisting {font-weight:bold; font-style:normal;}

.header {position:relative;}
.header a:hover {background:transparent;}
.headerShadow {position:relative; padding:4.5em 0 1em 1em; left:-1px; top:-1px;}
.headerForeground {position:absolute; padding:4.5em 0 1em 1em; left:0; top:0;}

.siteTitle {font-size:3em;}
.siteSubtitle {font-size:1.2em;}

#mainMenu {position:absolute; left:0; width:10em; text-align:right; line-height:1.6em; padding:1.5em 0.5em 0.5em 0.5em; font-size:1.1em;}

#sidebar {position:absolute; right:3px; width:16em; font-size:.9em;}
#sidebarOptions {padding-top:0.3em;}
#sidebarOptions a {margin:0 0.2em; padding:0.2em 0.3em; display:block;}
#sidebarOptions input {margin:0.4em 0.5em;}
#sidebarOptions .sliderPanel {margin-left:1em; padding:0.5em; font-size:.85em;}
#sidebarOptions .sliderPanel a {font-weight:bold; display:inline; padding:0;}
#sidebarOptions .sliderPanel input {margin:0 0 0.3em 0;}
#sidebarTabs .tabContents {width:15em; overflow:hidden;}

.wizard {padding:0.1em 1em 0 2em;}
.wizard h1 {font-size:2em; font-weight:bold; background:none; padding:0; margin:0.4em 0 0.2em;}
.wizard h2 {font-size:1.2em; font-weight:bold; background:none; padding:0; margin:0.4em 0 0.2em;}
.wizardStep {padding:1em 1em 1em 1em;}
.wizard .button {margin:0.5em 0 0; font-size:1.2em;}
.wizardFooter {padding:0.8em 0.4em 0.8em 0;}
.wizardFooter .status {padding:0 0.4em; margin-left:1em;}
.wizard .button {padding:0.1em 0.2em;}

#messageArea {position:fixed; top:2em; right:0; margin:0.5em; padding:0.5em; z-index:2000; _position:absolute;}
.messageToolbar {display:block; text-align:right; padding:0.2em;}
#messageArea a {text-decoration:underline;}

.tiddlerPopupButton {padding:0.2em;}
.popupTiddler {position: absolute; z-index:300; padding:1em; margin:0;}

.popup {position:absolute; z-index:300; font-size:.9em; padding:0; list-style:none; margin:0;}
.popup .popupMessage {padding:0.4em;}
.popup hr {display:block; height:1px; width:auto; padding:0; margin:0.2em 0;}
.popup li.disabled {padding:0.4em;}
.popup li a {display:block; padding:0.4em; font-weight:normal; cursor:pointer;}
.listBreak {font-size:1px; line-height:1px;}
.listBreak div {margin:2px 0;}

.tabset {padding:1em 0 0 0.5em;}
.tab {margin:0 0 0 0.25em; padding:2px;}
.tabContents {padding:0.5em;}
.tabContents ul, .tabContents ol {margin:0; padding:0;}
.txtMainTab .tabContents li {list-style:none;}
.tabContents li.listLink { margin-left:.75em;}

#contentWrapper {display:block;}
#splashScreen {display:none;}

#displayArea {margin:1em 17em 0 14em;}

.toolbar {text-align:right; font-size:.9em;}

.tiddler {padding:1em 1em 0;}

.missing .viewer,.missing .title {font-style:italic;}

.title {font-size:1.6em; font-weight:bold;}

.missing .subtitle {display:none;}
.subtitle {font-size:1.1em;}

.tiddler .button {padding:0.2em 0.4em;}

.tagging {margin:0.5em 0.5em 0.5em 0; float:left; display:none;}
.isTag .tagging {display:block;}
.tagged {margin:0.5em; float:right;}
.tagging, .tagged {font-size:0.9em; padding:0.25em;}
.tagging ul, .tagged ul {list-style:none; margin:0.25em; padding:0;}
.tagClear {clear:both;}

.footer {font-size:.9em;}
.footer li {display:inline;}

.annotation {padding:0.5em; margin:0.5em;}

* html .viewer pre {width:99%; padding:0 0 1em 0;}
.viewer {line-height:1.4em; padding-top:0.5em;}
.viewer .button {margin:0 0.25em; padding:0 0.25em;}
.viewer blockquote {line-height:1.5em; padding-left:0.8em;margin-left:2.5em;}
.viewer ul, .viewer ol {margin-left:0.5em; padding-left:1.5em;}

.viewer table, table.twtable {border-collapse:collapse; margin:0.8em 1.0em;}
.viewer th, .viewer td, .viewer tr,.viewer caption,.twtable th, .twtable td, .twtable tr,.twtable caption {padding:3px;}
table.listView {font-size:0.85em; margin:0.8em 1.0em;}
table.listView th, table.listView td, table.listView tr {padding:0 3px 0 3px;}

.viewer pre {padding:0.5em; margin-left:0.5em; font-size:1.2em; line-height:1.4em; overflow:auto;}
.viewer code {font-size:1.2em; line-height:1.4em;}

.editor {font-size:1.1em;}
.editor input, .editor textarea {display:block; width:100%; font:inherit;}
.editorFooter {padding:0.25em 0; font-size:.9em;}
.editorFooter .button {padding-top:0; padding-bottom:0;}

.fieldsetFix {border:0; padding:0; margin:1px 0px;}

.zoomer {font-size:1.1em; position:absolute; overflow:hidden;}
.zoomer div {padding:1em;}

* html #backstage {width:99%;}
* html #backstageArea {width:99%;}
#backstageArea {display:none; position:relative; overflow: hidden; z-index:150; padding:0.3em 0.5em;}
#backstageToolbar {position:relative;}
#backstageArea a {font-weight:bold; margin-left:0.5em; padding:0.3em 0.5em;}
#backstageButton {display:none; position:absolute; z-index:175; top:0; right:0;}
#backstageButton a {padding:0.1em 0.4em; margin:0.1em;}
#backstage {position:relative; width:100%; z-index:50;}
#backstagePanel {display:none; z-index:100; position:absolute; width:90%; margin-left:3em; padding:1em;}
.backstagePanelFooter {padding-top:0.2em; float:right;}
.backstagePanelFooter a {padding:0.2em 0.4em;}
#backstageCloak {display:none; z-index:20; position:absolute; width:100%; height:100px;}

.whenBackstage {display:none;}
.backstageVisible .whenBackstage {display:block;}
/*}}}*/
/***
StyleSheet for use when a translation requires any css style changes.
This StyleSheet can be used directly by languages such as Chinese, Japanese and Korean which need larger font sizes.
***/
/*{{{*/
body {font-size:0.8em;}
#sidebarOptions {font-size:1.05em;}
#sidebarOptions a {font-style:normal;}
#sidebarOptions .sliderPanel {font-size:0.95em;}
.subtitle {font-size:0.8em;}
.viewer table.listView {font-size:0.95em;}
/*}}}*/
/*{{{*/
@media print {
#mainMenu, #sidebar, #messageArea, .toolbar, #backstageButton, #backstageArea {display: none !important;}
#displayArea {margin: 1em 1em 0em;}
noscript {display:none;} /* Fixes a feature in Firefox 1.5.0.2 where print preview displays the noscript content */
}
/*}}}*/
<!--{{{-->
<div class='toolbar' role='navigation' macro='toolbar [[ToolbarCommands::ViewToolbar]]'></div>
<div class='title' macro='view title'></div>
<div class='subtitle'><span macro='view modifier link'></span>, <span macro='view modified date'></span> (<span macro='message views.wikified.createdPrompt'></span> <span macro='view created date'></span>)</div>
<div class='tagging' macro='tagging'></div>
<div class='tagged' macro='tags'></div>
<div class='viewer' macro='view text wikified'></div>
<div class='tagClear'></div>
<!--}}}-->
<<ListTagged 0>>
[<img[images/png/1-form.png]]A ''1-form'', or ''//cotangent vector//'', $\f{f}$, is a geometric object that acts on a [[tangent vector]] at a point, $p$, to give a real number.  It may be written in terms of the [[coordinate basis 1-forms]] as
\[ \f{f} = f_i \f{dx^i} \in T_{p}^{*}M \]
It is a linear operator, and so may be written as a function of a vector or more simply as a [[vector-form contraction|vector-form algebra]] (product),
\[ \f{f}(\ve{v}) = {\bf i}_{\ve{v}} \f{f} = \ve{v} \f{f} = v^j f_i \ve{\pa_j} \f{dx^i} = v^j f_i \de_j^i= v^i f_i \in \Re \]
The vector space of [[1-form]]s at each point, $p$, of a [[manifold]], $M$, is the ''cotangent space'', $T_{p}^* M$, and is spanned by the $\f{dx^i}$.
A ''2-sphere'' embedded in flat 3-space is a two dimensional [[manifold]], $M$, defined by the equation $x^e x^f \de_{ef} = r^2$ -- it is the surface of constant $r=r$ in [[spherical coordinates]]. The angular spherical coordinates, $(\th,\ph)$, cover patches of the manifold (other patches are needed for the poles). The [[metric]] induced on the 2-sphere is $g_{ij} = {\rm diag}(r^2, r^2 \sin^2(\th))$. The simplest [[frame]] compatible with this metric is
$$
\f{e} = \f{d \th} \, r \si_1 + \f{d \ph} \, r \sin(\th) \si_2
$$
in which $\si_{1/2}$ are the [[Clifford basis vectors]] for [[Cl(2,0)|Clifford matrix representation]]. The coframe is
$$
\ve{e} = \si^1 \fr{1}{r} \ve{\pa_\th} + \si^2 \fr{1}{r \sin(\th)} \ve{\pa_\ph}
$$
The [[torsion]]less [[spin connection]], found by solving [[Cartan's equation]], $0=\f{d} \f{e} + \f{\om} \times \f{e}$, is
$$
\f{\om} = - \ve{e} \times \f{d} \f{e} + \fr{1}{4} \lp \ve{e} \times \ve{e} \rp \lp \f{e} \cdot \f{d} \f{e} \rp = - \f{d \ph} \cos(\th) \si_{12}
$$
The [[Clifford vector bundle]] curvature is
$$
\ff{F} = \f{d} \f{\om} + \ha \f{\om} \f{\om} = \f{d \th} \f{d \ph} \sin(\th) \si_{12}
$$
The [[Clifford-Ricci curvature]] is
$$
\f{R} = \ve{e} \times \ff{F} = \f{d \ph} \fr{1}{r} \sin(\th) \si_2 + \f{d \th} \fr{1}{r} \si_1 = \fr{1}{r^2} \f{e}
$$
showing that the 2-sphere is an [[Einstein space|Einstein's equation]] with cosmological constant $\La = 0$ (as do all two dimensional spaces). The [[Clifford curvature scalar]] is $R = \ve{e} \cdot \f{R} = \fr{2}{r^2}$.
A ''3-sphere'' embedded in flat 4-space of positive signature is a three dimensional [[manifold]], $M$, defined by the equation $x^w x^x \de_{wx} = r^2$ -- it is the surface of constant $r=r$ in $4d$ [[hyperspherical coordinates]],
\begin{eqnarray}
x^1 &=& r \cos(a^1) \\
x^2 &=& r \sin(a^1) \cos(a^2) \\
x^3 &=& r \sin(a^1) \sin(a^2) \cos(a^3) \\
x^4 &=& r \sin(a^1) \sin(a^2) \sin(a^3)
\end{eqnarray}
The angular hyperspherical coordinates, $(a^1,a^2,a^3)$, cover patches of the manifold (other patches are needed for the poles). The [[metric]] induced on the 3-sphere is $g_{ij} = {\rm diag}(r^2, r^2 \sin^2(a^1),r^2 \sin^2(a^1) \sin^2(a^2))$. The simplest [[frame]] compatible with this metric is
$$
\f{e} = \f{d a^1} \, r \si_1 + \f{d a^2} \, r \sin(a^1) \si_2 + \f{d a^3} \, r \sin(a^1) \sin(a^2) \si_3
$$
in which $\si_{1/2/3}$ are the [[Clifford basis vectors]] for [[Cl(3,0)|Cl(3)]]. The coframe is
$$
\ve{e} = \si^1 \fr{1}{r} \ve{\pa_1} + \si^2 \fr{1}{r \sin(a^1)} \ve{\pa_2} + \si^3 \fr{1}{r \sin(a^1) \sin(a^2)} \ve{\pa_3}
$$
The [[torsion]]less [[spin connection]], found by solving [[Cartan's equation]], $0=\f{d} \f{e} + \f{\om} \times \f{e}$, is
\begin{eqnarray}
\f{\om} &=& - \ve{e} \times \f{d} \f{e} + \fr{1}{4} \lp \ve{e} \times \ve{e} \rp \lp \f{e} \cdot \f{d} \f{e} \rp \\
&=& - \f{d a^2} \cos(a^1) \si_{12} - \f{d a^3} \cos(a^1) \sin(a^2) \si_{13} - \f{d a^3} \cos(a^2) \si_{23}
\end{eqnarray}
The [[Clifford vector bundle]] curvature is
\begin{eqnarray}
\ff{F} &=& \f{d} \f{\om} + \ha \f{\om} \f{\om} \\
&=& \f{d a^1} \f{d a^2} \sin(a^1) \si_{12} + \f{d a^1} \f{d a^3} \sin(a^1) \sin(a^2) \si_{13} + \f{d a^2} \f{d a^3} \sin^2(a^1) \sin(a^2) \si_{23}
\end{eqnarray}
The [[Clifford-Ricci curvature]] is
\begin{eqnarray}
\f{R} &=& \ve{e} \times \ff{F} \\
&=& \f{d a^1} \fr{2}{r} \si_1 + \f{d a^2} \fr{2}{r} \sin(a^1) \si_2 + \f{d a^3} \fr{2}{r} \sin(a^1) \sin(a^2) \si_3 \\
&=& \fr{2}{r^2} \f{e}
\end{eqnarray}
showing that the 3-sphere is an [[Einstein space|Einstein's equation]] with cosmological constant $\La = \fr{1}{2 r^2}$. The [[Clifford curvature scalar]] is $R = \ve{e} \cdot \f{R} = \fr{8}{r^2}$.
<<tiddler HideTags>>@@display:block;text-align:center;
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<img src="talks/StAnth09/images/bubblechamber3.png" width="585" height="440">
</center></html>
$\p{{}_{\small (}^{(}}$&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
[[Garrett Lisi]] '86&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Cate Centennial@@
<<tiddler HideTags>>@@display:block;text-align:center;
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<img src="talks/StAnth09/images/bubblechamber3.png" width="585" height="440">
</center></html>
$\p{{}_{\small (}^{(}}$&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
[[Garrett Lisi]]&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Goldman Lecture, UCF, April 14, 2011@@
<<tiddler HideTags>>@@display:block;text-align:center;



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<img src="talks/CSUF09/images/D8PSsGna.png" width="300" height="300">
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$\p{{}_{\small (}^{(}}$



[[Garrett Lisi]]&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; CSUF 4/24/09@@
<<tiddler HideTags>>@@display:block;text-align:center;
<html><center>
<img src="talks/StAnth09/images/bubblechamber3.png" width="585" height="440">
</center></html>
$\p{{}_{\small (}^{(}}$&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
[[Garrett Lisi]]&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; IEEE Aerospace Conference, Big Sky, Montana, March 6, 2011@@
<<tiddler HideTags>>$$
\begin{array}{rclclc}
\f{\om} \!\!&\!\!=\!\!&\!\! \f{dx^k} \ha \om_k^{\p{k}\mu\nu} \ga_{\mu\nu} \!&\!\! \in \!\!&\! \f{Cl}^2(3,1)
&
\quad
\f{e} = \f{dx^k} (e_k)^\mu \ga_\mu \, \in \, \f{Cl}^1(3,1) \vp{|_{(}} \\

\f{W} \!\!&\!\!=\!\!&\!\! \f{dx^k} W_k^{\p{i}\pi} \fr{i}{2} \si_\pi  \!&\!\! \in \!\!&\! \f{su}(2)
&
\quad
\lb \matrix{
\ph_+ \\ \ph_0
} \rb
\qquad
\lb \matrix{
\nu_{eL} \\ e_L
} \rb
\\

\f{B} \!\!&\!\!=\!\!&\!\! \f{dx^k} B_k i \!&\!\! \in \!\!&\! \f{u}(1)
&
\quad
Y \\

\f{g} \!\!&\!\!=\!\!&\!\! \f{dx^k} g_k^{\p{k}A} \fr{i}{2} \la_A  \!&\!\! \in \!\!&\! \f{su}(3)
&
\quad
\lb u^r, u^g, u^b \rb \vp{|^{(^(}_{(}}
\end{array}
\begin{array}{c}
\quad
\lb \matrix{
e_L^\wedge \\ e_L^\vee \\ e_R^\wedge \\ e_R^\vee
} \rb
\; \\
\; \\
\end{array}
$$
$$
\updownarrow \vp{{\huge(}_{\big(}}
$$
$$
\begin{array}{rcl}
\udf{A} \!\!&\!\!=\!\!&\!\! {\small \frac{1}{2}} \f{\om} + {\small \frac{1}{4}} \f{e} \ph + \f{W} + \f{B}  + \f{g} + ( \ud{\nu}{}_e + \ud{e} + \ud{u} + \ud{d} ) \\
&& + \, (\ud{\nu}{}_\mu + \ud{\mu} + \ud{c} + \ud{s}) + (\ud{\nu}{}_\ta + \ud{\ta} + \ud{t} + \ud{b}) \vp{|_{\Big(}}
\end{array}
$$
$$
\udff{F} = \f{d} \udf{A} + {\scriptsize \frac{1}{2}} \big[ \udf{A}, \udf{A} \big]
$$
This hasn't been around long enough for any questions asked to be frequent, but I'll try to anticipate some.

!!Who?
The site is principally authored by me, [[Garrett Lisi]].  I may open it up for collaboration in the future.

As to who it's good for... well, mostly it's for my own use.  But if you have a background in physics and math, with at least some graduate level work under your belt, most of what's here should be accessible to you, and some may even be of interest.

!!What?
It's sort of a "choose your own adventure" book in theoretical physics &mdash; only the book is being written day-by-day and no one knows the ending, or if there is one.  It's my real-time research tiddlerbook, made available to public view.  I hope to make it comparable to an open ended [[Living Reviews in Relativity|http://relativity.livingreviews.org/]] article in spirit and quality, but updated more frequently and navigable as a wiki.  My long term goal is to construct a concise and beautiful theoretical description of reality unifying General Relativity, Quantum Field Theory, and the Standard Model using the foundations and language of basic differential geometry.  Such a theory may not exist, but that's what I'm after.  And here you can watch me walk down every dark alleyway looking for it &mdash; until I find it, or at least some interesting stuff along the way.  This evolving search tree will grow and be pruned in ways I can't now predict.  But I expect the information contained to be equivalent to a book and several overlapping research papers, wikified and presented as they are written.  It's open source physics.

!!Why?
I needed a way to organize my physics tiddlers.  And I was simultaneously contemplating the best way to present and navigate theoretical physics.  [[Semantic Networks|http://www.jfsowa.com/pubs/semnet.htm]] provide a natural structure for relating abstract conceptual information,  and I considered building a graphical system to do what I want along those lines... but a wiki is a good practical equivalent.  It allows a reader to quickly learn new concepts in digestible pieces, and trace forwards or backwards to the implications or foundations of those concepts &mdash; while allowing an author, or authors, to easily expand the content.  I am very impressed with the way [[Wikipedia|http://en.wikipedia.org/wiki/Main_Page]] works and evolves, and this is my own personal version, for research.

!!How?
The two main pieces from which this site is built are [[TiddlyWiki|http://www.tiddlywiki.com/]], created by Jeremy Ruston, and [[jsMath|http://www.math.union.edu/~dpvc/jsMath/]], made by Davide P. Cervone.  Both of these excellent open source software packages are under continuing development and have supportive communities.  I owe thanks to many people for building the pieces used for this site, and for helping out with technical details.  To delve more into the nitty-gritty of how it's put together, and see who contributed which plugins, go check out the [[Configuration]].  If things don't work perfectly, it's probably my fault. You can get everything you need to set up a similar wiki for yourself from this [[downloads directory|http://deferentialgeometry.org/download/]].

To use it... try things.  Click on buttons and see what happens, you'll figure it out.

!!What about money for food?

My research is supported entirely by private contributions. Support is always appreciated, is tax deductible, and may be contributed by contacting me or [[Theiss Research|http://www.theissresearch.org/]], a 501(c)(3) corporation.

!!Where
This site is served from a closet in San Jose, California (thanks Rich!).  It's currently mirrored from a laptop on a volcanic island in the middle of the Pacific, but the laptop follows its owner everywhere... except out surfing &mdash; it hates that.

!!When
Now.

----
<<slider chkSliderAbout 'About (slider)' 'More questions and answers >' 'Click to see more questions and answers'>>
!!How did you come up with the title, "Deferential Geometry"?
My favorite interpretation is that it's about geometry in the service of physics.  There is a lot of bad theoretical physics out there without math, and a lot of good math without physics; good physics uses math, and this site is about using only the math needed by physics. There shouldn't be any mathematical tangents here without physics ideas motivating them &mdash; the geometry is deferential to the physics.
<<tiddler HideTags>>Bosonic connection:
$$\f{H} = {\textstyle \ha} \f{\om} + {\textstyle \fr{1}{4}} \f{e} \ph + \f{G} \;\;\;\; \in spin(3,1) + 4 \! \times \! 10 + spin(10) = spin(3,11)$$
Curvature:
$$\ff{F} = \f{d} \f{H} + \f{H} \f{H} = {\textstyle \ha} (\ff{R} - {\textstyle \fr{1}{8}} \f{e}\f{e} \ph^2) + {\textstyle \fr{1}{4}} (\f{T} \ph - \f{e} \f{D} \ph) + \ff{F}{}^G \vp{A_{\big(}}$$
Riemann: $\;\; \ff{R} = \f{d} \f{\om} + \ha \f{\om} \f{\om} \;\;\;\;\;\;$ Torsion: $\;\; \ff{T} = \f{d} \f{e} + \ha \f{\om} \f{e} + \ha \f{e} \f{\om} \;\;\;\;\;\; $ Covariant: $\;\; \f{D} \ph = \f{d} \ph + \f{G} \ph - \ph \f{G} \vp{A_{\big(}}$
$spin(3,1)$ duality operator: $\;\; \ep = \Ga_1 \Ga_2 \Ga_3 \Ga_4 \;\;\;\;\;\;\;\;$ Hodge duality operator: $\;\; * = \ff{\vv{\ep}} = \big< \f{e} \f{e} \ep \ve{e} \ve{e} \big> \vp{A_{\Big(}}$
Boson action:
\begin{eqnarray}
S_H &=& \int \big< {\textstyle \fr{-1}{\pi G}} \ff{F} \ep \ff{F} + {\textstyle \fr{1}{4g^2}} \ff{F} \ff{\vv{\ep}} \ff{F} \big> \\
&=& \int \big< {\textstyle \fr{-1}{4 \pi G}} \ff{R} \ff{R} \ep + {\textstyle \fr{1}{16 \pi G}} \ph^2 \ff{R} \f{e} \f{e} \ep + {\textstyle  \fr{3}{32\pi G}} \ph^4 \nf{e}
+ {\textstyle \fr{1}{64 g^2}} \ph^2 \ff{T} \ff{\vv{\ep}} \ff{T} + {\textstyle \fr{1}{64 g^2}} \f{e} \f{D} \ph \ff{\vv{\ep}} \f{e} \f{D} \ph
+ {\textstyle \fr{1}{4g^2}} \ff{F}^G \ff{\vv{\ep}} \ff{F}^G \big> \\
&\sim& \int \big< {\textstyle \fr{1}{16\pi G}} \nf{e} \big( R - {\textstyle \fr{3}{2}} \ph^2 \big) \ph^2 + {\textstyle \fr{1}{4g^2}} \f{D} \ph \fff{*D} \ph + {\textstyle \fr{1}{4g^2}} \ff{F}^G \ff{*F}^G \big>
\end{eqnarray}
A possible $spin(3,11)$ invariant action:
$$
S_H = \int \big< \ff{B} \ff{F} - \ff{B} \Ph \ff{B} + {\textstyle \fr{\al}{3}} \ff{B} \Ph^3 \ff{B} \big>
$$
Symmetry breaking: $\;\;\;\;\;\;\;\;\; \Ph^0 = \fr{\pi G}{4} \ep - g^2 \ff{\vv{\ep}} \;\;\;\;\;\;\;\;\;\; \f{H}^0 = \fr{1}{4}\f{e}\ph^0$
<<tiddler HideTags>>Bosonic connection:
$$\f{H} = {\textstyle \ha} \f{\om} + {\textstyle \fr{1}{4}} \f{e} \ph + \f{A} \;\;\;\; \in spin(1,3) \,\oplus\, 4 \! \otimes \! 10 \,\oplus\, spin(10) = spin(11,3)$$
Curvature:
$$\ff{F} = \f{d} \f{H} + \f{H} \f{H} = {\textstyle \ha} (\ff{R} - {\textstyle \fr{1}{8}} \f{e}\f{e} \ph^2) + {\textstyle \fr{1}{4}} (\f{T} \ph - \f{e} \f{D} \ph) + \ff{F}{}^A \vp{A_{\big(}}$$
Riemann: $\;\; \ff{R} = \f{d} \f{\om} + \ha \f{\om} \f{\om} \;\;\;\;\;\;$ Torsion: $\;\; \ff{T} = \f{d} \f{e} + \ha \f{\om} \f{e} + \ha \f{e} \f{\om} \;\;\;\;\;\; $ Covariant: $\;\; \f{D} \ph = \f{d} \ph + \f{A} \ph - \ph \f{A} \vp{A_{\big(}}$
$spin(1,3)$ duality operator: $\;\; \ep = \Ga_1 \Ga_2 \Ga_3 \Ga_4 \;\;\;\;\;\;\;\;$ Hodge duality operator: $\;\; * = \ff{\vv{\ep}} = \big< \f{e} \f{e} \ep \ve{e} \ve{e} \big> \vp{A_{\Big(}}$
Boson action:
\begin{eqnarray}
S_H &=& \int \big< {\textstyle \fr{-1}{\pi G}} \ff{F} \ep \ff{F} + {\textstyle \fr{1}{4g^2}} \ff{F} \ff{\vv{\ep}} \ff{F} \big> \\
&=& \int \big< {\textstyle \fr{-1}{4 \pi G}} \ff{R} \ff{R} \ep + {\textstyle \fr{1}{16 \pi G}} \ph^2 \ff{R} \f{e} \f{e} \ep + {\textstyle  \fr{3}{32\pi G}} \ph^4 \nf{e}
+ {\textstyle \fr{1}{64 g^2}} \ph^2 \ff{T} \ff{\vv{\ep}} \ff{T} + {\textstyle \fr{1}{64 g^2}} \f{e} \f{D} \ph \ff{\vv{\ep}} \f{e} \f{D} \ph
+ {\textstyle \fr{1}{4g^2}} \ff{F}^A \ff{\vv{\ep}} \ff{F}^A \big> \\
&\sim& \int \big< {\textstyle \fr{1}{16\pi G}} \nf{e} \big( R - {\textstyle \fr{3}{2}} \ph^2 \big) \ph^2 + {\textstyle \fr{1}{4g^2}} \f{D} \ph \fff{*D} \ph + {\textstyle \fr{1}{4g^2}} \ff{F}^A \ff{*F}^A \big>
\end{eqnarray}
A possible $spin(11,3)$ invariant action:
$$
S_H = {\textstyle \fr{1}{g}} \int \big< \ff{B} \ff{F} + \ff{B} \Ph \ff{B} + {\textstyle \fr{1}{3}} \ff{B} \Ph^3 \ff{B} \big>
$$
Symmetry breaking: $\;\;\;\;\;\;\;\;\; \Ph_0 = \ff{\vv{\ep}} \;\;\;\;\;\;\;\;\;\; \f{H}{}_0 = \fr{1}{4}\f{e}{}_0 \ph_0$
Modified BF action, using $\ff{\od{B}} = \ff{B} + \fff{\od{B}} \,$:

\begin{eqnarray}
S &=& \int \big< \ff{\od{B}} \udff{F} + \nf{\Phi} ( \f{H}{}_1, \f{H}{}_2, \ff{B} ) \big> \\
&=& \int \big< \fff{\od{B}} \f{D} \ud{\Ps}
+ \ff{B} \ff{F} + {\scriptsize \frac{\pi G}{4}} \ff{B}{}_G \ff{B}{}_G \ga + \ff{B'} \ff{*B'} \big> \\
&=& \int \big< \fff{\od{B}} \f{D} \ud{\Ps}
+ \nf{e} \fr{1}{16 \pi G} \ph^2  \big( R - \fr{3}{2} \ph^2 \big) + \fr{1}{4} \ff{F'} \ff{*F'} \big>
\end{eqnarray}

Cosmological constant from the Higgs VEV: $\quad \La = \fr{3}{4} \ph^2$

Implies frame VEV is de Sitter: $\quad \ff{R} = \fr{\La}{6} \f{e} \f{e} \qquad R = 4 \La$

Vacuum expectation value of the curvature vanishes: $\quad \udff{F} = 0$
<<tiddler HideTags>>
<<tiddler HideTags>>$$
\udff{F} = \f{d} \udf{A} + \udf{A} \udf{A} = \big( \f{d} \f{H} + \f{H} \f{H} \big) + \big( \f{d} \f{G} + \f{G} \f{G} \big) + \big( \f{d} \ud{\ps} + \f{H} \ud{\ps} + \ud{\ps} \f{G} \big)
$$
Modified BF action for everything, using $\ff{\od{B}} = \ff{B} + \fff{\od{B}} \,$:
\begin{eqnarray}
S &=& \int \big< \ff{\od{B}} \udff{F} + \nf{\Phi} ( \f{H}, \f{G}, \ff{B} ) \big> \\
&=& \int \big< \fff{\od{B}} \big( \f{d} \ud{\ps} + \f{H} \ud{\ps} + \ud{\ps} \f{G} \big)
+ \ff{B} \ff{F} - {\scriptsize \frac{1}{4}} \ff{B_s} \ff{B_s} \ga + \ff{B_{m,h,G}} \ff{*B_{m,h,G}} \big>
\end{eqnarray}
Fermionic part, using [[anti-ghost|BRST technique]] [[Grassmann|Grassmann number]] 3-form, $\fff{\od{B}} = \nf{e} \od{\ps} \ve{e} \,$:
\begin{eqnarray}
S_f &=& \int \big< \fff{\od{B}} \big( \f{d} \ud{\ps} + \f{H} \ud{\ps} + \ud{\ps} \f{G} \big) \big> \\
&=& \int \big< \nf{e} \od{\ps} \ve{e} \big( \f{d} \ud{\ps} + {\scriptsize \frac{1}{2}} \f{\om} \ud{\ps} + {\scriptsize \frac{1}{4}} \f{e} \ph \ud{\ps} + \f{B} \ud{\ps} + \f{W} \ud{\ps} + \ud{\ps} \f{G} \big) \big> \\
&=& \int \nf{d^4 x} |e| \, \big< \od{\ps} \ga^\mu (e_\mu)^i \big( \pa_i \ud{\ps} + {\scriptsize \frac{1}{4}} \om_i^{\p{i} \mu \nu} \ga_{\mu \nu} \ud{\ps} + B_i \ud{\ps} + W_i \ud{\ps} - \ud{\ps} G_i \big) + \od{\ps} \, \ph \, \ud{\ps} \big>
\end{eqnarray}
<<tiddler HideTags>>
$$
\begin{array}{rcl}
L \!\!&\!\!=\!\!&\!\! \bar{\ps} \ve{e} \lp \f{\pa} + {\small \frac{1}{4}} \f{\om}^{a b} \ga_{a b} + \f{A} + \f{W} + \f{g} \rp \ps + \bar{\ps} \ph \ps
\end{array}
$$



<html>
<table class="gtable">

<tr>
<td>
<img SRC="images/png/fermion photon vertex.png" height=100px>
</td>
<td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>
<td>
<SPAN class="math">\ps = 
\lb \matrix{
\ps_1 \\ \ps_2 \\ \ps_3 \\ \vdots
} \rb
</SPAN>
</td>
</tr>

</table>
</html>
<<options>>
/***
name: AllTagsExceptPlugin
author: Garrett
version: 0.1.0
This is a revision of Clint Checketts' allTagsExcept plugin, which lists all tags except those listed.

<<option chkDisableExcept>> show hidden system tags

!!Usage
{{{
<<AllTagsExcept tag1 tag2 ...>>
}}}
!!!Code
***/
/*{{{*/
version.extensions.AllTagsExcept = {major: 0, minor: 1, revision: 0};

if (!config.options.chkDisableExcept) config.options.chkDisableExcept=false; // default to standard action

config.macros.AllTagsExcept = {tooltip: "Show tiddlers tagged with '%0'",noTags: "There are no tags to display"};

config.macros.AllTagsExcept.handler = function(place,macroName,params)
{
	var tags = store.getTags();
	var theDateList = createTiddlyElement(place,"ul");
	if(tags.length == 0)
		createTiddlyElement(theDateList,"li",null,"listTitle",this.noTags);
	for(var t=0; t<tags.length; t++)
		{
		var includeTag = true;
            	for (var p=0;p<params.length; p++) if ((tags[t][0] == params[p])&&(!config.options.chkDisableExcept)) includeTag = false;
            	if (includeTag)
			{
			var theListItem =createTiddlyElement(theDateList,"li");
			var theTag = createTiddlyButton(theListItem,tags[t][0] + " (" + tags[t][1] + ")",this.tooltip.format([tags[t][0]]),onClickTag);
			theTag.setAttribute("tag",tags[t][0]);
			}
		}
}
/*}}}*/
author: [[Garrett Lisi]]
arxiv: http://arxiv.org/abs/0711.0770
locally: [[AESToE|papers/AESToE.pdf]]
abstract:
All fields of the [[standard model]] and [[gravity|modified BF gravity]] are unified as an [[E8]] [[principal bundle]] [[connection]]. A non-compact real form of the [[E8 Lie algebra|e8]] has [[G2]] and [[F4]] subalgebras which break down to strong [[su(3)]], electroweak [[su(2)]] x u(1), gravitational [[so(3,1)|spacetime]], the [[frame]]-Higgs, and three generations of fermions related by [[triality]]. The interactions and dynamics of these [[1-form]] and [[Grassmann|Grassmann number]] valued parts of an E8 [[superconnection]] are described by the [[curvature]] and action over a four dimensional base [[manifold]].

A [[talk for ILQGS 07]] on 11/13/07 was presented on this paper.

Internet discussion of the paper, in chronological order: 
*[[Backreaction|http://backreaction.blogspot.com/2007/11/theoretically-simple-exception-of.html]]
*[[Physics Forums|http://www.physicsforums.com/showthread.php?t=196498]]
*[[The Reference Frame|http://motls.blogspot.com/2007/11/exceptionally-simple-theory-of.html]]
*[[Hidden Variables|http://blog.domenicdenicola.com/post/2007/11/Criteria-for-a-Theory-of-Everything.aspx]]
*[[Not Even Wrong|http://www.math.columbia.edu/~woit/wordpress/?p=617]]
*[[Arcadian Functor|https://www.blogger.com/comment.g?blogID=28857369&postID=5548882952979522971]]
*[[Freedom of Science|http://globalpioneering.com/wp02/an-exceptionally-simple-theory-of-everything/]]
*[[Theoreman Egregium|http://egregium.wordpress.com/2007/11/10/physics-needs-independent-thinkers/]]
*[[Science Forums|http://www.scienceforums.net/forum/showthread.php?t=29522]]

And previously:
*[[This Week's Finds 253]]
<<tiddler HideTags>>@@display:block;text-align:center;
<html><center>
<img src="talks/StAnth09/images/bubblechamber3.png" width="585" height="440">
</center></html>
$\p{{}_{\small (}^{(}}$&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
[[Garrett Lisi]]&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;UGM, 2012@@
<<tiddler HideTags>>@@display:block;text-align:center;
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$\p{{}_{\small (}^{(}}$&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
[[Garrett Lisi]]&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;UAT, 2013@@
<<tiddler HideTags>>The BRST technique accounts for gauge symmetries by introducing new fields with anticommuting coefficients and dynamics that fixes the original local gauge symmetry and introduces a new global (super) symmetry -- the BRST transformation.

Connection: $\;\;\;\; \f{A} = \f{H} + \f{K} \;\; \in \; H + K = G\vp{A_{\big(}} \;\;\;\;$ with $H$ reductive in $G$.
Action: $\;\;\;\; S = \int \big< \ff{B} \ff{F} + \nf{V}(B^H) \big> \vp{A_{\big(}} \;\;\;\;$ purely gauge (topological) in $\f{K}$.
BRST transformation: (make gauge parameter, $\ps = \ps^{\, \io}(x) T_\io \in K$, an anticommuting BRST field)
$$
\begin{array}{rclcrclcrcl}
\ud{\de} \f{K} \!\!&\!\!=\!\!&\!\! -\f{D} \ud{\ps} & \;\;\; & \ud{\de} \ud{\ps} \!\!&\!\!=\!\!&\!\! -\ha \big[ \ud{\ps}, \ud{\ps} \big]  & & & & \\
\ud{\de} \ff{B} \!\!&\!\!=\!\!&\!\! \big[ \ff{B}, \ud{\ps} \big] & \;\;\; & \ud{\de} \fff{\od{B}} \!\!&\!\!=\!\!&\!\! \fff{\la} & \;\; & \ud{\de} \fff{\la} \!\!&\!\!=\!\!&\!\! 0 
\end{array}
\;\;\;\;\;\;\;\;\;\;\;\;\; \Longrightarrow \;\;\;
\ud{\de} S = 0, \;\; \ud{\de} \ud{\de} = 0
$$
Choose a ''BRST potential'', $\od{\Ps} = \int \big< \fff{\od{B}} \f{K} \big>$, and use it to make the BRST action,
$$
S' = \ud{\de} \od{\Ps} + S = \int \big< \fff{\la} \f{K} + \fff{\od{B}} \f{D} \ud{\ps} + \ff{B} \ff{F} + \nf{V}(B^H) \big>
$$
Varying $\fff{\la}$ fixes the gauge to $\f{K} = 0$, giving the effective action,
$$
S^{\mbox{eff}} = \int \big< \fff{\od{B}} \f{D} \ud{\ps} + \ff{B} \ff{F}{}^H + \nf{V}(B^H) \big>
$$
which can be the Dirac action for a suitable algebra, and $\fff{\od{B}}= \fr{1}{4!} \bar{\ps} \f{e} \f{e} \f{e} \ep$. The literature mentions the BRST superconnection, $\udf{A} = \f{H} + \ud{\ps}$, with $\ud{\ps}$ a "1-form in the space of connections," related to TQFT, and $(\f{d}+\ud{\de})$ relating to BRST cohomology and anomalies.
<<tiddler HideTags>>Start with $E8$ principal bundle connection and its curvature,
$$
\f{A} = \f{H} + \f{\Ps} \qquad \quad
\ff{F} = (\f{d} \f{H} + \f{H} \f{H} + \f{\Ps} \f{\Ps})
+ (\f{d} \f{\Ps} + \f{H} \f{\Ps} + \f{\Ps} \f{H})
$$
Action such that $\f{\Ps}$ part is pure gauge,
$$
S = \int \big< \ff{B} \ff{F}
+ {\scriptsize \frac{\pi G}{4}} \ff{B}{}_G \ff{B}{}_G \ga + \ff{B'} \ff{*B'} \big>
$$
BRST: Replace $\f{\Ps}$ part with ghosts, $\ud{\Ps}$, in extended connection, 
$$
\udf{A} = \f{H} + \ud{\Ps} \qquad \quad
\udff{F} = \big( \f{d} \f{H} + \f{H} \f{H} \big) + \big( \f{d} \ud{\Ps} + [ \f{H}, \ud{\Ps} ] \big)
= \ff{F}{}_H + \f{D} \ud{\Ps}
$$
Effective action for gauge fields, ghosts, and anti-ghosts:
\begin{eqnarray}
S &=& \int \big< \ff{\od{B}} \udff{F}
+ {\scriptsize \frac{\pi G}{4}} \ff{B}{}_G \ff{B}{}_G \ga + \ff{B'} \ff{*B'} \big> \\
&=& \int \big< \fff{\od{B}} \f{D} \ud{\Ps}
+ \nf{e} \fr{1}{16 \pi G} \ph^2  \big( R - \fr{3}{2} \ph^2 \big) + \fr{1}{4} \ff{F'} \ff{*F'} \big>
\end{eqnarray}
<<tiddler HideTags>>$\de \nf{L} = 0$ under [[gauge transformation]]: &nbsp;&nbsp; $\de \f{A} = - \f{\na} C = -\f{d} C - \big[ \f{A}, C \big]$
Account for gauge part of $\f{A}$ by introducing [[Grassmann|Grassmann number]] valued ''ghosts'', $\ud{C} \in \ud{\rm Lie}(G)_g$, ''anti-ghosts'', $\nf{\od{B}}$, ''partners'', $\nf{\la}$, and [[BRST transformation|BRST technique]]:
$$
\begin{array}{rclcrcl}
\ud{\de} \f{A} &=& - \f{\na} \ud{C} & \;\;\;\;\;\;\;\;\; & \ud{\de} \ud{C} &=& - \ha \big[ \ud{C}, \ud{C} \big] \\
\ud{\de} \ff{B} &=& \big[ \ff{B}, \ud{C} \big] & \;\;\;\;\;\;\;\;\; & \ud{\de} \nf{\od{B}} &=& \nf{\la} \\
\ud{\de} \nf{\la} &=& 0 & \;\;\;\;\;\;\;\;\; & & &
\end{array}
$$
This satisfies $\ud{\de} \nf{L} = 0_{\phantom{\big(}}$and $\ud{\de} \ud{\de} = 0$.
Choose a ''BRST potential'', $\nf{\od{\Ps}} = \big< \nf{\od{B}} \f{A} \big>$, to get new Lagrangian:
$$
\nf{L'} = \nf{L} + \ud{\de} \nf{\od{\Ps}} = \nf{L} + \big< \nf{\la} \f{A_g} \big> + \big< \nf{\od{B}} \f{\na} \ud{C} \big>
$$
BRST partners act as Lagrange multipliers; ''effective Lagrangian'':
$$
\nf{L^{\rm eff}} = \nf{L}[\ff{B'},\f{A'}] + \big< \nf{\od{B}} \f{\na'} \ud{C} \big>
$$
//This is a speculative description of the [[BRST technique]] based on conversations with [[Michael Edwards]]//

Start with a [[connection]] 1-form, $\f{\om}$, defined over the entire space of a [[fiber bundle]], and some fiber bundle section, $\si$. The connection field over the base manifold is the pullback of the connection along the section,
$$
\f{A} = \si^* \f{\om}
$$
A BRST transformation may be a way of describing how $\f{A}$ changes under a change of section. (Though I think this is just a gauge transformation, with a funny pair of Grassmann valued parameters.) Consider a vector field,
$$
\ve{\va} = \va^A(x) \ve{\xi_A}(p)
$$
on the entire space, with $\ve{\xi_A}$ the flow fields corresponding to the group generators, $T_A$. The gauge transformation parameters can be written in terms of a Grassmann valued parameter and Grassmann valued ghost fields as $\va^A(x)= \va C^A(x)$. The BRST transformation then is
$$
\f{\de A} = - \si^* L_{\ve{\va}} \f{\om} = \va s \f{A} = - \va (\f{d} C + \f{A} \times C)
$$
in which $C=C^A T_A$.
//Hmm, this seems to give the change in A from flowing $\om$ under $\si$...// 


Another idea, from Picken. Instead of pulling the [[Ehresmann connection]] back along a section, use the surface [[vector projection]] on the E conn to project to the gauge 1-form on the surface and a ''ghost'' -- a 1-form off of the surface. This ghost is the projection of the [[Maurer-Cartan form]], and its value determines the shape of the section. May be able to connect this with other descriptions, like the one above.


Nah, none of this is going to work right. Have to work in the space of connections.

variational bicomplex

Refs:
*M. Ghiotti
**[[Gauge fixing and BRST formalism in non-Abelian gauge theories|papers/Ghiotti - Gauge fixing and BRST formalism in non-Abelian gauge theories.pdf]]
***Excellent new thesis.
*G. Catren and J. Devoto
**[[Extended Connection in Yang-Mills Theory|http://arxiv.org/abs/0710.5698]]
*Bonora and Cotta-Ramusino
**[[Some Remarks on BRS Transformations, Anomalies and the Cohomology of the Lie Algebra of the Group of Gauge Transformations|papers/1103922136.pdf]]
***This is one of the first, and probably the best, description of ghosts as 1-forms in the space of connections.
***${\cal G}$ is group of vertical [[automorphism]]s of $E$, equals group of [[gauge transformation]]s.
*Stora and Kastler
**[[A Differential Geometric Setting for BRS Transformations and Anomalies|papers/A Differential Geometric Setting for BRS Transformations and Anomalies.pdf]]
***detailed exposition.
***gauge transformation bundle
***old (hard to read scanned text) but good. lengthy.
***same gauge trasf as Viallet (below)
*Viallet
**[[The Geometry of the Space of Fields in Yang-Mills theory|papers/The Geometry of the Space of Fields in Yang-Mills theory.pdf]]
***space of fields as bundle, physical fields as base
***strange definition for group automorphism, which disagrees with mine and Wikipedia's.
***gauge transf are ''equivariant'' automorphisms, $f(p)=p\ph(p)$, satisfying $f(ph)=f(p)h$ and hence $\ph(ph)=h^- \ph(p) h$.
*J.W. van Holten
**[[Aspects of BRST Quantization|papers/JHolten_BRST.pdf]]
***excellent elementary practical intro
***ghost to M-C form mapping not an identification
**[[The BRST Complex and the Cohomology of Compact Lie Algebras|papers/The BRST Complex and the Cohomology of Compact Lie Algebras.pdf]]
***BRST analysis analogous to [[Hodge decomposition]]
***this paper's content is included in the paper above
*http://en.wikipedia.org/wiki/BRST_Quantization
*[[Principal Bundles, Connections and BRST Cohomology|papers/9408003.pdf]]
**(//read this now//)
**BRST cohomology in the space of connections
**mathematically dense, but I'm hacking it so far
**[[lots of geometric brst papers from spires|http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=FIND+c+cmpha,87,589&SKIP=0]]
*Jeffrey A. Harvey
**[[TASI 2004 Lectures on Anomalies|papers/0509097.pdf]]
***for particle physicists
***ghosts are 1-forms in the space of gauge transformations
***BRST operator is exterior derivative in this space
*Barnich, Brandt, and Henneaux
**[[Local BRST cohomology in gauge theories|papers/0002245.pdf]]
***these guys are the big shots in the field, but I don't like this antifield approach yet.
*[[Four-Dimensional Yang-Mills Theory as a Deformation of Topological BF Theory|papers/9705123.pdf]]
**pretty good and succinct intros, plus treats BF
**agrees with Viallet's def of equivariant automorphism.
*Moritsch, Sorella, et al
**[[Algebraic characterization of gauge anamolies on a nontrivial bundle|papers/9611168.pdf]]
***nice algebraic treatment, with generalized connection
**[[Algebraic characterization of the Wess-Zumino consistency conditions in gauge theory|papers/9302136.pdf]]
***Sorrella's paper introducing $\de$. Seems to work with jets without calling them that.
**[[Algebraic structure of gravity in Ashtekar variables|papers/9409046.pdf]]
***Blaga, using $\de$.
*Jim Stasheff
**[[The (secret?) homological algebra of the Batalin-Vilkovisky approach|papers/9712157.pdf]]
***abstract mathematical overview (jets) of relations between physics and math structures
***ghost = Chevelley-Eilenberg generator
***anti-ghost = Tate generator
***anti-field = Koszul generator
*Yang, Lee
**[[Lie algebra cohomology and group structure of gauge theories|papers/9503204.pdf]]
***maybe or maybe not useful
*Kelnhofer
**[[On the Geometrical Structure of Covariant Anomalies in Yang-Mills Theory|paper/9302012.pdf]]
***damn this is a (unavoidable) mess
***universal bundle
*Thomas Schucker
**see his book on Amazon for introductory material:
***[[Differential Geometry, Gauge Theories and Gravity|http://www.amazon.com/Differential-Geometry-Cambridge-Monographs-Mathematical/dp/0521378214/ref=si3_rdr_bb_product/104-9709999-3726336]]
**[[The Cohomological Construction of Stora's Solutions|papers/The Cohomological Construction of Stora's Solutions.pdf]]
*Picken: http://www.iop.org/EJ/abstract/0305-4470/19/5/001
**//(scan this)//
*J. P. Zwart
**[[BRST Reduction and Quantization of Constrained Hamiltonian Systems|papers/zwart98brst.pdf]]
*Witten
**[[Topological Quantum Field Theory|papers/1104161738.pdf]]
**differential forms on the space of connections
*Laurent Baulieu
**[[On the Cohomological Structure of Gauge Theories|papers/On the Cohomological Structure of Gauge Theories.pdf]]
***adds two Grassmann coordinates to spacetime
*Rudolf Schmid
**[[Local Cohomology in Gauge Theories BRST Tansformations and Anomalies|papers/localbrst.pdf]]
***mathematically abstract, but geometric
***ack, [[jet]]s. 
***connects ghosts to [[Maurer-Cartan form]]
**[[A Few BRST Bicomplexes|papers/nankai.pdf]]
*discussion with [[Michael Edwards]] on
**[[Not Even Wrong|http://www.math.columbia.edu/~woit/wordpress/?p=436]]
*Ian Anderson
**[[The Variational Bicomplex|papers/The Variational Bicomplex.pdf]]
The [[BRST technique]] fixes and accounts for [[gauge symmetries|gauge transformation]] by introducing new fields with [[Grassmann valued|Grassmann number]] coefficients having dynamics and interactions with existing fields that breaks the original local gauge symmetry but includes a new global (super) symmetry -- the BRST transformation -- that's a "rotation" between old and new fields. This method of gauge fixing is an indispensable tool in the application of path integral methods in the quantum field theory of non-abelian gauge fields ([[principal bundle]] connections), and has a natural extension to describe the existence and dynamics of fermionic [[spinor]] fields.

A restricted BF Lagrangian,
$$
\nf{L} = \li \nf{B} \ff{F} + \nf{\Phi}(\f{A},\nf{B}) \ri
$$
invariant, $\delta_{G} \nf{L} = 0$, under some subset of the gauge transformation, $G \in {\rm Lie}(H) \subset {\rm Lie}(G)$, is amenable to the BRST technique. A ''ghost field'', $\ud{C} = \ud{C^A} T_A \in \ud{\rm Lie}(H)$, is introduced with Grassmann coefficients multiplying [[Lie algebra]] elements, along with an anti-Grassmann valued $(n-1)$-form ''antighost field'', $\nf{\od{B}} = \nf{\od{B}{}^A} T_A$, and a real valued $(n-1)$-form ''BRST partner field'', $\nf{\lambda} = \nf{\lambda^A} T_A$. This new system is equipped with a global ''BRST transformation'' -- a ''supersymmetry rotation'' between real and Grassmann valued variables,
$$
\begin{array}{rclcrcl}
\ud{\de} \f{A} &=& -\f{\nabla} \ud{C} & \;\;\; & \ud{\de} \ud{C} &=& -\ha \lb \ud{C}, \ud{C} \rb \\
\ud{\de} \nf{B} &=& \lb \nf{B}, \ud{C} \rb & \;\;\; & \ud{\de} \nf{\od{B}} &=& \nf{\la} \\
\ud{\de} \nf{\la} &=& 0 & & & &
\end{array}
$$
that is nilpotent, $\ud{\de} \ud{\de} = 0$, and leaves the Lagrangian invariant (''BRST [[closed]]''), $\ud{\de} \nf{L} = 0$. Physical observables are in the [[cohomology]] of this ''BRST operator'', $\ud{\de}$. Dynamics are introduced for the ghosts by adding a ''BRST [[exact]]'' term to get a ''BRST extended Lagrangian'',
$$
\nf{L'} = \nf{L} + \ud{\de} \nf{\od{\Psi}}
$$
with some ''BRST potential'', $\nf{\od{\Psi}}$, chosen. For example, choosing
$$
\nf{\od{\Psi}} = \li \nf{\od{B}} \f{A} \ri
$$
gives
$$
\ud{\de} \nf{\od{\Psi}} = \li \nf{\la} \f{A} \ri + \li \nf{\od{B}} \f{\nabla} \ud{C} \ri
$$
The BRST partner field, $\nf{\la}$, acts as a Lagrange multiplier constraining the gauge freedom of the connection, so the ''gauge fixed connection'' is $\f{A} = \f{A'}$, with $\nf{\la} \f{A'} = 0$. The resulting ''effective Lagrangian'' is
$$
\nf{L^{\rm eff}} = \li \nf{B'} \ff{F'} + \nf{\Phi}(\f{A'},\nf{B'}) \ri 
+ \li \nf{\od{B}} \f{\nabla'} \ud{C} \ri
$$

This form of the Lagrangian suggests the introduction of a ''BRST extended connection'',
$$
\udf{A} = \f{A'} + \ud{C}
$$
with ''BRST extended curvature'',
$$
\udff{F} = \f{d} \udf{A} + \ha \lb \udf{A} , \udf{A} \rb = \ff{F'} + \f{\nabla'} \ud{C} +  \ha \lb \ud{C} , \ud{C} \rb
$$
allowing the effective Larangian to be written as
$$
\nf{L^{\rm eff}} = \li \nf{\od{B'}} \udff{F} + \nf{\Phi}(\f{A'},\nf{B'}) \ri
$$
with $\nf{\od{B'}} = \nf{B'} + \nf{\od{B}}$.

Ref:
*J.W. van Holten
**[[Aspects of BRST Quantization|papers/JHolten_BRST.pdf]]
***Good modern introduction.
*Laurent Baulieu
**[[Perturbative Gauge Theories|papers/LBaulieu_BRST.pdf]]
***Early reference, including superconnection.
<<tiddler HideTags>>$$\begin{array}{rcl}
\f{H} \!\!&\!\!=\!\!&\!\! \ha \f{\om} + \fr{1}{4}\f{e}\ph + \f{B} + \f{W} =
{\scriptsize
\lb \begin{array}{cccc} 
\ha \f{\om_L} \!+\! i \f{W^3} & i \f{W^1} \!+\! \f{W^2} & - \fr{1}{4} \f{e_R} \ph_0^* & \fr{1}{4} \f{e_R} \ph_+ \\
i \f{W^1} \!-\! \f{W^2} & \ha \f{\om_L} \!-\! i \f{W^3} & \p{-} \fr{1}{4} \f{e_R} \ph_+^* & \fr{1}{4} \f{e_R} \ph_0 \\
- \fr{1}{4} \f{e_L} \ph_0 & \fr{1}{4} \f{e_L} \ph_+ & \ha \f{\om_R} \!+\! i \f{B} & \\
\p{-} \fr{1}{4} \f{e_L} \ph_+^* & \fr{1}{4} \f{e_L} \ph_0^* & & \ha \f{\om_R} \!-\! i \f{B}
\end{array} \rb_{\p{(}}
} \\
\!\!&\!\!=\!\!&\!\! \f{dx^a} \ha h_a^{\p{a} \al\be} \ga_{\al\be} \;\; \in \;\; \f{so}(1,7) = \f{Cl}^2(1,7) \subset \f{\mathbb{C}}(8\times8)
\end{array}$$
@@display:block;text-align:center;[[Clifford bivector|Clifford algebra]] parts:@@$$
\begin{array}{rcl}
\f{\om} \!\!&\!\!=\!\!&\!\! \f{dx^a} \ha \om_a^{\p{a} \mu \nu} \ga_{\mu \nu}
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\leftarrow \text{spin connection} \\
\f{e} \ph \!\!&\!\!=\!\!&\!\! \f{dx^a} \lp e_a \rp^\mu \ph^\ph \ga_{\mu \ph}
\, \left\{
\begin{array}{rcl}
\f{e} \!\!&\!\!=\!\!&\!\! \f{dx^a} \lp e_a \rp^\mu \ga_\mu
\;\;\;\;\;\;\;\;\;\;\;\;\;
\leftarrow \text{frame (vierbein)} \\
\ph \!\!&\!\!=\!\!&\!\! \ph^\ph \ga_\ph
\, \left\{
\begin{array}{rcl}
\ph_+ \!\!&\!\!=\!\!&\!\! (-\ph^5 \!+\! i \ph^6) \\
\ph_0 \!\!&\!\!=\!\!&\!\! (\ph^7 \!+\! i \ph^8)
\end{array}
\right\}
\begin{array}{c}
\;\;\;\;
\leftarrow \text{Higgs} \\
\ph \ph = -M^2
\end{array}
\end{array}
\rd
\end{array}
$$
$$
\begin{array}{rcl}
\f{B} \!\!&\!\!=\!\!&\!\! - \! \f{dx^a} \ha B_a \big( \ga_{56} - \ga_{78} \big) 
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,
\leftarrow \; \downarrow \text{electroweak gauge fields} \\
\f{W} \!\!&\!\!=\!\!&\!\! - \! \ha \f{W^1} \big( \ga_{67} + \ga_{58} \big)
- \ha \f{W^2} \big(-\ga_{57} + \ga_{68} \big)
- \ha \f{W^3} \big( \ga_{56} + \ga_{78} \big) 
\;\;\;\;\;\;\;\, \\
\end{array}
$$
@@display:block;text-align:center;[[indices]]: $\;\;\;\; 0 \le a,b \le 3 \;\;\;\;\;\; 0 \le \mu,\nu \le 3 \;\;\;\;\;\; 5 \le \ph,\ps \le 8 $@@
<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
<embed src="talks/TED08/images/Soft Coral_620.jpg" width="827" height="620"></embed>
</center></html>@@
<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
<embed src="talks/TED08/images/Soft Coral_620.jpg" width="827" height="620"></embed>
</center></html>@@
Creating bulleted lists is simple.
* Just add an asterisk
* at the beginning of a line.
** If you want to create sub-bullets
** start the line with two asterisks
*** And if you want yet another level
*** use three asterisks
* You can also do [[Numbered Lists]]
{{{
Creating bulleted lists is simple.
* Just add an asterisk
* at the beginning of a line.
** If you want to create sub-bullets
** start the line with two asterisks
*** And if you want yet another level
*** use three asterisks
* You can also do [[Numbered Lists]]
}}}
Ref:
*M. Berg, C. DeWitt-Morette, S. Gwo, E. Kramer, [[The Pin Groups in Physics: C, P, and T|papers/The Pin Groups in Physics- C, P, and T.pdf]] 
[[arxiv|http://arxiv.org/abs/0706.0217]]
S. Raby, A. Wingerter
Abstract: We investigate whether the hypercharge assignments in the Standard Model can be interpreted as a hint at Grand Unification in the context of heterotic string theory. To this end, we introduce a general method to calculate U(1)_Y for any heterotic orbifold and compare our findings to the cases where hypercharge arises from a GUT. Surprisingly, in the overwhelming majority of 3-2 Standard Models, a non-anomalous hypercharge direction can be defined, for which the spectrum is vector-like. For these models, we calculate sin^2 theta to see how well it agrees with the standard GUT value. We find that 12% have sin^2 theta = 3/8, while all others have values which are less. Finally, 89% of the models with sin^2 theta = 3/8 have U(1)_Y in SU(5). 

*computation to find hypercharge directions in E8xE8 root system
[>img[images/person/Carlo Rovelli.jpg]]Homepage: http://www.cpt.univ-mrs.fr/~rovelli/rovelli.html
*Location: Marseille
*CV: http://www.cpt.univ-mrs.fr/%7Erovelli/vita.pdf
*arXiv papers: http://arxiv.org/find/gr-qc/1/au:+Rovelli_C/0/1/0/all/0/1

Selected work:
*[[Quantum Gravity|http://www.cpt.univ-mrs.fr/%7Erovelli/book.pdf]]
*[[Graviton propagator in loop quantum gravity|http://arxiv.org/abs/gr-qc/0604044]]
**nice treatment. includes basic example of canonical and path integral QM, field theory, then does LQG via GFT. 
A ''Cartan H-bundle'', with total space $E_H$, is a [[principal bundle]] with $n_H$ dimensional [[Lie group]], $H$, as the typical fiber (and structure group) and $n_M$ dimensional base, $M$. This bundle is not [[associated]] to the [[Ehresmann Cartan geometry]], $E_G$, since the structure group of $E_H$ is $H \subset G$; however, $E_H$ does serve as a base space under $E_G$, and the [[Ehresmann Cartan connection]] over $E_G$ does pull back to give a connection over $E_H$.

The Cartan H-bundle is mapped into a section (a [[submanifold]]), $E'_H$, of the Ehresmann Cartan geometry, $E_G$, by the reference section of the [[Cartan homogeneous space bundle]],
\begin{eqnarray}
\si'^S &:& E_H \to E_G \\
\si'^S(x,y) &=& (x,x_{s\si}(x),y)
\end{eqnarray}
The [[Ehresmann Cartan connection]] form [[pulls back|pullback]] along this map to give the ''Ehresmann Cartan H-connection form'' over $E_H$,
\begin{eqnarray}
\f{{\cal C}_H}(x,y) &=& \si'^{S*} \f{\cal C} = \lp \f{C^J}(x) L^I{}_J(x_{s\si}(x),y) + \si'^{S*} \f{\xi_R^I}(x_{s\si}(x),y) \rp T_I \\
&=& g^-(x_{s\si}(x),y) \, \f{C}(x) \, g(x_{s\si}(x),y) + g^-(x_{s\si}(x),y) \, \f{d} \, g(x_{s\si}(x),y) \\
&=& h^-(y) \Big( r^-(x_{s\si}(x)) \, \f{C}(x) \, r(x_{s\si}(x)) + r^-(x_{s\si}(x)) \, \f{d} \, r(x_{s\si}(x)) \Big) h(y) + h^-(y) \, \f{d} h(y) 
\end{eqnarray}
in which the [[coset representative section|homogeneous space]], $r:G/H \to G$, is used to write $g(x_s,y)=r(x_s) \, h(y)$. Choosing the homogeneous space bundle zero reference section, $r(x_{s\si_0}(x)) = r(0) = 1$, this gives
$$
\f{{\cal C}_H}(x,y) = h^-(y) \, \f{C}(x) \, h(y) + h^-(y) \, \f{d} \, h(y)
$$
Pulling this back along the [[canonical reference section|Ehresmann principal bundle connection]] gives the [[Cartan connection|Cartan geometry]], $\si_0^{H*} \f{{\cal C}_H} = \f{C}$, over $M$.

If $H$ is [[reductive]] in $G$ (as is usually assumed) the Ehresmann Cartan H-connection form splits into the ''Ehresmann Cartan H-connection frame form'' and '' Ehresmann H-connection form'',
\begin{eqnarray}
\f{{\cal C}_H} &=& \f{{\cal E}_H} + \f{{\cal A}_H} \\ 
\f{{\cal E}_H} &=& h^-(y) \, \f{e}(x) \, h(y) \in \f{\rm Lie}(G/H) \\
\f{{\cal A}_H} &=& h^-(y) \, \f{A}(x) \, h(y) + h^-(y) \, \f{d} \, h(y) \in \f{\rm Lie}(H)
\end{eqnarray}
The Ehresmann H-connection form, $\f{{\cal A}_H}(x,y)$, over $E_H$ is an [[Ehresmann principal bundle connection]] form for the bundle.

When the Ehresmann Cartan H-connection form equals the [[Maurer-Cartan form]], $\f{{\cal C}_H} = \f{\cal I}$, the Cartan H-bundle is an [[Ehresmann homogeneous space geometry]], $E_H = G$. In this way, the Cartan H-bundle may be considered to be a [[reductive Lie group geometry]], $G$, that has gone wavy along $G/H$ -- with $\f{{\cal C}_H}$ deviating from $\f{\cal I}$ to give the new [[frame]] 1-forms, $\f{{\cal C}_H^J} = \f{E^J}$, of the [[Cartan tangent bundle geometry]] over what was $G$.
A ''Cartan geometry'' is a [[Lie group geometry]], $G$, that's allowed to go wavy while maintaining some of its symmetry, represented by a subgroup, $H \subset G$, usually assumed to be [[reductive]] in $G$. The wavy ''Cartan geometry base manifold'', $M$, is ''modeled'' on the [[homogeneous space]], $M \sim S=G/H$, and has the same dimension, $n_S = (n_G - n_H)$. The ''Cartan connection'' over $M$,
$$
\f{C}(x) = \f{e} + \f{A} \in \f{\rm Lie}(G)
$$
is a [[Lieform]] modeled on the [[Maurer-Cartan frame|homogeneous space]], $\f{C} \sim \f{I} = r^- \f{d} r(x)$, and splits (for $H$ reductive in $G$) into the ''Cartan frame'', $\f{e}(x) = \f{e^A} K_A \in \f{\rm Lie}(G/H)$, and ''Cartan H-connection'', $\f{A}(x) = \f{A^P} H_P \in \f{\rm Lie}(H)$, which (unlike their homogeneous space counterparts) may vary freely. 

The ''Cartan [[curvature]]'' of the connection is
\begin{eqnarray}
\ff{F}(x) &=& \f{d} \f{C} + \ha \lb \f{C}, \f{C} \rb \\
&=& \f{d} \f{e} + \f{d} \f{A} + \ha \lb \f{e}, \f{e} \rb + \lb \f{A}, \f{e} \rb + \ha \lb \f{A}, \f{A} \rb \\
&=& \ff{F^A} K_A + \ff{F^P} H_P 
\end{eqnarray}
which (like the [[homogeneous space curvature|homogeneous space tangent bundle geometry]]) splits into
\begin{eqnarray}
\ff{F^A} &=& \f{d} \f{e^A} + \f{A^P} \f{e^B} C_{PB}{}^A + \ha \f{e^C} \f{e^B} C_{CB}{}^A \\  
\ff{F^P} &=& \ff{F_H^P} + \ha \f{e^C} \f{e^D} C_{CD}{}^P
\end{eqnarray}
with the ''curvature of the Cartan H-connection'' defined by:
$$
\ff{F_H^P} = \f{d} \f{A^P} + \ha \f{A^Q} \f{A^R} C_{QR}{}^P
$$
Note that $\ha \f{e^C} \f{e^B} C_{CB}{}^A = 0$ if $G/H$ is a [[symmetric space]].

There are many relationships between a Cartan geometry and other structures. A [[natural]] [[Ehresmann Cartan geometry]] is a description of a Cartan geometry as an [[Ehresmann principal bundle connection]] for a [[G-bundle|principal bundle]] -- and this description splits via $G/H$ into the [[Cartan H-bundle]] and [[Cartan homogeneous space bundle]]. These two bundles relate to the way the [[Lie group tangent bundle geometry]] of $G$ turns wavy, described by the [[Cartan tangent bundle geometry]].

Refs:
*http://en.wikipedia.org/wiki/Cartan_connection
*[[Differential Geometry of Cartan Connections|papers/9412232.pdf]]
**by [[Peter Michor]] and Alekseevsky (tiddler $G \subset H$)
*[[The Works of Charles Ehresmann on Connections: From Cartan Connections to Connections on Fibre Bundles|papers/CMMarle.pdf]]
**Ehresmann version of Cartan, nicely explained. See p9 for main def.
**http://www.math.jussieu.fr/~marle/
*[[MacDowell-Mansouri Gravity and Cartan Geometry|papers/0611154.pdf]]
**a new paper by Derek Wise
*[[Natural Operations on the Bundle of Cartan Connections|papers/Natural Operations on the Bundle of Cartan Connections.pdf]]
*[[The Existance of Cartan Connections and Geometrizable Principal Bundles|papers/0206136.pdf]]
**a very concise and interesting mathematical treatment.
*[[Gravity, Cartan geometry, and idealized waywisers|http://arxiv.org/abs/1203.5709]]
**a nice description by Hans Westman and Tom Zlosnik
A ''Cartan homogeneous space bundle'', with total space $E_S$, is a [[fiber bundle]] with $n_S$ dimensional [[homogeneous space]], $F=S=G/H$, as the typical fiber and $n_M = n_S$ dimensional base, $M$. This bundle may be visualized as the set of homogeneous spaces tangent to the base space. (If $n_M \neq n_S$ this is a ''generalized Cartan homogeneous space bundle''.) The structure group, $G$, of the bundle is the subset of [[homogeneous space geometry symmetries]] corresponding to the [[left action|group]] of $G$ on the space.

The Cartan homogeneous space bundle, $E_S$, is [[associated]] to the [[Ehresmann Cartan geometry]], $E_G$, and $E_S$ also serves as a base space under $E_G$. The $n_M$ coordinates, $x^a$, cover patches of the base manifold, $M$, and the $n_S$ homogeneous space coordinates, $x_s^a$, cover patches of $S$ -- the combined coordinates, $(x,x_s)$, cover patches of $E_S$. The [[reference section|Ehresmann gauge transformation]], $\si^S : M \to E_S$, of the Cartan homogeneous space bundle determines the ''points of tangency'' -- the points, $x_{s\si}(x)$, of the $S_x$ in contact with $x$. Since a homogeneous space has a natural zero point, we often use the ''zero point reference section'', $\si_0^S(x) = (x,0)$.

The Cartan homogeneous space bundle, $E_S$, is mapped into a section (a [[submanifold]]), $E'_S$, of the Ehressmann Cartan geometry, $E_G$, by the reference section of the [[Cartan H-bundle]],
$$
\si'^H(x,x_s) = (x,x_s,y_\si(x))
$$
The [[Ehresmann Cartan connection]] form [[pulls back|pullback]] along this map to give the ''Cartan homogeneous space connection form'' over $E_S$,
\begin{eqnarray}
\f{{\cal C}_S}(x,x_s) &=& \si'^{H*} \f{\cal C} = \lp \f{C^J}(x) L^I{}_J(x_s,y_\si(x)) + \si'^{H*} \f{\xi_R^I}(x_s,y_\si(x)) \rp T_I \\
&=& g^-(x_s,y_\si(x)) \, \f{C}(x) \, g(x_s,y_\si(x)) + g^-(x_s,y_\si(x)) \, \f{d} \, g(x_s,y_\si(x)) \\
&=& h^-(y_\si(x)) \Big( r^-(x_s) \, \f{C}(x) \, r(x_s) + r^-(x_s) \, \f{d} \, r(x_s) \Big) h(y_\si(x)) + h^-(y_\si(x)) \, \f{d} h(y_\si(x)) 
\end{eqnarray}
in which the [[coset representative section|homogeneous space]], $r:S \to G$, is used to write $g(x_s,y)=r(x_s) \, h(y)$. Choosing the canonical H-bundle reference section, $h(y_{\si_0}(x)) = h(0) = 1$, this gives
$$
\f{{\cal C}_S}(x,x_s) = r^-(x_s) \, \f{C}(x) \, r(x_s) + r^-(x_s) \, \f{d} \, r(x_s)
$$
Pulling this back along the zero point reference section gives the [[Cartan connection|Cartan geometry]], $\si_0^{S*} \f{{\cal C}_S} = \f{C}$, over $M$.
<<tiddler HideTags>>Mutually [[commuting|commutator]] set of $r$ [[Lie algebra]] generators:
$$
\left\{ T_1, T_2, ..., T_r \right\} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; [ T_i, T_j ] = 0
$$
[[Cartan subalgebra|Lie algebra structure]]: $\;\;\; C=c^i T_i \;\; \in {\rm Lie(G)} \p{{}_{(}}$
[[Eigenvalues|eigen]], $\al^a$, and [[eigenvectors|eigen]], $V_a \in {\rm Lie(G)}$, using the Lie bracket:
$$
[ C , V_a ] = \al^a V_a = \sum_i c^i \al_i^a V_a
$$
Unique eigenvalue for each of the $(n-r)$ eigenvectors, corresponding to $(n-r)$ ''roots'', $\al_i^a$, in $r$ dimensional vector space.

Cartan subalgebra of the standard model and gravity:
$$
C = {\scriptsize \frac{1}{2}} \om^{01} \ga_{01} + {\scriptsize \frac{1}{2}} \om^{12} \ga_{12} + W^3 i \Si_3 + B i Y + G^3 i \la_3 + G^8 i \la_8 
$$
Eigenvectors are elementary particles, roots are their charges:
$$
\al(e_L) = ( \pm {\scriptsize \frac{1}{2}}, \mp {\scriptsize \frac{1}{2}}, -1, -1, 0, 0 ) \p{{}_{\Big(}}
$$
The [[Riemann curvature]] for the [[Cartan tangent bundle geometry]] is calculated from the [[Cartan tangent bundle spin connection]],
$$
\ff{R}^J{}_I = \f{d} \f{W}^J{}_I + \f{W}^J{}_K \f{W}^K{}_I
$$
We'll tackle this in pieces. Using a [[left-right rotator]] identity,
$$
\f{d} \lp L^h \rp^J{}_I = \f{e_H^P} C_P{}^J{}_K \lp L^h \rp^K{}_I
$$
the [[exterior derivative]]s are:
\begin{eqnarray}
\f{d} \f{W}^B{}_A &=& \f{d} \lp \f{\nu}^E{}_F \lp L^h \rp^B{}_E \lp L^h \rp_A{}^F - \ha \lp \f{A^Q} + \f{e_H^P} \lp L^h \rp_P{}^Q \rp F^H_{DEQ} \lp L^h \rp^{BD} \lp L^h \rp_A{}^E - \f{e_H^P} C_P{}^B{}_A \rp \\
&=& \lp \f{d} \f{\nu}^E{}_F \rp \lp L^h \rp^B{}_E \lp L^h \rp_A{}^F - \f{\nu}^E{}_F \f{e_H^P} C_P{}^B{}_D \lp L^h \rp^D{}_E \lp L^h \rp_A{}^F - \f{\nu}^E{}_F \lp L^h \rp^B{}_E \f{e_H^P} C_P{}^F{}_C \lp L^h \rp^C{}_A \\
&-& \ha \lp \f{d} \f{A^Q} + \lp \f{d} \f{e_H^P} \rp \lp L^h \rp_P{}^Q - \f{e_H^P} \f{e_H^R} C_{RP}{}^T \lp L^h \rp_T{}^Q \rp F^H_{DEQ} \lp L^h \rp^{BD} \lp L^h \rp_A{}^E \\
&+& \ha \lp \f{A^Q} + \f{e_H^P} \lp L^h \rp_P{}^Q \rp
\lp \lp \f{d} F^H_{DEQ} \rp \lp L^h \rp^{BD} \lp L^h \rp_A{}^E
+ F^H_{DEQ} \f{e_H^R} C_R{}^B{}_C \lp L^h \rp^{CD} \lp L^h \rp_A{}^E
+ F^H_{DEQ} \lp L^h \rp^{BD} \f{e_H^R} C_R{}_{AC} \lp L^h \rp^{CE} \rp \\
&-& \f{d} \f{e_H^P} C_P{}^B{}_A \\
\\
\f{d} \f{W}^B{}_R &=& \f{d} \lp \ha \f{e^D} F^H_{DEQ} \lp L^h \rp^{BE} \lp L^h \rp_R{}^Q \rp \\
&=& \ha \lp \f{d} F^H_{DEQ} \rp \lp L^h \rp^{BE} \lp L^h \rp_R{}^Q 
- \ha \f{e^D} F^H_{DEQ} \f{e_H^P} C_P{}^B{}_C \lp L^h \rp^{CE} \lp L^h \rp_R{}^Q 
- \ha \f{e^D} F^H_{DEQ} \lp L^h \rp^{BE} \f{e_H^P} C_{PRS} \lp L^h \rp^{SQ} \\
\\
\f{d} \f{W}^Q{}_R &=& - \ha \f{d} \lp \f{A^S} \, \lp L^h \rp^P{}_S + \f{e_H^P} \rp C_P{}^Q{}_R \\
&=& - \ha \lp \lp \f{d} \f{A^S} \rp \lp L^h \rp^P{}_S
- \f{A^S} \f{e_H^U} C_U{}^P{}_T \lp L^h \rp^T{}_S
+ \f{d} \f{e_H^P} \rp C_P{}^Q{}_R 
\end{eqnarray}
The pieces quadratic in the spin connection are:
\begin{eqnarray}
\f{W}^B{}_K \f{W}^K{}_A &=& \f{W}^B{}_C \f{W}^C{}_A + \f{W}^B{}_P \f{W}^P{}_A \\
&=&
\lp \f{\nu}^E{}_F \lp L^h \rp^B{}_E \lp L^h \rp_C{}^F - \ha \lp \f{A^Q} + \f{e_H^P} \lp L^h \rp_P{}^Q \rp F^H_{DEQ} \lp L^h \rp^{BD} \lp L^h \rp_C{}^E - \f{e_H^P} C_P{}^B{}_C \rp \\
&\times&
\lp \f{\nu}^G{}_H \lp L^h \rp^C{}_G \lp L^h \rp_A{}^H - \ha \lp \f{A^R} + \f{e_H^S} \lp L^h \rp_S{}^R \rp F^H_{HGR} \lp L^h \rp^{CH} \lp L^h \rp_A{}^G - \f{e_H^Q} C_Q{}^C{}_A \rp \\
&-& \lp \ha \f{e^D} F^H_{DEQ} \lp L^h \rp^{BE} \lp L^h \rp_P{}^Q \rp
\lp \ha \f{e^C} F^H_{CFR} \lp L^h \rp_A{}^F \lp L^h \rp^{PR} \rp \\
\\
\f{W}^B{}_K \f{W}^K{}_R &=& \f{W}^B{}_C \f{W}^C{}_R + \f{W}^B{}_P \f{W}^P{}_R \\
&=& \lp \f{\nu}^E{}_F \lp L^h \rp^B{}_E \lp L^h \rp_C{}^F - \ha \lp \f{A^Q} + \f{e_H^P} \lp L^h \rp_P{}^Q \rp F^H_{DEQ} \lp L^h \rp^{BD} \lp L^h \rp_C{}^E - \f{e_H^P} C_P{}^B{}_C \rp
\lp \ha \f{e^G} F^H_{GHS} \lp L^h \rp^{CH} \lp L^h \rp_R{}^S \rp \\
&-& \lp \ha \f{e^D} F^H_{DEQ} \lp L^h \rp^{BE} \lp L^h \rp_P{}^Q \rp
 \ha \lp \f{A^S} \, \lp L^h \rp^T{}_S + \f{e_H^T} \rp C_T{}^P{}_R \\
\\
\f{W}^Q{}_K \f{W}^K{}_R &=& \f{W}^Q{}_C \f{W}^C{}_R + \f{W}^Q{}_P \f{W}^P{}_R \\
&=& - \lp \ha \f{e^A} F^H_{AFR} \lp L^h \rp_C{}^F \lp L^h \rp^{QR} \rp
\lp \ha \f{e^D} F^H_{DEQ} \lp L^h \rp^{CE} \lp L^h \rp_R{}^Q \rp \\
&+&
\ha \lp \f{A^S} \, \lp L^h \rp^T{}_S + \f{e_H^T} \rp C_T{}^Q{}_P
\ha \lp \f{A^U} \, \lp L^h \rp^V{}_U + \f{e_H^V} \rp C_V{}^P{}_R
\end{eqnarray}
Combining these gives the curvature,
\begin{eqnarray}
\ff{R}^B{}_A &=& \lp \f{d} \f{\nu}^E{}_F \rp \lp L^h \rp^B{}_E \lp L^h \rp_A{}^F - \f{\nu}^E{}_F \f{e_H^P} C_P{}^B{}_D \lp L^h \rp^D{}_E \lp L^h \rp_A{}^F - \f{\nu}^E{}_F \lp L^h \rp^B{}_E \f{e_H^P} C_P{}^F{}_C \lp L^h \rp^C{}_A \\
&-& \ha \lp \f{d} \f{A^Q} + \lp \f{d} \f{e_H^P} \rp \lp L^h \rp_P{}^Q - \f{e_H^P} \f{e_H^R} C_{RP}{}^T \lp L^h \rp_T{}^Q \rp F^H_{DEQ} \lp L^h \rp^{BD} \lp L^h \rp_A{}^E \\
&+& \ha \lp \f{A^Q} + \f{e_H^P} \lp L^h \rp_P{}^Q \rp
\lp \lp \f{d} F^H_{DEQ} \rp \lp L^h \rp^{BD} \lp L^h \rp_A{}^E
+ F^H_{DEQ} \f{e_H^R} C_R{}^B{}_C \lp L^h \rp^{CD} \lp L^h \rp_A{}^E
+ F^H_{DEQ} \lp L^h \rp^{BD} \f{e_H^R} C_R{}_{AC} \lp L^h \rp^{CE} \rp \\
&-& \f{d} \f{e_H^P} C_P{}^B{}_A \\
&+& \lp \f{\nu}^E{}_F \lp L^h \rp^B{}_E \lp L^h \rp_C{}^F - \ha \lp \f{A^Q} + \f{e_H^P} \lp L^h \rp_P{}^Q \rp F^H_{DEQ} \lp L^h \rp^{BD} \lp L^h \rp_C{}^E - \f{e_H^P} C_P{}^B{}_C \rp \\
&\times&
\lp \f{\nu}^G{}_H \lp L^h \rp^C{}_G \lp L^h \rp_A{}^H - \ha \lp \f{A^R} + \f{e_H^S} \lp L^h \rp_S{}^R \rp F^H_{HGR} \lp L^h \rp^{CH} \lp L^h \rp_A{}^G - \f{e_H^Q} C_Q{}^C{}_A \rp \\
&-& \lp \ha \f{e^D} F^H_{DEQ} \lp L^h \rp^{BE} \lp L^h \rp_P{}^Q \rp
\lp \ha \f{e^C} F^H_{CFR} \lp L^h \rp_A{}^F \lp L^h \rp^{PR} \rp \\
\\
\ff{R}^B{}_R &=& \ha \lp \f{d} F^H_{DEQ} \rp \lp L^h \rp^{BE} \lp L^h \rp_R{}^Q 
- \ha \f{e^D} F^H_{DEQ} \f{e_H^P} C_P{}^B{}_C \lp L^h \rp^{CE} \lp L^h \rp_R{}^Q 
- \ha \f{e^D} F^H_{DEQ} \lp L^h \rp^{BE} \f{e_H^P} C_{PRS} \lp L^h \rp^{SQ} \\
&+& \lp \f{\nu}^E{}_F \lp L^h \rp^B{}_E \lp L^h \rp_C{}^F - \ha \lp \f{A^Q} + \f{e_H^P} \lp L^h \rp_P{}^Q \rp F^H_{DEQ} \lp L^h \rp^{BD} \lp L^h \rp_C{}^E - \f{e_H^P} C_P{}^B{}_C \rp
\lp \ha \f{e^G} F^H_{GHS} \lp L^h \rp^{CH} \lp L^h \rp_R{}^S \rp \\
&-& \lp \ha \f{e^D} F^H_{DEQ} \lp L^h \rp^{BE} \lp L^h \rp_P{}^Q \rp
 \ha \lp \f{A^S} \, \lp L^h \rp^T{}_S + \f{e_H^T} \rp C_T{}^P{}_R \\
\\
\ff{R}^Q{}_R &=& - \ha \lp \lp \f{d} \f{A^S} \rp \lp L^h \rp^P{}_S
- \f{A^S} \f{e_H^U} C_U{}^P{}_T \lp L^h \rp^T{}_S
+ \f{d} \f{e_H^P} \rp C_P{}^Q{}_R \\
&+&
\ha \lp \f{A^S} \, \lp L^h \rp^T{}_S + \f{e_H^T} \rp C_T{}^Q{}_P
\ha \lp \f{A^U} \, \lp L^h \rp^V{}_U + \f{e_H^V} \rp C_V{}^P{}_R
\end{eqnarray}

From this ugly mess, the [[Ricci curvature]], $\f{R}{}_I = \ve{E_J} \ff{R}^J{}_I$, is
\begin{eqnarray}
\f{R}{}_B &=& \ve{E_A} \ff{R}^A{}_B + \ve{E_R} \ff{R}^R{}_B \\
\f{R}{}_R &=& \ve{E_B} \ff{R}^B{}_R + \ve{E_Q} \ff{R}^Q{}_R \\
\end{eqnarray}

//ack, I've got to work on something else for awhile.//

[[curvature scalar]]

Check that it matches [[reductive Lie group tangent bundle geometry]] as special case.
A ''Cartan tangent bundle geometry'' is a [[reductive Lie group tangent bundle geometry]] that has gone a little wavy. The [[frame]] 1-forms, $\f{E^J}$, over what was the Lie group manifold, $E_H \sim G$, split in adapted coordinates as
\begin{eqnarray}
\f{E^A}(x,y) &=& \f{e^B}(x) \, \lp L^h\rp^A{}_B(y) \\
\f{E^P}(x,y) &=& \f{A^Q}(x) \, \lp L^h \rp^P{}_Q(y) + \f{e_H^P}(y)
\end{eqnarray}
in which $\f{e^B}$ and $\f{A^Q}$ are the [[Cartan frame|Cartan geometry]] forms and [[Cartan H-connection|Cartan geometry]] forms, $\lp L^h \rp^A{}_B = \lp H^A, h^- H_B h(y) \rp$ is the [[left-right rotator]] over $H$, and $\f{e_H^P}$ are the frame 1-forms over $H$. These frame 1-forms are components of the Ehresmann Cartan H-connection form, $\f{E^J} = \f{{\cal C}_H^J}$, over the [[Cartan H-bundle]], $E_H$, with $\f{E^A} = \f{{\cal E}_H^A}$ and $\f{E^P} = \f{{\cal A}_H^P}$. The Cartan tangent bundle, $TE_H$ IS the bundle of tangent vectors over the Cartan H-bundle, $E^H$, but from the point of view of treating the Ehresmann Cartan H-connection forms as a frame. Holding this point of view, we need to figure out what the [[Cartan tangent bundle spin connection]], $\f{W}{}^J{}_K$, is from this frame, and its curvature.
The [[tangent bundle spin connection|tangent bundle connection]] for a [[Cartan tangent bundle geometry]] is determined by insisting the [[torsion]] vanishes over $E_H$, giving [[Cartan's equation]],
$$
\ff{T^J} = 0 = \f{d} \f{E^J} + \f{W}{}^J{}_K \f{E^K}
$$
which may be solved for the ''Cartan tangent bundle spin connection'', $\f{W}{}^J{}_K$. To construct the solution, we first compute the [[exterior derivative]] of the [[frame]] 1-forms,
\begin{eqnarray}
\f{d} \f{E^A} &=& \lp \f{d} \f{e^B} \rp \lp L^h\rp^A{}_B - \f{e^B} \f{d} \lp L^h\rp^A{}_B \\
&=& \lp \f{d} \f{e^B} \rp \lp L^h\rp^A{}_B - \f{e^B} \f{e_H^P} \lp L^h \rp^D{}_B C_{DP}{}^A \\
\f{d} \f{E^P} &=& \lp \f{d} \f{A^Q} \rp \lp L^h \rp^P{}_Q - \f{A^Q} \f{d} \lp L^h \rp^P{}_Q + \f{d} \f{e_H^P} \\
&=& \lp \f{d} \f{A^Q} \rp \lp L^h \rp^P{}_Q - \f{A^Q} \f{e_H^R} \lp L^h \rp^T{}_Q C_{TR}{}^P - \ha \f{e_H^Q} \f{e_H^R} C_{QR}{}^P
\end{eqnarray}
using the [[left-right rotator]], $\lp L^h\rp^I{}_J = \lp T^I, h^- T_J h \rp$, and the [[Maurer-Cartan equation|Maurer-Cartan form]] over $H$. Next we write down the [[orthonormal basis vectors|frame]] (satisfying $\ve{E_J} \f{E^K} = \de_J^K$),
\begin{eqnarray}
\ve{E_A}(x,y) &=& \lp L^h \rp_A{}^B \, \ve{e_B}(x) - \lp L^h \rp_A{}^C \lp \ve{e_C} \f{A^Q} \rp \lp L^h \rp^P{}_Q \, \ve{e^H_P}  \\
\ve{E_P}(x,y) &=& \ve{e^H_P}(y)
\end{eqnarray}
and compute the [[anholonomy|Cartan's equation]] coefficients, $f_{IJ}{}^K = \ve{E_J} \ve{E_I} \lp \f{d} \f{E^K} \rp$, getting
\begin{eqnarray}
f_{AB}{}^C &=& \ve{e_E} \ve{e_D} \lp \f{d} \f{e^F} \rp \lp L^h \rp_B{}^E \lp L^h \rp_A{}^D \lp L^h \rp^C{}_F 
+ 2 \lp - \lp L^h \rp_{\lb B \rd}{}^E \, \ve{e_E} \lp L^h \rp_{\ld A \rb}{}^G \lp \ve{e_G} \f{A^Q} \rp \lp L^h \rp^R{}_Q \, \ve{e^H_R} \rp \lp - \f{e^F} \f{e_H^P} \lp L^h \rp^D{}_F C_{DP}{}^C \rp \\
&=& f^M_{DE}{}^F \lp L^h \rp_A{}^D \lp L^h \rp_B{}^E \lp L^h \rp^C{}_F 
- 2 \lp \ve{e_E} \f{A^Q} \rp \lp L^h \rp^R{}_Q \lp L^h \rp_{\lb A \rd}{}^E \, C_{\ld B \rb R}{}^C \\
f_{AQ}{}^C &=& - C_{AQ}{}^C \\
f_{AB}{}^R &=& \ve{e_E} \ve{e_D} \lp \f{d} \f{A^Q} \rp \lp L^h \rp_B{}^E \lp L^h \rp_A{}^D \lp L^h \rp^R{}_Q
- \lp L^h \rp_A{}^E \lp \ve{e_E} \f{A^Q} \rp \lp L^h \rp^P{}_Q \lp L^h \rp_B{}^D \lp \ve{e_D} \f{A^S} \rp \lp L^h \rp^T{}_S C_{TP}{}^R \\
&=& \ve{e_E} \ve{e_D} \lp \ff{F_H^Q} \rp \lp L^h \rp_B{}^E \lp L^h \rp_A{}^D \lp L^h \rp^R{}_Q \\
&=& F^H_{DE}{}^Q \lp L^h \rp_A{}^D \lp L^h \rp_B{}^E \lp L^h \rp^R{}_Q \\
f_{AQ}{}^R &=& 0 \\
f_{PQ}{}^C &=& 0 \\
f_{PQ}{}^R &=& - C_{PQ}{}^R
\end{eqnarray}
in which $f^M_{DE}{}^F(x)$ is the anholonomy for $\f{e^A}$ and $\ff{F_H^Q}(x) = \f{d} \f{A^Q} + \ha \f{A^P} \f{A^R} C_{PR}{}^Q$ is the [[curvature]] for $\f{A^Q}$. Using these, the solution to Cartan's equation, $W_{IJK} = \ha \lp f_{IJK} - f_{JKI} + f_{KIJ} \rp$, gives the Cartan tangent bundle spin connection coefficients,
\begin{eqnarray}
W_{ABC} &=& \nu_{DEF} \lp L^h \rp_A{}^D \lp L^h \rp_B{}^E \lp L^h \rp_C{}^F - \lp L^h \rp_A{}^E \lp \ve{e_E} \f{A^Q} \rp \lp L^h \rp^R{}_Q C_{B R C} \\
W_{ABR} &=& \ha f_{ABR} = \ha F^H_{DEQ} \lp L^h \rp_A{}^D \lp L^h \rp_B{}^E \lp L^h \rp_R{}^Q \\
W_{AQR} &=& 0 \\
W_{PBC} &=& - C_{PBC} - \ha f_{BCP} = - C_{PBC} - \ha F^H_{DEQ} \lp L^h \rp_B{}^D \lp L^h \rp_C{}^E \lp L^h \rp_P{}^Q \\
W_{PQC} &=& 0 \\
W_{PQR} &=& - \ha C_{PQR}
\end{eqnarray}
with $\nu_{DEF}(x)$ the coefficients of the torsionless spin connection for $\f{e^A}$. From these, the ''Cartan tangent bundle spin connection'', $\f{W}{}_{JK} = \f{E^I} W_{IJK}$, is
\begin{eqnarray}
\f{W}{}_{BC} &=& \f{E^A} W_{ABC} + \f{E^P} W_{PBC} \\
&=& \f{e^D} \lp  \nu_{DEF} \lp L^h \rp_B{}^E \lp L^h \rp_C{}^F - \lp \ve{e_D} \f{A^Q} \rp \lp L^h \rp^R{}_Q C_{B R C} \rp 
+ \lp \f{A^R} \, \lp L^h \rp^P{}_R + \f{e_H^P} \rp \lp - C_{PBC} - \ha F^H_{DEQ} \lp L^h \rp_B{}^D \lp L^h \rp_C{}^E \lp L^h \rp_P{}^Q \rp \\
&=& \f{\nu}{}_{EF} \lp L^h \rp_B{}^E \lp L^h \rp_C{}^F 
- \ha \lp \f{A^Q} + \f{e_H^P} \lp L^h \rp_P{}^Q \rp F^H_{DEQ} \lp L^h \rp_B{}^D \lp L^h \rp_C{}^E - \f{e_H^P} C_{PBC} \\
\f{W}{}_{BR} &=& \f{E^A} W_{ABR} =  \ha \f{e^D} F^H_{DEQ} \lp L^h \rp_B{}^E \lp L^h \rp_R{}^Q \\
\f{W}{}_{QR} &=& \f{E^P} W_{PQR} = - \ha \lp \f{A^S} \, \lp L^h \rp^P{}_S + \f{e_H^P} \rp C_{PQR}
\end{eqnarray}
This may be used to calculate the [[Cartan tangent bundle curvature]].
''Cartan's equation'' relates the [[spin connection]] to the [[exterior derivative]] of the [[frame]] by asserting that the [[torsion]] is zero,
$$
0 = \f{d} \f{e} + \f{\om} \times \f{e} 
$$
or, equivalently,
$$
0 = \f{d} \f{e^\al} + \f{\om}^\al{}_\be \f{e^\be} 
$$
This equation may be solved in closed form for the spin or [[tangent bundle connection]] coefficients. To generalize, lets solve
$$
\f{\om} \times \f{e} = -\ff{f}
$$
for $\f{\om}$ in terms of the frame and an arbitrary Clifford vector valued 2-form, $\ff{f}$. In components, using the [[index bracket]], this is
$$
\om_{\lb i \rd}{}^{\al \be} \lp e_{\ld j \rb} \rp_\be = -\ha f_{ij}{}^\al
$$
Using the frame to change from coordinate to Clifford indices, and using antisymmetry of the last two spin connection coefficient indices, this may be expressed simply as
$$
\om_{\lb \be \ga \rb \al} = \ha f_{\be \ga \al}
$$
By once again juggling spin connection indices, we see from this that
$$
\om_{\lp \be \ga \rp \al} = \om_{\lb \al \be \rb \ga} + \om_{\lb \al \ga \rb \be}  = f_{\al \lp \be \ga \rp}
$$
Adding these last two expressions gives the explicit solution for the spin connection coefficients:
$$
\om_{\al \be \ga} = \om_{\lb \al \be \rb \ga} + \om_{\lp \al \be \rp \ga} = \ha \lp f_{\al \be \ga} - f_{\be \ga \al} + f_{\ga \al \be} \rp
$$
Putting these indices in their more familiar positions is done using the frame and [[Minkowski metric]]: $\om_{i}{}^{\de \ep} = \lp e_i \rp^\al \et^{\de \be} \et^{\ep \ga} \om_{\al \be \ga}$. Cartan's equation is solved by simply plugging $\ff{f}=\f{d} \f{e}$ into the above equation -- in coefficients,
$$
f_{\al \be \ga} = \lp e_\al \rp^i \lp e_\be \rp^j \et_{\ga \de} \lp \pa_i \lp e_j \rp^\de - \pa_j \lp e_i \rp^\de \rp
= \ve{e_\be} \ve{e_\al} \lp \f{d} \f{e}{}_\ga \rp
$$

Note that this last tensor, the ''anholonomy'', may also be expressed using the [[Lie bracket|Lie derivative]] of the orthonormal basis vectors,
$$
\lb \ve{e_\al} , \ve{e_\be} \rb_L = 2 \lb \lp e_{\lb \al \rd} \rp^j \pa_j \lp e_{\ld \be \rb} \rp^i \rb \ve{\pa_i}
= - 2 \lb \lp e_{\lb \al \rd} \rp^j \lp e_{\ld \be \rb} \rp^k \lp e_\ga \rp^i \pa_j \lp e_k \rp^\ga \rb \ve{\pa_i}
= - f_{\al \be}{}^\ga \ve{e_\ga} 
$$

It is possible to express the solution to Cartan's equation in a particularly pretty, index free way using [[Clifform algebra]]:
$$
\f{\om} = - \ve{e} \times \ff{f} + \fr{1}{4} \lp \ve{e} \times \ve{e} \rp \lp \f{e} \cdot \ff{f} \rp
$$
(A cute, if not particularly useful expression.)
[[Lie algebra]]
http://en.wikipedia.org/wiki/Casimir_invariant

related to [[Laplacian]]
For a $2 \times 2$ square matrix or $2$ or $3$ dimensional [[Clifford algebra]] element, $A$, using the [[trace]] and products gives
$$
0 = A^2 - \li A \ri A + \ha \lp \li A \ri^2 - \li A^2 \ri \rp
$$
For a $3 \times 3$ square matrix, $A$,
$$
0 = A^3 - \li A \ri A^2 + \ha \lp \li A \ri^2 - \li A^2 \ri \rp A - \fr{1}{6} \lp \li A \ri^3 - 3 \li A^2 \ri \li A \ri + 2 \li A^3 \ri \rp
$$
This generalizes to formula for larger matrices,
http://arxiv.org/hep-th/0701116
A ''Chern-Simons form'', $\nf{\om_p}$, is a grade $p$ [[differential form]] defined (for odd $p$) to satisfy
$$
\f{d} \nf{\om_p} = Tr\lp \ff{F}^{\fr{p+1}{2}} \rp
$$
in which $\ff{F}=\f{d} \f{A} + \f{A} \f{A}$ is the [[curvature]] for some [[principal bundle]] [[connection]], $\f{A}$. The first few Chern-Simons forms are
\begin{eqnarray}
\f{\om_1} &=& Tr\lp \f{A} \rp \\
\nf{\om_3} &=& Tr\lp \ff{F} \f{A} - \fr{1}{3} \f{A} \f{A} \f{A} \rp \\
\nf{\om_5} &=& Tr\lp \ff{F} \ff{F} \f{A} - \fr{1}{2} \ff{F} \f{A} \f{A} \f{A} + \fr{1}{10} \f{A} \f{A} \f{A} \f{A} \f{A} \rp \\
\nf{\om_7} &=& Tr\lp \ff{F} \ff{F} \ff{F} \f{A} + ? \ff{F} \ff{F} \f{A} \f{A} \f{A} + ? \ff{F} \f{A} \f{A} \f{A} \f{A} \f{A} + ? \f{A} \f{A} \f{A} \f{A} \f{A} \f{A} \f{A} \rp
\end{eqnarray}

The [[integral|integration]] of a Chern-Simons p-form over a $p$ dimensional [[manifold]] is a homotopy invariant called the ''Chern number'',
$$
c_p = \int \nf{\om_p}
$$
corresponding to the topology of the manifold. For a $(p+1)$ dimensional manifold with a boundary,
$$
\int Tr\lp \ff{F}^{\fr{p+1}{2}} \rp = \int \f{d} \nf{\om_p} = \int_\pa \nf{\om_p} = c_p 
$$

Also of potential interest is the relationship to the ''Pfaffian'',
$$
\ff{F}^{\fr{p+1}{2}} = Pf\lp F \rp \nf{d^{p+1}x}
$$
where $Pf(F) = \sqrt{\ll F \rl}$
ref:
http://en.wikipedia.org/wiki/Chern-Simons_form
Using the alternative notation for the [[covariant derivative]] employing the [[tangent bundle connection]] and [[cotangent bundle connection]], the covariant derivative of a suitably indexed tensor is written as
$$
D_{i}T^{k}{}_{j}=\partial_{i}T^{k}{}_{j}+\Gamma^{k}{}_{im}T^{m}{}_{j}-\Gamma^{m}{}_{ij}T^{k}{}_{m}
$$
with the ''Christoffel symbols'', $\Ga^k{}_{ij}$, defined as tangent bundle connection coefficients, 
$$
\na_i \ve{\pa_j} = \Ga^k{}_{ij} \ve{\pa_k} 
$$
The Christoffel symbols are determined from the assumptions that the [[torsion]] vanishes,
$$
\Ga^k{}_{\lb ij \rb} = 0
$$
and that the covariant derivative is ''[[metric]] compatible'',
$$
0 = D_i g_{jk} = \pa_i g_{jk} - \Gamma^{m}{}_{ij} g_{mk} - \Gamma^{m}{}_{ik} g_{jm}
$$
It is then computed explicitly in terms of the metric, metric inverse, and its partial derivatives as
$$
\Ga^k{}_{ij} = \ha g^{km} \lp \pa_j g_{mi} + \pa_i g_{jm} - \pa_m g_{ij} \rp
$$
Computing the Christoffel symbols from vanishing torsion and metric compatibility is equivalent to calculating the [[spin connection]] from [[Cartan's equation]].

The Christoffel symbols (with torsion) may alternatively be computed from the [[tangent bundle spin connection|tangent bundle connection]], using the expression for the covariant derivative of the [[orthonormal basis vectors|frame]],
$$
\na_i \ve{e_\al} = \lp \pa_i \lp e_\al \rp^j + \lp e_\al \rp^k \Ga^j{}_{ik} \rp \ve{\pa_j} = w_{i}{}^\be{}_\al \ve{e_\be}
= w_{i}{}^\be{}_\al \lp e_\be \rp^j \ve{\pa_j}
$$
to get
$$
\Ga^j{}_{ik} = \lp e_k \rp^\al \lp w_{i}{}^\be{}_\al \lp e_\be \rp^j - \pa_i \lp e_\al \rp^j \rp = \lp e_\be \rp^j \lp w_{i}{}^\be{}_\al \lp e_k \rp^\al + \pa_i \lp e_k \rp^\be \rp
$$
This last expression may be used to easily determine how the Christoffel symbols, which do not constitute a tensor, transform under a [[coordinate change]] to
\begin{eqnarray}
\Ga^n{}_{ml} &=& \fr{\pa x^k}{\pa y^l} \lp e_k \rp^\al \fr{\pa x^i}{\pa y^m} \lp w_{i}{}^\be{}_\al \lp e_\be \rp^j \fr{\pa y^n}{\pa x^j} - \pa_i \lp \lp e_\al \rp^j \fr{\pa y^n}{\pa x^j} \rp \rp \\
&=& \fr{\pa y^n}{\pa x^j} \fr{\pa x^i}{\pa y^m} \fr{\pa x^k}{\pa y^l} \Ga^j{}_{ik} - \fr{\pa x^i}{\pa y^m} \fr{\pa x^j}{\pa y^l} \fr{\pa^2 y^n}{\pa x^i \pa x^j} 
\end{eqnarray}
From the last term we see that it's possible to choose a set of coordinates in which the Christoffel symbols vanish if and only if the torsion vanishes, $\Ga^k{}_{\lb ij \rb} = 0$. It is sometimes argued, along the lines of the [[equivalence principle|frame]], that such a choice should be possible and hence torsion should vanish.  

Using the Christoffel symbols is quite old fashioned, but sometimes practical. Things may be fancied up a bit by defining the ''Christoffel 1-form''s, $\f{\Ga^k{}_j} = \f{dx^i} \Ga^k{}_{ij}$, and using the [[vector-form algebra]] and [[partial derivative]] to get $\f{\na} \ve{\pa_j} = \f{\Ga^k{}_j} \ve{\pa_k}$ and
$$
\f{\na} \ve{e_\al} = \f{\na} \lp e_\al \rp^k \ve{\pa_k} = \f{\pa} \ve{e_\al} + \lp e_\al \rp^k \f{\Ga^j{}_k} \ve{\pa_j} = \f{\om^\be{}_\al} \ve{e_\be}
$$
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The $n=6$ dimensional [[Clifford algebra]], ''Cl(0,6)'', is built from 3 negative norm [[Clifford basis vectors]], $\Ga_\al$.

This algebra has many [[Clifford matrix representation]]s in real or complex $8\times8$ matrices. One particularly nice real representation, built using the [[Kronecker product]] of [[Pauli matrices]], is
\begin{eqnarray}
\Ga_1 &=& i \, \si^P_1 \otimes \si^P_2 \otimes \si^P_1 \\
\Ga_2 &=& i \, \si^P_3 \otimes \si^P_2 \otimes \si^P_0 \\
\Ga_3 &=& i \, \si^P_1 \otimes \si^P_2 \otimes \si^P_3 \\
\Ga_4 &=& i \, \si^P_3 \otimes \si^P_1 \otimes \si^P_2 \\
\Ga_5 &=& i \, \si^P_1 \otimes \si^P_0 \otimes \si^P_2 \\
\Ga_6 &=& i \, \si^P_3 \otimes \si^P_3 \otimes \si^P_2
\end{eqnarray}
A real, chiral representation of $Cl(0,8)$ is
\begin{eqnarray}
\Ga_1 &=& i \si^P_1 \otimes \si^P_3 \otimes \si^P_0 \otimes \si^P_2 \\
\Ga_2 &=& i \si^P_1 \otimes \si^P_1 \otimes \si^P_0 \otimes \si^P_2 \\
\Ga_3 &=& i \si^P_1 \otimes \si^P_2 \otimes \si^P_3 \otimes \si^P_0 \\
\Ga_4 &=& i \si^P_1 \otimes \si^P_2 \otimes \si^P_1 \otimes \si^P_0 \\
\Ga_5 &=& i \si^P_1 \otimes \si^P_0 \otimes \si^P_2 \otimes \si^P_3 \\
\Ga_6 &=& i \si^P_1 \otimes \si^P_0 \otimes \si^P_2 \otimes \si^P_1 \\
\Ga_7 &=& i \si^P_1 \otimes \si^P_2 \otimes \si^P_2 \otimes \si^P_2 \\
\Ga_8 &=& i \si^P_2 \otimes \si^P_0 \otimes \si^P_0 \otimes \si^P_0
\end{eqnarray}
The $n=4$ dimensional ''[[spacetime]] [[Clifford algebra]]'', Cl(1,3), is built from 4 anti-commuting, [[Clifford basis vectors]], $\ga_\mu$, with positive time [[Minkowski norm|Minkowski metric]],
$$
\ga_\mu \cdot \ga_\nu = \ha \lp \ga_\mu \ga_\nu + \ga_\nu \ga_\mu \rp = \et_{\mu \nu}
$$
The full algebra has $2^4 = 16$ [[Clifford basis elements]],
| !Element(s) | !Grade | !Multiplicity |!Names |
| $1$ | $0$ | $1$ |scalar |
| $\ga_\mu$ | $1$ | $4$ |vector |
| $\ga_{\mu \nu}$ | $2$ | $6$ |bivector |
| $\ga_{\mu \nu \ka } = \fr{1}{3!} \ep_{\mu \nu \ka \la} \ga^\la \ga $ | $3$ | $4$ |trivector |
| $\ga_{\mu \nu \ka \la} = \ep_{\mu \nu \ka \la} \ga$ | 4 | $1$ |4-vector, pseudoscalar |
The ''spacetime [[pseudoscalar]]'', $\ga = \ga_0 \ga_1 \ga_2 \ga_3$, satisfies $\ga \ga = -1$ and anti-commutes with odd-graded elements. This algebra has several nice [[Clifford matrix representation]]s in real or complex $4\times4$ matrices -- the [[Dirac matrices]].
The 6 bivector [[Clifford basis elements]] of the spacetime Clifford algebra, [[Cl(1,3)]] may be represented by multiplying the 4 basis vectors, in the [[Weyl representation|Dirac matrices]], to get:
\begin{eqnarray}
\ga_{01} &=&  \si^P_3 \otimes \si^P_1 \\
\ga_{02} &=&  \si^P_3 \otimes \si^P_2 \\
\ga_{03} &=&  \si^P_3 \otimes \si^P_3 \\
\ga_{12} &=& -i 1 \otimes \si^P_3 \\
\ga_{13} &=& +i 1 \otimes \si^P_2 \\
\ga_{23} &=& -i 1 \otimes \si^P_1
\end{eqnarray}
Any ''Cl(1,3) bivector'' may thus be represented as
\begin{eqnarray}
B &=& \ha B^{\mu \nu} \ga_{\mu \nu} =
\lb \begin{array}{cc}
B_L & 0 \\
0 & B_R
\end{array} \rb
=
\lb \begin{array}{cc}
B^{0 \va} \si^P_\va - i \ha B^{\va \ze} \ep_{\va \ze \ta} \si^P_\ta & 0 \\
0 & - B^{0 \va} \si^P_\va - i \ha B^{\va \ze} \ep_{\va \ze \ta} \si^P_\ta
\end{array} \rb \\
&=&
\lb \begin{array}{cccc}
B^{03}- i B^{12} & B^{01}+B^{13}-i B^{02}- i B^{23} & 0 & 0 \\
B^{01}- B^{13}+i B^{02}-i B^{23} & -B^{03}+i B^{12} & 0 & 0 \\
0 & 0 & -B^{03}- i B^{12} & -B^{01}+B^{13}+i B^{02}- i B^{23} \\
0 & 0 & -B^{01}- B^{13}-i B^{02}-i B^{23} & B^{03}+i B^{12}
\end{array} \rb
\end{eqnarray}
in which $B_{L/R}$ are the ''left and right [[chiral]] bivector parts'', projected out by the [[left/right chirality projector]], and $\ep_{\va \ze \ta}$ is the three dimensional [[permutation symbol]]. These $2\times2$ matrices satisfy $B_L^\dagger = - B_R$, using Hermitian conjugation. Note that a bivector is completely determined by one of its chiral parts. The bivectors of [[Cl(3,1)]] have the same expression, with signs reversed since the expressions of all vectors pick up $i$'s. These $4 \times 4$ and $2 \times 2$ matrices are representations of the $spin(1,3)$ [[Lie algebra]]. 
A real, chiral representation for $Cl(8,8)$ may be built by starting with a real, chiral [[Cl(0,8)]] rep and picking out one of the vectors, such as $\Ga_8$, and using it to build the ''real, chiral Cl(8,8) basis vectors'':
\begin{eqnarray}
\ga^{\lp16\rp}_\al &=& \Ga_\al \otimes 1 \\
\ga^{\lp16\rp}_8 &=& \Ga_8 \otimes \Ga \\
\ga^{\lp16\rp}_{\lp\al+8\rp} &=& \Ga_8 \otimes \Ga_\al \\
\ga^{\lp16\rp}_{(16)} &=& \Ga_8 \otimes \Ga_8
\end{eqnarray}
with $1 \le \al \le 7$.
The $n=16$ dimensional [[Clifford algebra]], $Cl(16,0)$, is built from 16 anti-commuting, positive norm [[Clifford basis vectors]], $\ga^{\lp16\rp}_\al$. The full algebra has $2^{16} = 65,536$ [[Clifford basis elements]]. This algebra has many [[Clifford matrix representation]]s in real or complex $256\times256$ matrices. One particularly nice representation, built using the [[Kronecker product]] of [[Cl(8)]] elements, is
\begin{eqnarray}
\ga^{\lp16\rp}_\al &=& \Ga_\al \otimes 1 \\
\ga^{\lp16\rp}_{\lp\al+8\rp} &=& \Ga \otimes \Ga_\al
\end{eqnarray}
with $1 \le \al \le 8$. (Note that this rep is not [[chiral]].) These $16$ ''Cl(16) basis vectors'' may be multiplied to get the $120$ ''Cl(16) basis bivectors'',
\begin{eqnarray}
\ga^{\lp16\rp}_{\al \be} &=& \Ga_{\al \be} \otimes 1 \\
\ga^{\lp16\rp}_{\lp\al+8\rp \lp\be+8\rp} &=& 1 \otimes \Ga_{\al \be} \\
\ga^{\lp16\rp}_{\al \lp\be+8\rp} = -\ga^{\lp16\rp}_{\lp\be+8\rp \al} &=& \lp \Ga_\al \Ga \rp \otimes \Ga_{\be}
\end{eqnarray}
The [[pseudoscalar]] in this rep, $\ga^{\lp16\rp} = \Ga \otimes \Ga$, satisfies $\ga^{\lp16\rp} \ga^{\lp16\rp} = 1$ and anti-commutes with odd-graded elements.

A chiral representation for $Cl(16)$ may be built by starting with a chiral [[Cl(8)]] rep and picking out one of the vectors, such as $\Ga_8$, and using it to build the ''chiral Cl(16) basis vectors'':
\begin{eqnarray}
\ga^{\lp16\rp}_\al &=& \Ga_\al \otimes 1 \\
\ga^{\lp16\rp}_8 &=& \Ga_8 \otimes \Ga \\
\ga^{\lp16\rp}_{\lp\al+8\rp} &=& \Ga_8 \otimes \Ga_\al \\
\ga^{\lp16\rp}_{(16)} &=& \Ga_8 \otimes \Ga_8
\end{eqnarray}
with $1 \le \al \le 7$. The pseudoscalar in this rep is
$$
\ga^{\lp16\rp} = ( \Ga \otimes \Ga ) ( 1 \otimes \Ga ) = \Ga \otimes 1
$$
The (1st level) chirality projector is $P_{\pm} = \ha \lp 1 \mp \ga^{\lp16\rp} \rp = \ha \lp 1 \mp \Ga \rp \otimes 1$. The basis vectors may be used to build the ''chiral Cl(16) basis bivectors'',
\begin{eqnarray}
\ga^{\lp16\rp}_{\al \be} &=& \Ga_{\al \be} \otimes 1 \\
\ga^{\lp16\rp}_{\al 8} &=& \Ga_{\al 8} \otimes \Ga \\
\ga^{\lp16\rp}_{\al \lp\be+8\rp} &=& \Ga_{\al 8} \otimes \Ga_\be \\
\ga^{\lp16\rp}_{\al (16)} &=& \Ga_{\al 8} \otimes \Ga_8 \\
\ga^{\lp16\rp}_{\lp\al+8\rp \lp\be+8\rp} &=& 1 \otimes \Ga_{\al \be} \\
\ga^{\lp16\rp}_{\lp\al+8\rp (16)} &=& 1 \otimes \Ga_{\al 8} \\
\ga^{\lp16\rp}_{8 \lp\be+8\rp} &=& 1 \otimes \Ga \Ga_\be \\
\ga^{\lp16\rp}_{8 (16)} &=& 1 \otimes \Ga \Ga_8
\end{eqnarray}
The three dimensional [[Clifford algebra]], Cl(3,0), is generated by three [[Clifford basis vectors]], $\si_\io$. These basis vectors have a matrix representation as the three [[Pauli matrices]], $\si_\io=\si_\io^P$. The eight [[Clifford basis elements]] are formed by all possible products of these Clifford basis vectors. The complete multiplication table for the algebra, calculated from the general [[Clifford basis product identities]], is (row header times column header equals entry):
|              | !$1$ | !$\si_1$ | !$\si_2$ | !$\si_3$ | !$\si_{12}$ | !$\si_{13}$ | !$\si_{23}$ | !$\si$ |
| !$1$      |  $1$ |  $\si_1$ |  $\si_2$ |  $\si_3$ |  $\si_{12}$ |  $\si_{13}$ |  $\si_{23}$ |  $\si$ |
| !$\si_1$ |  $\si_1$ |  $1$ |bgcolor(#a0ffa0):  $\si_{12}$ |bgcolor(#a0ffa0):  $\si_{13}$ |  $\si_2$ |  $\si_3$ |  $\si$ |bgcolor(#a0ffa0):  $\si_{23}$ |
| !$\si_2$ |  $\si_2$ |bgcolor(#a0ffa0):  $-\si_{12}$ |  $1$ |bgcolor(#a0ffa0):  $\si_{23}$ |  $-\si_1$ |  $-\si$ |  $\si_3$ |bgcolor(#a0ffa0):  $-\si_{13}$ |
| !$\si_3$ |  $\si_3$ |bgcolor(#a0ffa0):  $-\si_{13}$ |bgcolor(#a0ffa0):  $-\si_{23}$ |  $1$ |  $\si$ |  $-\si_1$ |  $-\si_2$ |bgcolor(#a0ffa0):  $\si_{12}$ |
| !$\si_{12}$ |  $\si_{12}$ |  $-\si_2$ | $\si_1$ |  $\si$ |bgcolor(#88ccff):  $-1$ |bgcolor(#88ccff):  $-\si_{23}$ |bgcolor(#88ccff):  $\si_{13}$ |  $-\si_3$ |
| !$\si_{13}$ |  $\si_{13}$ |  $-\si_3$ |  $-\si$ |  $\si_1$ |bgcolor(#88ccff):  $\si_{23}$ |bgcolor(#88ccff):  $-1$ |bgcolor(#88ccff):  $-\si_{12}$ |  $\si_2$ |
| !$\si_{23}$ |  $\si_{23}$ |  $\si$ |  $-\si_3$ |  $\si_2$ |bgcolor(#88ccff):  $-\si_{13}$ |bgcolor(#88ccff):  $\si_{12}$ |bgcolor(#88ccff):  $-1$ |  $-\si_1$ |
| !$\si$ |  $\si$ |bgcolor(#a0ffa0):  $\si_{23}$ |bgcolor(#a0ffa0):  $-\si_{13}$ |bgcolor(#a0ffa0):  $\si_{12}$ |  $-\si_3$ |  $\si_2$ |  $-\si_1$ |  $-1$ |
The blue square shows the bivector subalgebra. This bivector subalgebra is the [[three dimensional special unitary group Lie algebra|su(2)]], with the identification $T_A = \si \si_A = \epsilon_{ABC} \si_{BC} = i \si_A^P$ giving the three $su(2)$ generators,
$$
\begin{array}{ccc}
T_1 = i \sigma_{1}^{P} = \si_{23}
&
T_2 = i \sigma_{2}^{P} = -\si_{13}
&
T_3 = i \sigma_{3}^{P} = \si_{12}
\end{array}
$$
which form a closed subalgebra under the [[commutator]]. The green entries illustrate the two ways the bivector basis elements can be represented -- as the product of vectors, or as the product of vector and pseudoscalar. The [[pseudoscalar]], $\si=\si_1 \si_2 \si_3$, squares to $-1$ and has the matrix representation $\si = i$.

The ''three dimensional Clifford algebra of negative signature'', $Cl(0,3)$, is obtained by using $\si'_\io = i \si_\io$ as the basis vectors.
The $n=4$ dimensional ''[[spacetime]] [[Clifford algebra]]'', Cl(3,1), is built from 4 anti-commuting, [[Clifford basis vectors]], $\ga_\mu$, with negative time [[Minkowski norm|Minkowski metric]],
$$
\ga_\mu \cdot \ga_\nu = \ha \lp \ga_\mu \ga_\nu + \ga_\nu \ga_\mu \rp = \et_{\mu \nu}
$$
The full algebra has $2^4 = 16$ [[Clifford basis elements]],
| !Element(s) | !Grade | !Multiplicity |!Names |
| $1$ | $0$ | $1$ |scalar |
| $\ga_\mu$ | $1$ | $4$ |vector |
| $\ga_{\mu \nu}$ | $2$ | $6$ |bivector |
| $\ga_{\mu \nu \ka } = \fr{1}{3!} \ep_{\mu \nu \ka \la} \ga^\la \ga $ | $3$ | $4$ |trivector |
| $\ga_{\mu \nu \ka \la} = \ep_{\mu \nu \ka \la} \ga$ | 4 | $1$ |4-vector, pseudoscalar |
The ''spacetime [[pseudoscalar]]'', $\ga = \ga_0 \ga_1 \ga_2 \ga_3$, satisfies $\ga \ga = -1$ and anti-commutes with odd-graded elements. This algebra has several nice [[Clifford matrix representation]]s in real or complex $4\times4$ matrices -- the [[Dirac matrices]].
The $n=6$ dimensional [[Clifford algebra]], ''Cl(3,3)'', is built from 3 positive norm and 3 negative norm [[Clifford basis vectors]], $\Ga_\al$.

This algebra has many [[Clifford matrix representation]]s in real or complex $8\times8$ matrices. One particularly nice, [[chiral]], real representation, built using the [[Kronecker product]] of [[Pauli matrices]], is
\begin{eqnarray}
\Ga_1 &=& i \, \si^P_2 \otimes \si^P_1 \otimes \si^P_0 \\
\Ga_2 &=& i \, \si^P_2 \otimes \si^P_2 \otimes \si^P_2 \\
\Ga_3 &=& i \, \si^P_2 \otimes \si^P_3 \otimes \si^P_0 \\
\Ga_4 &=& \p{i \,} \si^P_1 \otimes \si^P_0 \otimes \si^P_0 \\
\Ga_5 &=& \p{i \,} \si^P_2 \otimes \si^P_2 \otimes \si^P_3 \\
\Ga_6 &=& \p{i \,} \si^P_2 \otimes \si^P_2 \otimes \si^P_1
\end{eqnarray}
The first four of these come from realifying the Weyl representation of the [[Dirac matrices]].
The $n=8$ dimensional [[Clifford algebra]], ''Cl(4,4)'', is built from 4 positive norm and 4 negative norm [[Clifford basis vectors]], $\Ga_\al$. It is the same as [[Cl(8)]] except for the signature.

This algebra has many [[Clifford matrix representation]]s in real or complex $16\times16$ matrices. One particularly nice, [[chiral]], real representation, built using the [[Kronecker product]] of [[Pauli matrices]], is
\begin{eqnarray}
\Ga_1 &=& i \, \si^P_2 \otimes \si^P_2 \otimes \si^P_1 \otimes \si^P_2 \\
\Ga_2 &=& -i \, \si^P_2 \otimes \si^P_2 \otimes \si^P_2 \otimes \si^P_0 \\
\Ga_3 &=& i \, \si^P_2 \otimes \si^P_2 \otimes \si^P_3 \otimes \si^P_2 \\
\Ga_0 = \Ga_4 &=& \si^P_2 \otimes \si^P_1 \otimes \si^P_0 \otimes \si^P_2 \\
\Ga_5 &=& \si^P_1 \otimes \si^P_2 \otimes \si^P_2 \otimes \si^P_1 \\
\Ga_6 &=& \si^P_2 \otimes \si^P_3 \otimes \si^P_0 \otimes \si^P_2 \\
\Ga_7 &=& \si^P_1 \otimes \si^P_2 \otimes \si^P_2 \otimes \si^P_3 \\
\Ga_8 &=& i \, \si^P_1 \otimes \si^P_0 \otimes \si^P_0 \otimes \si^P_2
\end{eqnarray}
giving $\Ga = - \, \si^P_3 \otimes \si^P_0 \otimes \si^P_0 \otimes \si^P_0$.
The $n=8$ dimensional [[Clifford algebra]], ''Cl(5,3)'', is built from 5 positive norm and 3 negative norm [[Clifford basis vectors]], $\Ga_\al$. It is the same as [[Cl(8)]] except for the signature.

This algebra has many [[Clifford matrix representation]]s in real or complex $16\times16$ matrices. One particularly nice, [[chiral]], complex representation, built using the [[Kronecker product]] of [[Pauli matrices]], is
\begin{eqnarray}
\Ga_1 &=& i \si^P_2 \otimes \si^P_3 \otimes \si^P_0 \otimes \si^P_1 \\
\Ga_2 &=& i \si^P_2 \otimes \si^P_3 \otimes \si^P_0 \otimes \si^P_2 \\
\Ga_3 &=& i \si^P_2 \otimes \si^P_3 \otimes \si^P_0 \otimes \si^P_3 \\
\Ga_0 = \Ga_4 &=& \si^P_1 \otimes \si^P_0 \otimes \si^P_0 \otimes \si^P_0 \\
\Ga_5 &=& \si^P_2 \otimes \si^P_1 \otimes \si^P_1 \otimes \si^P_0 \\
\Ga_6 &=& \si^P_2 \otimes \si^P_1 \otimes \si^P_2 \otimes \si^P_0 \\
\Ga_7 &=& \si^P_2 \otimes \si^P_1 \otimes \si^P_3 \otimes \si^P_0 \\
\Ga_8 &=& \si^P_2 \otimes \si^P_2 \otimes \si^P_0 \otimes \si^P_0
\end{eqnarray}
giving $\Ga = - i \, \si^P_3 \otimes \si^P_0 \otimes \si^P_0 \otimes \si^P_0$.

Standard model using this Cl(5,3) rep:
Has correct Higgs.
Ah, no good -- would give negative [[cosmological constant]].
$$
{\scriptsize
\lb \begin{array}{cccccccc}
\ha \f{\om_L} \!+\! i \f{W^3} & i \f{W^1} \!+\! \f{W^2} & - \fr{1}{4} \f{e_R} \ph_0^* & \fr{1}{4} \f{e_R} \ph_+ &
\ud{\nu_L} & \ud{u_L^r} & \ud{u_L^b} & \ud{u_L^g} \\

i \f{W^1} \!-\! \f{W^2} & \ha \f{\om_L} \!-\! i \f{W^3} & \fr{1}{4} \f{e_R} \ph_+^* & \fr{1}{4} \f{e_R} \ph_0 &
\ud{e_L} & \ud{d_L^r} & \ud{d_L^b} & \ud{d_L^g} \\

\fr{1}{4} \f{e_L} \ph_0 & -\fr{1}{4} \f{e_L} \ph_+ & \ha \f{\om_R} \!+\! i \f{B} & &
\ud{\nu_R} & \ud{u_R^r} & \ud{u_R^b} & \ud{u_R^g} \\

-\fr{1}{4} \f{e_L} \ph_+^* & -\fr{1}{4} \f{e_L} \ph_0^* & & \ha \f{\om_R} \!-\! i \f{B} &
\ud{e_R} & \ud{d_R^r} & \ud{d_R^b} & \ud{d_R^g} \\

& & & & i \f{B} & &  &  \\
& & & &  & \fr{-i}{3} \f{B} \!+\! i \f{G^{3+8}} & i\f{G^1} \!-\! \f{G^2} & i\f{G^4} \!-\! \f{G^5} \\
& & & &  & i\f{G^1} \!+\! \f{G^2} & \fr{-i}{3} \f{B} \!-\! i \f{G^{3+8}} & i\f{G^6} \!-\! \f{G^7} \\
& & & &  & i\f{G^4} \!+\! \f{G^5} & i\f{G^6} \!+\! \f{G^7} & \fr{-i}{3} \f{B} \!-\!\! \fr{2i}{\sqrt{3}}\f{G^8}
\end{array} \rb
}
$$
The $n=8$ dimensional [[Clifford algebra]], $Cl(8,0)$, is built from 8 anti-commuting, positive norm [[Clifford basis vectors]], $\Ga_\al$. The full algebra has $2^8 = 256$ [[Clifford basis elements]],
| !Element(s) | !Grade | !Multiplicity |!Names |
| $1$ | $0$ | $1$ |scalar |
| $\Ga_\al$ | $1$ | $8$ |vector |
| $\Ga_{\al \be}$ | $2$ | $28$ |bivector |
| $\Ga_{\al \be \ga}$ | $3$ | $56$ |trivector |
| $\Ga_{\al \be \ga \de}$ | $4$ | $70$ |4-vector |
| $\Ga_{\al \be \ga \de \ep}$ | $5$ | $56$ |5-vector |
| $\Ga_{\al \dots \be}$ | $6$ | $28$ |6-vector |
| $\Ga_{\al \dots \be}$ | $7$ | $8$ |7-vector |
| $\Ga_{\al \dots \be} = \ep_{\al \dots \be} \Ga$  | $8$ | $1$ |8-vector, psuedoscalar |
The [[pseudoscalar]], $\Ga$, satisfies $\Ga \Ga = 1$ and anti-commutes with odd-graded elements. This algebra has many [[Clifford matrix representation]]s in real or complex $16\times16$ matrices. One particularly nice, [[chiral]], complex representation, built using the [[Kronecker product]] of [[Pauli matrices]], is
\begin{eqnarray}
\Ga_1 &=& \si^P_2 \otimes \si^P_3 \otimes \si^P_0 \otimes \si^P_1 \\
\Ga_2 &=& \si^P_2 \otimes \si^P_3 \otimes \si^P_0 \otimes \si^P_2 \\
\Ga_3 &=& \si^P_2 \otimes \si^P_3 \otimes \si^P_0 \otimes \si^P_3 \\
\Ga_4 &=& \si^P_2 \otimes \si^P_2 \otimes \si^P_0 \otimes \si^P_0 \\
\Ga_5 &=& \si^P_2 \otimes \si^P_1 \otimes \si^P_1 \otimes \si^P_0 \\
\Ga_6 &=& \si^P_2 \otimes \si^P_1 \otimes \si^P_2 \otimes \si^P_0 \\
\Ga_7 &=& \si^P_2 \otimes \si^P_1 \otimes \si^P_3 \otimes \si^P_0 \\
\Ga_0 = \Ga_8 &=& \si^P_1 \otimes \si^P_0 \otimes \si^P_0 \otimes \si^P_0
\end{eqnarray}
giving $\Ga = \si^P_3 \otimes \si^P_0 \otimes \si^P_0 \otimes \si^P_0$. These may all be expressed in a $16\times16$ matrix (using $2\times2$ sub-matrices) as
\begin{eqnarray}
& & \Ga_\pi + z \Ga_4 + a \Ga_5 + b \Ga_6 + c \Ga_7 + \Ga_8 = \\
& & \lb
\begin{array}{cccccccc}
& & & & 1-i\si^p_\pi & & -z-ic & -b-ia \\
& & & & & 1-i\si^p_\pi & b-ia & -z+ic \\
& & & & z-ic & -b-ia & 1+i\si^p_\pi & \\
& & & & b-ia & z+ic & & 1+i\si^p_\pi \\
1+i\si^p_\pi & & z+ic & b+ia & & & & \\
& 1+i\si^p_\pi & -b+ia & z-ic & & & & \\
-z+ic & b+ia & 1-i\si^p_\pi & & & & & \\
-b+ia & -z-ic & & 1-i\si^p_\pi & & & &
\end{array}
\rb
\end{eqnarray}

A nice chiral, real representation of $Cl(8,0)$ is
\begin{eqnarray}
\Ga_1 &=& \si^P_2 \otimes \si^P_3 \otimes \si^P_0 \otimes \si^P_2 \\
\Ga_2 &=& \si^P_2 \otimes \si^P_1 \otimes \si^P_0 \otimes \si^P_2 \\
\Ga_3 &=& \si^P_2 \otimes \si^P_2 \otimes \si^P_3 \otimes \si^P_0 \\
\Ga_4 &=& \si^P_2 \otimes \si^P_2 \otimes \si^P_1 \otimes \si^P_0 \\
\Ga_5 &=& \si^P_2 \otimes \si^P_0 \otimes \si^P_2 \otimes \si^P_3 \\
\Ga_6 &=& \si^P_2 \otimes \si^P_0 \otimes \si^P_2 \otimes \si^P_1 \\
\Ga_7 &=& \si^P_2 \otimes \si^P_2 \otimes \si^P_2 \otimes \si^P_2 \\
\Ga_8 &=& \si^P_1 \otimes \si^P_0 \otimes \si^P_0 \otimes \si^P_0
\end{eqnarray}
And another one, matched to [[su(3)]], is
\begin{eqnarray}
\Ga_1 &=& \si^P_1 \otimes \si^P_0 \otimes \si^P_0 \otimes \si^P_0 \\
\Ga_2 &=&-\si^P_2 \otimes \si^P_0 \otimes \si^P_0 \otimes \si^P_2 \\
\Ga_3 &=& \si^P_2 \otimes \si^P_3 \otimes \si^P_2 \otimes \si^P_1 \\
\Ga_4 &=&-\si^P_2 \otimes \si^P_0 \otimes \si^P_2 \otimes \si^P_3 \\
\Ga_5 &=&-\si^P_2 \otimes \si^P_2 \otimes \si^P_0 \otimes \si^P_1 \\
\Ga_6 &=& \si^P_2 \otimes \si^P_2 \otimes \si^P_3 \otimes \si^P_3 \\
\Ga_7 &=& \si^P_2 \otimes \si^P_1 \otimes \si^P_3 \otimes \si^P_1 \\
\Ga_8 &=&-\si^P_2 \otimes \si^P_2 \otimes \si^P_1 \otimes \si^P_3
\end{eqnarray}
The ''chirality operator for Cl(8)'' is $P^{\lp8\rp}_\pm = \ha \lp 1 \pm \Ga \rp$.
The 28 bivector [[Clifford basis elements]] of [[Cl(8,0)|Cl(8)]] may be represented by multiplying the 8 basis vectors, in the complex rep, to get:
\begin{eqnarray}
\Ga_{01} &=& i \si^P_3 \otimes \si^P_3 \otimes \si^P_0 \otimes \si^P_1 \\
\Ga_{02} &=& i \si^P_3 \otimes \si^P_3 \otimes \si^P_0 \otimes \si^P_2 \\
\Ga_{03} &=& i \si^P_3 \otimes \si^P_3 \otimes \si^P_0 \otimes \si^P_3 \\
\Ga_{12} &=& i \si^P_0 \otimes \si^P_0 \otimes \si^P_0 \otimes \si^P_3 \\
\Ga_{13} &=& -i \si^P_0 \otimes \si^P_0 \otimes \si^P_0 \otimes \si^P_2 \\
\Ga_{23} &=& i \si^P_0 \otimes \si^P_0 \otimes \si^P_0 \otimes \si^P_1 \\
&\,& \\
\Ga_{04} &=& i \si^P_3 \otimes \si^P_2 \otimes \si^P_0 \otimes \si^P_0 \\
\Ga_{05} &=& i \si^P_3 \otimes \si^P_1 \otimes \si^P_1 \otimes \si^P_1 \\
\Ga_{06} &=& i \si^P_3 \otimes \si^P_1 \otimes \si^P_2 \otimes \si^P_0 \\
\Ga_{07} &=& i \si^P_3 \otimes \si^P_1 \otimes \si^P_3 \otimes \si^P_0 \\
\Ga_{14} &=& -i \si^P_0 \otimes \si^P_1 \otimes \si^P_0 \otimes \si^P_1 \\
\Ga_{15} &=& i \si^P_0 \otimes \si^P_2 \otimes \si^P_1 \otimes \si^P_1 \\
\Ga_{16} &=& i \si^P_0 \otimes \si^P_2 \otimes \si^P_2 \otimes \si^P_1 \\
\Ga_{17} &=& i \si^P_0 \otimes \si^P_2 \otimes \si^P_3 \otimes \si^P_1 \\
\Ga_{24} &=& -i \si^P_0 \otimes \si^P_1 \otimes \si^P_1 \otimes \si^P_2 \\
\Ga_{25} &=& i \si^P_0 \otimes \si^P_2 \otimes \si^P_1 \otimes \si^P_2 \\
\Ga_{26} &=& i \si^P_0 \otimes \si^P_2 \otimes \si^P_2 \otimes \si^P_2 \\
\Ga_{27} &=& i \si^P_0 \otimes \si^P_2 \otimes \si^P_3 \otimes \si^P_2 \\
\Ga_{34} &=& -i \si^P_0 \otimes \si^P_1 \otimes \si^P_0 \otimes \si^P_3 \\
\Ga_{35} &=& i \si^P_0 \otimes \si^P_2 \otimes \si^P_1 \otimes \si^P_3 \\
\Ga_{36} &=& i \si^P_0 \otimes \si^P_2 \otimes \si^P_2 \otimes \si^P_3 \\
\Ga_{37} &=& i \si^P_0 \otimes \si^P_2 \otimes \si^P_3 \otimes \si^P_3 \\
&\,& \\
\Ga_{45} &=& -i \si^P_0 \otimes \si^P_3 \otimes \si^P_1 \otimes \si^P_0 \\
\Ga_{46} &=& -i \si^P_0 \otimes \si^P_3 \otimes \si^P_2 \otimes \si^P_0 \\
\Ga_{47} &=& -i \si^P_0 \otimes \si^P_3 \otimes \si^P_3 \otimes \si^P_0 \\
\Ga_{56} &=& i \si^P_0 \otimes \si^P_0 \otimes \si^P_3 \otimes \si^P_0 \\
\Ga_{57} &=& -i \si^P_0 \otimes \si^P_0 \otimes \si^P_2 \otimes \si^P_0 \\
\Ga_{67} &=& i \si^P_0 \otimes \si^P_0 \otimes \si^P_1 \otimes \si^P_0
\end{eqnarray}
A real, chiral representation for $Cl(8,8)$ may be built by starting with a real, chiral [[Cl(0,8)]] rep and picking out one of the vectors, such as $\Ga_8$, and using it to build the ''real, chiral Cl(8,8) basis vectors'':
\begin{eqnarray}
\ga^{\lp16\rp}_\al &=& \Ga_\al \otimes 1 \\
\ga^{\lp16\rp}_8 &=& \Ga_8 \otimes \Ga \\
\ga^{\lp16\rp}_{\lp\al+8\rp} &=& \Ga_8 \otimes \Ga_\al \\
\ga^{\lp16\rp}_{(16)} &=& \Ga_8 \otimes \Ga_8
\end{eqnarray}
with $1 \le \al \le 7$.
The ''Clifford adjoint'' transformation of a [[Clifford element]], $A$, by a [[Clifford group]] element, $U$, is
\[ A' = U A U^- \]
The ''Clifford inner [[automorphism]]'', a.k.a. //''similarity transformation''//, is the Clifford adjoint transformation of the [[Clifford basis vectors]], 
\[ \ga'_\al = U \ga_\al U^- \]
This subsequently produces the Clifford adjoint transformation of all Clifford elements built from these basis vectors, since
$$
\ga'_{\al \dots \be} = \ga'_\al \dots \ga'_\be = U \ga_\al U^- \dots U \ga_\be U^- = U \ga_{\al \dots \be} U^-
$$
It is an automorphism because it is a map, specified by $U \in Cl^*$, from the [[Clifford algebra]] into itself.

The adjoint transformation does not necessarily preserve the [[grade|Clifford grade]] of elements.  It does, however, preserve scalars, $\li A' B' \ri = \li UAU^- UBU^- \ri = \li A B \ri$.  The [[fundamental Clifford identity|Clifford basis vectors]], $\ga_\al \cdot \ga_\be = \et_{\al \be}$, is preserved by the Clifford automorphism, $\ga'_\al \cdot \ga'_\be = \et_{\al \be}$, preserving the structure of the Clifford algebra and the decomposition of [[Clifford element]]s even though the transformed basis "vectors", $\ga'_\al$, may no longer be grade 1 with respect to the old basis.

For Clifford group elements near the identity, $U \simeq 1 + \ha C$, the Clifford adjoint is approximately
\[ A' = U A U^- \simeq \lp 1 + \ha C \rp A \lp 1 - \ha C \rp \simeq A + C \times A \]
with the "small" Clifford element, $C$, acting via the [[cross product|antisymmetric bracket]].
An "$n$ dimensional" ''Clifford algebra'', $Cl(p,q)$, is a $2^n$ dimensional [[Lie algebra]] of [[Clifford element]]s consisting of coefficients multiplying [[Clifford basis elements]] constructed from $p$ positive norm and $q$ negative norm ($n=p+q$) [[Clifford basis vectors]], $\ga_\al$.  The Clifford algebra product of any two Clifford elements, equivalent to the [[matrix product in a suitable representation|Clifford matrix representation]], is non-commutative and decomposes into ''symmetric product'' (''//dot product//'') and [[antisymmetric product|antisymmetric bracket]] (''//cross product//'') parts,
\begin{eqnarray}
AB &=& A \cdot B + A \times B\\
A \cdot B &=& \ha \lp AB+BA \rp\\
A \times B &=& \ha \lp AB-BA \rp
\end{eqnarray}
The product is associative and distributive,
\begin{eqnarray}
A \lp B C \rp = \lp A B \rp C\\
A \lp B + C \rp = A B + A C
\end{eqnarray}
And, just as for matrices, [[almost all elements|Clifford group]] have an [[inverse]], $AA^-=1$.

A Clifford algebra is a graded "geometric algebra" in that the elements of [[Clifford grade]] $0,1,2,3,\dots$ may be considered as scalars, vectors, areas, volumes, ... and the Clifford product as operations between them.  For example, the product of two vectors is a scalar (their dot product) plus an area (their cross product).  The antisymmetric product of three vectors is a 3-vector, or volume.  The product and its decomposition are completely described by the [[Clifford basis product identities]].
Any two [[Clifford basis elements]] are orthogonal under the [[scalar part|Clifford grade]] operator.  Taking the scalar part of two multiplied basis elements of grade $r$ gives the orthogonality relation,
\[ \li \ga_{\al \dots \be} \ga^{\ga \dots \de}  \ri = r! \de^\ga_{\lb \be \rd} \dots \de^\de_{\ld \al \rb} \]
in which the basis element indices have been raised with the [[Minkowski metric]].  The scalar part of any two multiplied basis elements of unequal grade is 0.

The orthogonality relation may be used to determine the ''scalar product'' of any two multivectors.  For example, between a multivector, $A$, and bivector, $B$, the scalar product is
\[ \li A B \ri = \fr{1}{4} A^{\al \be} B_{\ga \de} \li \ga_{\al \be} \ga^{\ga \de} \ri = \fr{1}{4} A^{\al \be} B_{\ga \de} 2 \de^\ga_{\lb \be \rd} \de^\de_{\ld \al \rb} = \ha A^{\al \be} B_{\be \al} \]

The scalar part operator, and the orthogonality relations, is equivalent to the matrix [[trace]] and [[Lie algebra]] generator orthogonality through the [[Killing form]].
The $2^n$ ''Clifford basis elements'' are formed by all possible products of the $n$ [[Clifford basis vectors]].  Because of the [[fundamental Clifford identity|Clifford basis vectors]], basis elements are antisymmetric under the exchange of indices, like [[coordinate basis forms]], and may be written via the [[antisymmetric bracket]].  Each basis element has a [[grade|Clifford grade]], $q$, corresponding to the number of constituent basis vectors, and a multiplicity, ${n \choose q} = \fr{n!}{q!(n-q)!}$, equal to the number of their ordered combinations,
| !Element(s) | !Grade | !Multiplicity |!Names |
| $1$ | $0$ | ${n \choose 0} = 1$ |scalar, real number |
| $\ga_\al$ | $1$ | ${n \choose 1} = n$ |vector, [[Clifford basis vectors]]|
| $\ga_{\al \be} = \ga_{\lb \al \be \rb} = \ga_\al \ga_\be = \lb \ga_\al,\ga_\be \rb_A$ | 2 | ${n \choose 2} = \ha n (n-1)$ |bivector, 2-vector |
| $\ga_{\al \be \ga} = \ga_{\lb \al \be \ga \rb} = \ga_\al \ga_\be \ga_\ga = \lb \ga_\al,\ga_\be,\ga_\ga \rb_A$ | 3 | ${n \choose 3} = \fr{1}{3!} n (n-1)(n-2)$ |trivector, 3-vector |
| $\vdots$ |  $\vdots$ | $\vdots$ |$\vdots$ |
| $\ga_{\al \dots \be} = \ep_{\al \dots \be} \ga$  | $n$ | ${n \choose n} = 1$ |n-vector |
Each of these $\sum_{k=0}^n {n \choose k} =2^n$ Clifford basis elements is a [[Lie algebra]] generator, with structure coefficients corresponding to [[Clifford basis product identities]].  The Clifford basis elements also satisfy [[Clifford basis element orthogonality]].

Using [[Clifford dual]]ity, it is often convenient to express high grade basis elements in terms of the Clifford [[pseudoscalar]],
$$\ga = \ga_0 \ga_1 \dots \ga_{n-1}$$
and the [[permutation symbol]].  In this way, the basis r-vectors can be written as
$$\ga_{\al \dots \be} = \fr{1}{\lp n-r \rp!} \ep_{\al \dots \be \ga \dots \de} \ga^{\ga \dots \de} \ga$$
For example, the $n$ basis (n-1)-vectors are
$$\ga_{\al \dots \be} = \fr{1}{\lp n-1 \rp!} \ep_{\al \dots \be \ga} \ga^\ga \ga$$
This reduces the number of indices necessary to represent high grade [[Clifford element]]s.

The $2^n$ basis elements can also be written via a generalized index as $\ga_A$, with $A$ enumerating
each different antisymmetric combination of the usual Clifford indices.
The ''Clifford basis product identities'' are derived from the [[fundamental Clifford identity|Clifford basis vectors]] by splitting the product of two basis vectors into a scalar (a [[Minkowski metric]] component) plus a bivector,
\[ \ga_\al \ga_\be = \ga_\al \cdot \ga_\be + \ga_\al \times \ga_\be = \et_{\al \be} + \ga_{\al \be} \]
or going in reverse &mdash; rewriting a bivector as a scalar plus the product of two basis vectors.  By selectively applying this rule, all [[Clifford element]]s can be written as sums of coefficients times [[Clifford basis elements]].  The structure coefficients characterizing the [[Clifford algebra]] as a [[Lie algebra]] can be read off the [[cross product|antisymmetric bracket]] identities,
\begin{eqnarray}
\ga_\al \times \ga_\be &=& \ga_{\al \be}\\
\ga_\al \times \ga_{\be \ga} &=& \et_{\al \be} \ga_{\ga} - \et_{\al \ga} \ga_{\be}\\
\ga_{\al \be} \times \ga_{\ga \de} &=& - \et_{\al \ga} \ga_{\be \de} + \et_{\al \de} \ga_{\be \ga} + \et_{\be \ga} \ga_{\al \de} - \et_{\be \de} \ga_{\al \ga}\\
\ga_\al \times \ga_{\be \ga \de} &=& \ga_{\al \be \ga \de} \\
&\vdots& 
\end{eqnarray}
Equally useful identities arise for the symmetric product,
\begin{eqnarray}
\ga_\al \cdot \ga_\be &=& \et_{\al \be}\\
\ga_\al \cdot \ga_{\be \ga} &=& \ga_{\al \be \ga}\\
\ga_{\al \be} \cdot \ga_{\ga \de} &=& \lp \et_{\al \de} \et_{\be \ga} - \et_{\al \ga} \et_{\be \de} \rp + \ga_{\al \be \ga \de}\\
\ga_\al \cdot \ga_{\be \ga \de} &=& \et_{\al\be} \ga_{\ga\de} - \et_{\al\ga} \ga_{\be\de} + \et_{\al\de} \ga_{\be\ga} \\
 &\vdots& 
\end{eqnarray}
Continuing the series, the product of two basis elements of [[grade|Clifford grade]]s $p$ and $q$, such as
\begin{eqnarray}
\ga_{\al \be} \ga_{\ga \de} &=& \ga_{\al \be} \cdot \ga_{\ga \de} + \ga_{\al \be} \times \ga_{\ga \de}\\
 &=& \lp \et_{\al \de} \et_{\be \ga} - \et_{\al \ga} \et_{\be \de} \rp + \lp - \et_{\al \ga} \ga_{\be \de} + \et_{\al \de} \ga_{\be \ga} + \et_{\be \ga} \ga_{\al \de} - \et_{\be \de} \ga_{\al \ga} \rp + \ga_{\al \be \ga \de}
\end{eqnarray}
gives a result of mixed grades $|p-q|$ through $p+q \le n$ in steps of $2$.  For example, a bivector times a 3-vector typically gives a vector plus a 3-vector plus a 5-vector if $n$ is at least 5, otherwise just a vector plus a 3-vector.

The products of even or odd graded elements are
| !grade of $A$ | !grade of $B$ | !grade of $AB$ |
| even | even | even |
| odd | odd | even |
| odd | even | odd |
The cross product of anything with a bivector is grade preserving.
A rest [[frame]] exists at each point in a curved [[spacetime]].  A sufficiently small surrounding region is described locally by a diagonal [[Minkowski metric]], $\eta_{\al \be}$, with $p$ positive and $q$ negative unit entries.  This may be visualized by considering a set of $n$ ''orthonormal'' (orthogonal and unit length) geometric "vector" elements, the ''Clifford basis vectors'', or ''//Clifford algebra generators//'', $\ga_\al$.  These Clifford basis vectors provide a means for invariantly describing local geometric objects.

Like [[coordinate basis 1-forms]], two unequal Clifford basis vectors anti-commute, and their product represents a geometric area, or ''bivector'', element such as $\ga_1 \ga_2 = - \ga_2 \ga_1$, representing a unit area element spanned by $\ga_1$ and $\ga_2$. The orthonormality of Clifford basis vectors is expressed by the ''fundamental Clifford identity'',
$$
\ga_\al \cdot \ga_\be = \ha \lp \ga_\al \ga_\be + \ga_\be \ga_\al  \rp = \et_{\al \be}
$$
which gives a Clifford scalar (real number) as a result of the symmetric product of two Clifford vectors.  The antisymmetric product of every combination of two unequal Clifford vectors gives the $\ha n (n-1)$ bivector [[Clifford basis elements]],
$$
\ga_\al \times \ga_\be = \ha \lb \ga_\al, \ga_\be \rb = \ha \lp \ga_\al \ga_\be - \ga_\be \ga_\al  \rp =  \ga_\al \ga_\be = \ga_{\lb \al \be \rb} = \ga_{\al \be}
$$
The ''Clifford bundle'', $Cl M$, with base [[manifold]] $M$ is an [[automorphism bundle]] and a [[vector bundle]] with $2^n$ fiber basis elements equal to the [[Clifford basis elements]], $\ga_{\al \dots \be}$. The fiber at each base manifold point, $p$, is the space of [[Clifford element]]s.  The transition functions for the basis elements over overlapping patches, $U_1$ and $U_2$, are given by [[Clifford adjoint]]s,
$$
\ga_{\al \dots \be}^2 = U_{21} \ga_{\al \dots \be}^1 U_{21}^-
$$
which don't necessarily preserve [[Clifford grade]]. The structure group, $Aut(Cl)=Cl^*$, the automorphism group, is the [[Clifford group]] with adjoint action on the fiber. //(Is that true?)//

For a section, $C(x)$, transforming under the adjoint action [[gauge transformation]], $C \mapsto C'=U C U^-$, the [[covariant derivative]] is
$$
\f{\na} C = \f{d} C + \ha \f{A} C - \ha C \f{A} = \f{d} C + \f{A} \times C
$$
(defined with a $\ha$ in it to keep things pretty) with the [[Clifford connection]], $\f{A}$, applied using the [[cross product|Clifford algebra]].

Any fiber element, $C$, at $t=0$ may be [[parallel transport]]ed to $C(t)=U(t)CU^-$ along a path on the base by a parameter dependent Clifford element, the path holonomy, $U(t) = Pe^{- \ha \int_0^t \f{A}}$, satisfying the [[path holonomy]] equation,
$$
\fr{d}{dt} U(t) = - \ha \ve{v} \f{A} U
$$

Applying the covariant derivative twice (being careful with the signs of 1-forms hopping sides),
\begin{eqnarray}
\f{\na} \f{\na} C &=& \f{d} \lp \f{d} C + \ha \f{A} C - \ha C \f{A} \rp + \ha \f{A} \lp \f{d} C + \ha \f{A} C - \ha C \f{A} \rp  + \ha \lp \f{d} C + \ha \f{A} C - \ha C \f{A} \rp \f{A} \\
&=& \ha \lp \f{d} \f{A} \rp C - \ha \f{A} \f{d} C - \ha \lp \f{d} C \rp \f{A} - \ha C \f{d} \f{A} 
 + \ha \f{A} \lp \f{d} C + \ha \f{A} C - \ha C \f{A} \rp  + \ha \lp \f{d} C + \ha \f{A} C - \ha C \f{A} \rp \f{A} \\
&=& \ff{F} \times C
\end{eqnarray}
gives the [[Clifford curvature|Clifford-Riemann curvature]],
$$
\ff{F} = \f{d} \f{A} + \ha \f{A} \times \f{A}
$$
a Clifford valued 2-form. This expression for the curvature may alternatively be obtained from the [[holonomy]] (minding the new factor of $\ha$ in the path holonomy equation).

Under a gauge transformation, $C(x) \mapsto C'(x) = U(x) C(x) U^-(x)$, the covariant derivative changes to
\begin{eqnarray}
\f{\na'} C' &=& U \lp \f{\na} C \rp U^-\\
\f{d} \lp U C U^- \rp + \ha \f{A'} U C U^- - \ha U C U^- \f{A'} &=& U \lp \f{d} C \rp U^- + \ha U \f{A} C U^- - \ha U C \f{A} U^-
\end{eqnarray}
giving the transformation law for the connection,
$$
\f{A'} = U \f{A} U^- - 2 \lp \f{d} U \rp U^- = U \f{A} U^- + 2 U \lp \f{d} U^- \rp 
$$
An infinitesimal transformation, $U \simeq 1 + \ha C$, changes the connection to
$$
\f{A'} \simeq \f{A} - \f{d} C - \ha \f{A} C + \ha C \f{A} = \f{A} - \f{\na} C
$$
The curvature consequently transforms under a gauge transformation to
$$
\ff{F'} = \f{d} \f{A'} + \ha \f{A'} \times \f{A'} = U \ff{F} U^- \simeq \ff{F} + C \times \ff{F}
$$

The covariant derivative acting on a [[Clifform]] such as the curvature, transforming under the adjoint action, $\ff{F'} = U \ff{F} U^-$, is still 
$$
\f{\na} \ff{F} = \f{d} \ff{F} + \f{A} \times \ff{F} 
$$

The [[graded Clifford bundle|Clifford vector bundle]] has the same fiber as the [[Clifford bundle]], but the transition functions (which for the graded Clifford bundle are grade preserving) are [[Clifford rotation]]s.
One may carry out several unary operations on [[Clifford element]]s.

The [[inverse]] of a Clifford element, $A^-$, is most generally computed by working in a [[Clifford matrix representation]].  However, some cases may be handled easily, such as the inverse of a Clifford vector, $v^- = \fr{v}{v \cdot v}$.

The [[Clifford dual]] of an element, $A \ga^-$, is often a useful object.

The ''involution'' operator inverts the signs of all vectors in an element, producing a [[grade|Clifford grade]] dependent sign change for the parts of an element, $\hat{A^r} = \lp -1 \rp^r A^r$, also expressible as $\hat{A} = A^e - A^o$.

The ''reversion operator'', a.k.a. ''//reverse//'', reverses the order of all vectors multiplied in an element, producing $\tilde{A^r} = \lp -1 \rp^{\ha r(r-1)} A^r$.

''Clifford conjugation'' combines these last two, $\bar{A} = \tilde{\hat{A^r}} = \lp -1 \rp^{\ha r(r+1)} A^r$.

For a set of [[Dirac matrices]] in which $\ga_0$ is represented by a Hermitian matrix and all spatial [[Clifford basis vectors]] are represented by anti-Hermitian matrices, the ''Hermitian conjugate'' of a Clifford element is $A^\dagger = \ga_0 \tilde{A} \ga^0$.  When written as a matrix, this gives the transpose of the complex conjugate, $A^\dagger = A^{*T}$.

One often encounters the ''Dirac conjugate'', $\overline{A} = A^\dagger \ga_0 = \ga_0 \tilde{A}$, which shouldn't be confused with the Clifford conjugate.
The [[vector bundle connection]] for the [[Clifford bundle]] is defined through the operation of the suitable [[vector bundle covariant derivative|vector bundle connection]] on the [[Clifford basis vectors]] for the $Cl$ fiber. The structure group for the bundle is the [[Clifford group]], with group elements acting on the fiber through the [[Clifford adjoint]]. The covariant derivative may therefore be represented using a ''Clifford connection'', $\f{A} \in \f{Cl}$, acting on basis elements via the [[cross product|antisymmetric bracket]],
$$
\f{\na} \ga_\al = \f{A} \times \ga_\al 
$$
which gives the ''Clifford covariant derivative'' acting on any Clifford valued field (Clifford bundle section),
$$
\f{\na} C = \f{d} C + \f{A} \times C 
$$
 
Note that the covariant derivative for the Clifford bundle does not necessarily preserve [[Clifford grade]].
The ''Clifford curvature scalar'' is obtained by taking the [[scalar part|Clifford grade]] of the [[frame]] contracted twice with the [[Clifford bundle]] curvature,
$$
R = \li \ve{e} \ve{e} \ff{F} \ri
$$
If, specifically, we are working with the [[Clifford vector bundle]], the Clifford curvature scalar is then the result of the [[dot product|Clifford algebra]] of the frame with the [[Clifford-Ricci curvature]],
$$
R = \li \ve{e} \ve{e} \ff{R} \ri = \ve{e} \cdot \f{R} = \lp \ve{e} \times \ve{e} \rp \cdot \ff{R}
$$
and equals the [[curvature scalar]] written in terms of the [[spin connection]] and frame.

The Clifford curvature scalar also comes from the expression:
\begin{eqnarray}
\fr{2}{\lp n-2 \rp!} \li \lp \f{e} \rp^{n-2} \ff{R} \ga^- \ri
&=& \fr{2}{\lp n-2 \rp!} \f{e}^\al \dots \f{e}^\be \f{e}^\mu \f{e}^\nu \fr{1}{4} R_{\mu\nu}{}^{\rh\si} \li \ga_{\al \dots \be} \ga_{\rh \si} \ga^- \ri \\
&=& \fr{2}{\lp n-2 \rp!} \nf{e} \ep^{\al \dots \be \mu \nu} \fr{1}{4} R_{\mu\nu}{}^{\rh\si} \ep_{\al \dots \be \rh \si} \\
&=& \nf{e} \de_{\lb \rh \si \rb}^{\mu\nu} R_{\mu\nu}{}^{\rh \si}=\nf{e} R
\end{eqnarray}
using the [[volume form]] and [[permutation identities]].
The ''Clifford dual'' of any [[Clifford element]], $A$, is obtained by right multiplying it by the inverse [[pseudoscalar]], $A \ga^-$.  For a [[Clifford grade]] $r$ element, this gives a grade $(n-r)$ element,
\[ A^r \ga^- = \fr{1}{r!} A^{\al \dots \be} \ga_{\al \dots \be} \ga^- = \fr{1}{r! \lp n - r \rp !} A^{\al \dots \be} \li \ga_{\al \dots \be \ga \dots \de} \ga^- \ri \ga^{\ga \dots \de} = \fr{1}{r! \lp n - r \rp !} A^{\al \dots \be} \ep_{\al \dots \be \ga \dots \de} \ga^{\ga \dots \de} \]
in which $\ep_{\al \dots \be \ga \dots \de}$ is the [[permutation symbol]], and [[indices]] are raised with the [[Minkowski metric]].

The Clifford dual transformation is analogous to the [[Hodge dual]]. 
All [[Clifford algebra]] elements may be written as a sum of $2^n$ real coefficients multiplying [[Clifford basis elements]], with multiplicative factors included to account for the redundant sums over [[indices]],
\begin{eqnarray}
A &=& A^s + A^\al \ga_\al + \ha A^{\al \be} \ga_{\al \be} + \fr{1}{3!} A^{\al \be \ga} \ga_{\al \be \ga} + \dots + A^p \ga\\
&=& A^0 + A^1 + A^2 + A^3 + \dots + A^n
\end{eqnarray}
(Some people choose to limit the sums so they don't run over all index values &mdash; but this isn't done here.)  Like the coefficients of [[differential form]]s, the Clifford element coefficients are [[antisymmetric|index bracket]] in their indices, $A^{\al \dots \be}=A^{\lb \al \dots \be \rb}$.  Unlike differential forms, Clifford elements may be of mixed [[grade|Clifford grade]].

A clifford element has a geometric interpretation as a collection of variously sized scalar, vector, oriented area set, ..., and n-volume objects.

Clifford elements have a faithful [[matrix representation|Clifford matrix representation]].

The high grade terms of Clifford elements may be written with fewer indices by using the [[pseudoscalar]],
$$A^r = \fr{1}{r!} A^{\al \dots \be} \ga_{\al \dots \be} = \fr{1}{r!} A^{\al \dots \be} \fr{1}{\lp n-r \rp!} \ep_{\al \dots \be \ga \dots \de} \ga^{\ga \dots \de} \ga
= \fr{1}{\lp n-r \rp!} \lp \fr{1}{r!} A^{\al \dots \be} \ep_{\al \dots \be \ga \dots \de} \rp \ga^{\ga \dots \de} \ga
= \fr{1}{\lp n-r \rp!} A^r_{\ga \dots \de} \ga^{\ga \dots \de} \ga$$
So, for example, the pseudoscalar (n-vector, grade $n$) part is
$$A^n = \fr{1}{n!} A^{\al \dots \be} \ga_{\al \dots \be} = A^p \ga$$
and the (n-1)-vector part is
$$A^{n-1} = \fr{1}{\lp n-1 \rp!} A^{\al \dots \be} \ga_{\al \dots \be} = A^{n-1}_\al \ga^\al \ga$$
A ''Clifford [[gauge transformation|vector bundle gauge transformation]]'' is a change of the fiber basis elements for a [[Clifford bundle]], [[Clifford vector bundle]], or any graded Clifford bundle. The change may be induced by the action of an arbitrary, position dependent element of the fiber bundle's structure group -- a subgroup of the [[Clifford group]] acting on the [[Clifford basis elements]] via the [[Clifford adjoint]],
$$
\ga'_\al = U \ga_\al U^-
$$
This gauge transformation is an active transformation of bundle elements, and transforms any Clifford valued field (section), $\Ph$, to
$$
\Ph' = U \Ph U^-
$$
By definition, the [[Clifford covariant derivative|Clifford connection]] of any Clifford valued field transforms under a gauge transformation such that,
$$
\f{\na'} \Phi' = \lp \f{\na} \Phi \rp'
$$
Writing out the covariant derivative operators in this equation using the [[Clifford connection]],
\begin{eqnarray}
\f{\na'} \lp U \Phi U^- \rp &=& U \lp \f{\na} \Phi \rp U^- \\
\lp \f{d} U \rp \Phi U^- + U \lp \f{d} \Phi \rp U^- + U \Phi \lp \f{d} U^- \rp + \f{A'} \times \lp U \Phi U^-\rp &=& U \lp \f{d} \Phi + \f{A} \times \Phi \rp U^- \\
U^- \lp \f{d} U \rp \Phi + \Phi \lp \f{d} U^- \rp U + \ha U^- \f{A'} U \Phi - \ha \Phi U^- \f{A'} U &=&  \ha \f{A} \Phi - \ha \Phi \f{A}
\end{eqnarray}
gives the transformation law for the connection under a gauge transformation:
$$
\f{A'} = U \f{A} U^- - 2 \lp \f{d} U \rp U^- 
$$
For an infinitesimal gauge transformation, $U \simeq 1 + \ha C$, the connection changes to
$$
\f{A'} \simeq \f{A} - \f{d} C - \ha \f{A} C + \ha C \f{A} = \f{A} - \f{\na} C
$$
giving the change $\de \f{A} = - \f{\na} C$.
The ''grade'' of a [[Clifford element]] corresponds to the number of [[Clifford basis vectors]] used in the [[Clifford basis elements]] needed to represent it.  An element may be a single grade, $q$, in which case it is called a ''q-vector'', or it may be of mixed grade, and called a ''multivector''.  For example,
\[ t = \fr{1}{3!} t^{\al \be \ga} \ga_{\al \be \ga} \]
is a 3-vector, or ''trivector'', while
\[ w = w^s + \ha w^{\al \be} \ga_{\al \be} \]
is a multivector of grades 0 and 2.

The ''grade operator'', $\li A \ri_q = A^q$, acts as a filter, passing only the grade $q$ parts of $A$.  For example, the bivector part of $w$ is
\[ \li w \ri_2 = \ha w^{\al \be} \ga_{\al \be} \in \li Cl \ri_2 = Cl^2 \] 
The grade operator may also be used to filter the even or odd graded parts of an element, such as $\li w\ri_e = \li w\ri_2$ and $\li w\ri_o = 0$.  Of special interest is the grade 0 operator, $\li A\ri = \li A\ri_0 = A^0 = A^s$, or //''scalar part''// operator which gives the scalar part of $A$.  This operator is proportional to the [[trace]] of an element in a [[Clifford matrix representation]].  It is useful since the grade 0 (scalar) part of a Clifford element is a real number.
Combining [[Clifford basis product identities]] with the [[grade|Clifford grade]] operator gives a ''Clifford graded [[commutation|commutator]]'' relationship for two Clifford elements of grades $r$ and $s$,
\[ \li A^r B^s \ri_q = \lp -1 \rp^\ep \li B^s A^r \ri_q \]
with
\[ \ep = \ha \lp q^2 + r^2 + s^2 - q - r - s \rp \]
This relation implies that any two Clifford elements commute inside the scalar part operator, $\li AB \ri = \li BA \ri$.

The Clifford product of two elements of grades $r$ and $s$ can produce elements of various grades,
\[ A^r B^s = \li A^r B^s \ri_{\ll r - s \rl} + \li A^r B^s \ri_{\ll r - s \rl + 2} + \dots + \li A^r B^s \ri_{r + s} \]
The ''Clifford group'' consists of [[Clifford algebra]] elements having an inverse,
\[ Cl^* = \left\{ U \in Cl \mid \exists \; U^- \ni U U^- = 1 \right\} \]
It is the [[Lie group]] corresponding to [[exponentiation]] of the [[Clifford basis elements]],
$$U = e^{B^A \ga_A}$$
The [[Lie algebra]] corresponding to the Clifford group is the Clifford algebra.
Each [[Clifford algebra]] has a faithful representation in the real, quaternionic, or complex matrices, $GL(2^{[n/2]},\mathbb{C})$, with the Clifford product isomorphic to matrix multiplication. This corresponds to the traditional definition of [[Pauli matrices]] and [[Dirac matrices]] as the $\gamma_{\alpha}$ for the purpose of using matrix algebra to do Clifford Algebra calculations, or simply for writing Clifford elements as matrices. Various unary operations on Clifford elements, the [[Clifford conjugate]]s, are equivalent to various matrix conjugates. The possible representations for any $Cl(p,q)$ are described by the [[table of Clifford matrix representations]].

A Clifford algebra is built by starting with the basis vectors and creating all possible multiples. For a seed example, we can build a representation for ''Cl(2,0)'' by starting with two Pauli matrices as the two [[Clifford basis vectors]],
$$
\begin{array}{cc}
\si_1 = \sigma_{1}^{P} =
\left[\begin{array}{cc}
0 & 1\\
1 & 0
\end{array}\right]
&
\si_2 = \sigma_{2}^{P}=\left[\begin{array}{cc}
0 & -i\\
i & 0\end{array}\right]
\end{array}
$$
and multiplying to get the scalar and bivector,
$$
\begin{array}{cc}
1 = \si_1 \si_1 =
\left[\begin{array}{cc}
1 & 0\\
0 & 1\end{array}\right]
&
\si_{12} = \si_1 \si_2 =
\left[\begin{array}{cc}
i & 0\\
0 & -i
\end{array}\right]
= i \si_3^P
\end{array}
$$
completing the list of $Cl(2,0)$ [[Clifford basis elements]] represented as $2 \times 2$ complex matrices. To build larger Clifford algebras we can use the [[Kronecker product]] of any smaller Clifford algebras. For example, $Cl(2,2) = Cl(2,0) \otimes Cl(2,0)$. The tricky part is finding a set of orthogonal, anticommuting, ''Clifford basis vector matrix representatives'' after doing the product, such as picking out:
$$
\begin{array}{cc}
\ga_1 = \si_1 \otimes 1 =
\left[\begin{array}{cccc}
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1\\
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0
\end{array}\right]
&
\ga_2 = \si_2 \otimes \si_2 =
\left[\begin{array}{cccc}
0 & 0 & 0 & -1\\
0 & 0 & 1 & 0\\
0 & 1 & 0 & 0\\
-1 & 0 & 0 & 0
\end{array}\right]
\\
\ga_3 = \si_2 \otimes \si_1 =
\left[\begin{array}{cccc}
0 & 0 & 0 & -i\\
0 & 0 & -i & 0\\
0 & i & 0 & 0\\
i & 0 & 0 & 0
\end{array}\right]
&
\ga_4 = \si_2 \otimes \si_{12} =
\left[\begin{array}{cccc}
0 & 0 & 1 & 0\\
0 & 0 & 0 & -1\\
-1 & 0 & 0 & 0\\
0 & 1 & 0 & 0
\end{array}\right]
\end{array}
$$
A matrix representation, such as above for $Cl(3,1)$, allows any element to be represented by a $4 \times 4$ complex matrix. For example,
$$
a \, \ga_{12} + b \, \ga_{34} = a \, \si_{12} \otimes \si_2 + b \, 1 \otimes \si_2 =
\left[\begin{array}{cccc}
0 & a - i b & 0 & 0\\
-a + i b & 0 & 0 & 0\\
0 & 0 & 0 & -a - i b\\
0 & 0 & a + i b & 0
\end{array}\right]
$$
To get a different signature we can multiply any basis vector representaive by $i$, such as multiplying $\ga_3$ above by $i$ to get a ''real representation'' of $Cl(2,2)$ -- in which all basis vectors, and hence all elements, are represented by real matrices. And to represent a Clifford algebra of one less dimension we can discard a vector.

Everything done with Clifford algebra can be identified with the corresponding matrix manipulation; however, it will almost always be more geometrically revealing to deal with the Clifford algebra elements directly.

Refs:
*http://en.wikipedia.org/wiki/Representations_of_Clifford_algebras
*Andrzej Trautman
**[[Clifford Algebras and their Representations|papers/Clifford Algebras and their Representations.pdf]]
***p20 describes construction of reps for arbitrarily high dimension
Iteration of the [[cross product|antisymmetric bracket]] produces the ''Clifford Jacobi identity'',
\[ A \times \lp B \times C \rp + B \times \lp C \times A \rp + C \times \lp A \times B \rp = 0 \]
and the ''cross product distributive rule'',
\[ A \times \lp B C \rp = \lp A \times B \rp C + B \lp A \times C \rp \]
A combination of [[Clifford algebra]] dot and cross products is
\[ A \cdot \lp B \times C \rp + A \times \lp B \cdot C \rp = \ha \lp ABC - CBA \rp = \lp A \cdot B \rp \times C + \lp A \times B \rp \cdot C \]

A string of cross products without parenthesis, $A \times B \times C$, is not well defined because $A \times \lp B \times C \rp \ne \lp A \times B \rp \times C$; but a string of dot products, $A \cdot B \cdot C$, or a string of Clifford products, $ABC$, is well defined.  In general, parenthesis should always be used to group multiple operations when the cross product is employed.

To calculate Clifford products, it is best to use the [[Clifford basis product identities]]
A bivector crossed with a vector gives a vector orthogonal to the original, in the plane (or planes) of the bivector. Using the [[Clifford basis product identities]] and antisymmetry of bivector indices,
\begin{eqnarray}
B \times v &=& \ha B^{\al \be} v^\ga \ga_{\al \be} \times \ga_\ga = B^{\al \be} v^\ga \ga_{\lb \al \rd} \et_{\ld \be \rb \ga} = B^{\al \be} v_\be \ga_\al \\
v \cdot \lp B \times v \rp &=& v^\de B^{\al \be} v_\be \ga_\de \cdot \ga_\al = B^{\al \be} v_\al v_\be = 0
 \end{eqnarray}
A small rotational transformation in the plane (or planes) of a bivector may be carried out by
\[ v' = v + \fr{1}{N}B \times v \simeq \lp 1 + \fr{1}{2N} B \rp v \lp 1 - \fr{1}{2N} B \rp \]
for a large parameter, $N$.  A finite rotation comes from [[exponentiating|exponentiation]] the bivector,
\[ v' = \lim_{N \to \infty} \lp 1 + \fr{1}{2N} B \rp^N v \lp 1 - \fr{1}{2N} B \rp^N = e^{\ha B} v e^{- \ha B} = U v U^- \]
For example, rotating $v$ by an angle of $\th$ in the $\ga_{12}$ plane gives
\[ v' = e^{\ha \th \ga_{12}} v e^{- \ha \th \ga_{12}} = \lp \cos \ha \th + \ga_{12} \sin \ha \th \rp v \lp \cos \ha \th - \ga_{12} \sin \ha \th \rp \]
Since $U^- = e^{- \ha B} = \tilde{U}$ is the [[reverse|Clifford conjugate]] and the [[inverse]] of $U = e^{\ha B}$, Clifford rotation is a special case of the [[Clifford adjoint]].  Such a $U$ is sometimes called a ''rotor''.  Any [[Clifford element]] may be rotated by, $A' = U A U^-$ -- which preserves the [[Clifford grade]] of the element.

A Clifford rotation may be readily translated into the standard matrix coefficient notation for a [[Lorentz rotation]] via
\[ \ga'_\al = \ga_\be L^\be{}_\al = U \ga_\al U^- \]

The group of Clifford rotations, $\mbox{Spin}{}^+$, in any [[spacetime]] is a double cover of the [[special orthochronous Lorentz group|Lorentz group]]. Boosts along any [[spatial|indices]] direction, $\nu = \nu^\pi \ga_\pi$, are Clifford rotations in the $\ga_0 \nu$ plane,
\[ v' = e^{\ha \ga_{0 \pi} \nu^\pi} v e^{- \ha \ga_{0 \rh} \nu^\rh} \]
For example, a boost of $\nu$ along $\ga_3$ gives
\[ v' = e^{\ha \ga_{03} \nu} v e^{- \ha \ga_{03} \nu} = \lp \cosh \ha \nu + \ga_{03} \sinh \ha \nu \rp v \lp \cosh \ha \nu - \ga_{03} \sinh \ha \nu \rp \]

A simple rotor is defined as a rotor that can be written as the product of two vectors, $U_{s}=ab=e^{\frac{1}{2}i_{2}\theta}$, in which $i_{2}$ is a unit bivector of a rotation plane and $\theta$ is a rotation angle. A rotor may always be factored into a product of $\leq\frac{n}{2}$ simple rotors. A standard decomposition uses the choice of a non-singular vector, $v$, to factor a rotor into $U=\pm U'U_{s}$, in which $U'$ is a rotor that leaves $v$ invariant, $U'v=vU'$, and $U_{s}$ is a simple rotor that rotates $v$. As an example, in a four dimensional Lorentzian spacetime, a rotor can be factored using the time-like frame vector, $\gamma_{0}$, into the spatial rotation and Lorentz boost,\[
U=e^{\frac{1}{2}\gamma\gamma_{0}\gamma_{\pi}\theta^{\pi}}\: e^{\frac{1}{2}\gamma_{0}\gamma_{\rh}\nu^{\rh}}\]
in which $\th = \gamma_{\pi}\theta^{\pi}$ is the spatial rotation vector and $\nu = \gamma_{\rh}\nu^{\rh}$ is the spatial boost vector used to construct the [[Cl(1,3) bivector]] corresponding to the Clifford rotation.
The ''Clifford vector bundle'', $Cl^1 M$, with base [[manifold]] $M$ is a [[vector bundle]] with $n$ fiber basis elements equal to the [[Clifford basis vectors]], $\ga_\al$. The fiber at each base manifold point, $p$, is the space of grade 1 Clifford elements, $Cl^1 = \li Cl \ri_1$.  The transition functions for the basis elements over overlapping patches, $U_1$ and $U_2$, are given by [[Clifford rotation]]s,
$$
\ga_\al^2 = U_{12} \ga_\al^1 U_{12}^- = \lp t^{12} \rp_\al{}^\be \ga_\be^1 =  \lp L^{12} \rp^\be{}_\al \ga_\be^1
$$
Through equating the transition functions, $L^\be{}_\al$, and using the [[frame]], $\ve{e_\al} \f{e} = \ga_\al$, the Clifford vector bundle may be [[associated]], $\ga_\al \leftrightarrow \ve{e_\al}$, to the [[tangent bundle]], with a corresponding equivalence between all their geometric structures. The structure group of the Clifford vector bundle, $\mbox{Spin}{}^+$, is a double cover of the [[special orthochronous Lorentz group|Lorentz group]].  A Clifford vector field, $v = v(x) = v^\al(x) \ga_\al$, over the manifold is a section of the bundle, and gives a Clifford vector at each manifold point.

[[Clifford grade]] $p$ fields are sections of the ''Clifford p-vector bundle'', $Cl^p M$, which has the $\frac{n!}{\left(n-p\right)!p!}$ grade $p$ [[Clifford basis elements]], $\ga_{\la \dots \be}$, as basis. The combined collection of these Clifford vector product bundles is the ''graded Clifford bundle'', $Cl^g M = \bigoplus_{p=0}^{n} Cl^p M$, having dimension $2^{n}$. The transition functions for the graded Clifford bundle are also [[Clifford rotation]]s,
$$\ga_{\al \dots \be}^2 = U_{12} \ga_{\al \dots \be}^1 U_{12}^-$$
which preserve the grade of the basis elements. The graded Clifford bundle fiber, $Cl$, is the same as for the [[Clifford bundle]] &mdash; but the transition functions (which for the graded Clifford bundle are grade preserving) are in different groups for the two bundles -- the Clifford vector bundle is a Clifford bundle with a [[reduction of the structure group]].

For a section, $C(x)$, transforming under the Clifford rotation [[gauge transformation]], $C \mapsto C'=U C U^-$, the [[covariant derivative]] is
$$
\f{\na} C = \f{d} C + \ha \f{\om} C - \ha C \f{\om} = \f{d} C + \f{\om} \times C
$$
(defined with a $\ha$ in it to keep things pretty) with the [[spin connection]], $\f{\om}$, applied using the [[cross product|Clifford algebra]].

Any fiber element, $C$, at $t=0$ may be [[parallel transport]]ed to $C(t)=U(t)CU^-$ along a path on the base by a parameter dependent Clifford element, the path holonomy, $U(t) = Pe^{- \ha \int_0^t \f{\om}}$, satisfying the [[path holonomy]] equation,
$$
\fr{d}{dt} U(t) = - \ha \ve{v} \f{\om} U
$$

Applying the covariant derivative twice (being careful with the signs of 1-forms hopping sides),
\begin{eqnarray}
\f{\na} \f{\na} C &=& \f{d} \lp \f{d} C + \ha \f{\om} C - \ha C \f{\om} \rp + \ha \f{\om} \lp \f{d} C + \ha \f{\om} C - \ha C \f{\om} \rp  + \ha \lp \f{d} C + \ha \f{\om} C - \ha C \f{\om} \rp \f{\om} \\
&=& \ha \lp \f{d} \f{\om} \rp C - \ha \f{\om} \f{d} C - \ha \lp \f{d} C \rp \f{\om} - \ha C \f{d} \f{\om} 
 + \ha \f{\om} \lp \f{d} C + \ha \f{\om} C - \ha C \f{\om} \rp  + \ha \lp \f{d} C + \ha \f{\om} C - \ha C \f{\om} \rp \f{\om} \\
&=& \ff{R} \times C
\end{eqnarray}
gives (after using one of the [[Clifford basis product identities]]) the [[Clifford-Riemann curvature]],
$$
\ff{R} = \f{d} \f{\om} + \ha \f{\om} \times \f{\om}
$$
This expression for the curvature may alternatively be obtained from the [[holonomy]] (minding the new factor of $\ha$ in the path holonomy equation).

Under a gauge transformation, $C(x) \mapsto C'(x) = U(x) C(x) U^-(x)$, the covariant derivative changes to
\begin{eqnarray}
\f{\na'} C' &=& U \lp \f{\na} C \rp U^-\\
\f{d} \lp U C U^- \rp + \ha \f{\om'} U C U^- - \ha U C U^- \f{\om'} &=& U \lp \f{d} C \rp U^- + \ha U \f{\om} C U^- - \ha U C \f{\om} U^-
\end{eqnarray}
giving the transformation law for the spin connection,
$$
\f{\om'} = U \f{\om} U^- - 2 \lp \f{d} U \rp U^- = U \f{\om} U^- + 2 U \lp \f{d} U^- \rp 
$$
An infinitesimal transformation, $U \simeq 1 + \ha B$, in which $B$ is a Clifford bivector, changes the spin connection to
$$
\f{\om'} \simeq \f{\om} - \f{d} B - \ha \f{\om} B + \ha B \f{\om} = \f{\om} - \f{\na} B
$$
The curvature consequently transforms under a gauge transformation to
$$
\ff{R'} = \f{d} \f{\om'} + \ha \f{\om'} \times \f{\om'} = U \ff{R} U^- \simeq \ff{R} + B \times \ff{R}
$$
These expressions equate to those for a [[tangent bundle gauge transformation]].

The covariant derivative acting on a [[Clifform]] such as the curvature, transforming under a Clifford rotation, $\ff{F'} = U \ff{F} U^-$, is still 
$$
\f{\na} \ff{F} = \f{d} \ff{F} + \f{\om} \times \ff{F} 
$$

Clifford vector bundles or graded Clifford bundles may alternatively be defined as [[automorphism bundle]]s -- for which outer automorphisms may prove interesting.
The ''Clifford-Ricci curvature'' is a [[Clifform]] obtained by taking the [[cross product|Clifford algebra]] of the [[frame]] with the [[Clifford bundle]] curvature,
$$
\f{R} = \ve{e} \times \ff{F}
$$
If, specifically, we are working with the [[Clifford vector bundle]], the Clifford-Ricci curvature is then a Clifford vector valued 1-form,
$$
\f{R} = \f{dx^i} R_i{}^\al \ga_\al = \ve{e} \times \ff{R} = \ve{e} \times \lp \f{d} \f{\om} + \ha \f{\om} \times \f{\om} \rp
$$
with coefficients equal to those of the [[Ricci curvature]], $R_i{}^\al = \lp e_\be \rp^j R_{ji}{}^{\be \al} = \et^{\al \be} R_{i \be}$.
The ''Clifford curvature'' is a [[Clifform]] describing the [[curvature]] of a [[Clifford bundle]],
$$
\ff{F} = \f{d} \f{A} + \ha \f{A} \times \f{A}
$$
If, specifically, we are working with the [[Clifford vector bundle]], the Clifford curvature is then the ''Clifford-Riemann curvature'', a Clifford bivector valued 2-form calculated from the [[spin connection]],
$$
\ff{R} = \f{d} \f{\om} + \ha \f{\om} \times \f{\om} = \f{dx^i} \f{dx^j} \fr{1}{4} R_{ij}{}^{\al \be} \ga_{\al \be}
$$
$$
R_{ij}{}^{\al \be} = 2 \pa_{\lb i \rd} \om_{\ld j \rb}{}^{\al \be} + 2 \om_{\lb i \rd}{}^\al{}_\ga \om_{\ld j \rb}{}^{\ga \be}
$$
with coefficients equal to those of the [[Riemann curvature]], $R_{ij}{}^{\al\be}$, when the [[tangent bundle connection]] and spin connection coefficients are identified, $\f{w^{\al\be}}=\f{\om^{\al\be}}$.
A ''Clifform'' is a [[Clifford algebra]] valued [[differential form]], or, conversely, a [[Clifford element]] with form valued coefficients. A Clifform has a single form grade, $p$, but may consist of pieces with different Clifford grades. In terms of [[coordinate basis forms]] and [[Clifford basis elements]], an arbitrary Clifform may be written as
$$
\nf{A} = \f{dx^i} \dots \f{dx^k} \fr{1}{p!} \lp A_{i \dots k}{}^s + A_{i \dots k}{}^\al \ga_\al + \ha A_{i \dots k}{}^{\al \be} \ga_{\al \be} + \fr{1}{3!} A_{i \dots k}{}^{\al \be \ga} \ga_{\al \be \ga} + \dots + A_{i \dots k}{}^p \ga \rp
$$
For example, a bivector 2-form is written (using the coordinate or [[frame]] basis forms) as
$$
\ff{R} = \ff{R^2} = \f{dx^i} \f{dx^j} \fr{1}{4} R_{ij}{}^{\al \be} \ga_{\al \be}
= \f{e^\ga} \f{e^\de} \fr{1}{4} R_{\ga \de}{}^{\al \be} \ga_{\al \be}
$$
The form elements and Clifford elements act in different algebras. All scalar valued form elements commute with all Clifford basis elements. By convention, the form basis elements will be collected on the left and the Clifford basis elements on the right.

The product of Clifforms may be computed using [[Clifform algebra]]. A Clifform is a [[Lieform]] in which the [[Lie algebra]] generators are Clifford basis elements.
The algebra of [[Clifform]]s is the disjoint union of [[vector-form algebra]] and [[Clifford algebra]]. When performing calculations, it is best to move all [[coordinate basis 1-forms]] to the left of the expression (without commuting them) and all [[Clifford basis elements]] (and the operations between them) to the right.  Then the basis contractions and products play out in their independent algebraic sandboxes.  Clifford algebra operators like $\cdot$, $\times$, $[,]$, and $<>_q$ do not act on the forms, only on the Clifford basis elements.  As an example, the dot product of a bivector (-2)-form and a bivector 2-form is a scalar plus a 4-vector,
\begin{eqnarray}
\vv{L} \cdot \ff{R} &=& \lp \ve{\pa_i} \ve{\pa_j} \fr{1}{4} L^{i j \al \be} \ga_{\al \be} \rp \cdot \lp \f{dx^k} \f{dx^m} \fr{1}{4} R_{km}{}^{\ga \de} \ga_{\ga \de} \rp\\
&=& \lp \ve{\pa_i} \ve{\pa_j} \lp \f{dx^k} \f{dx^m} \rp \rp \fr{1}{16} L^{i j \al \be} R_{km}{}^{\ga \de} \lp \ga_{\al \be} \cdot  \ga_{\ga \de} \rp\\
&=& \lp - 2 \de_i^{\lb k \rd} \delta_j^{\ld m \rb} \rp \fr{1}{16} L^{i j \al \be} R_{km}{}^{\ga \de} \lp \lp \et_{\al \de} \et_{\be \ga} - \et_{\al \ga} \et_{\be \de} \rp + \ga_{\al \be \ga \de} \rp\\
&=& - \fr{1}{8} L^{i j \al \be} R_{ij}{}^{\ga \de} \lp \lp \et_{\al \de} \et_{\be \ga} - \et_{\al \ga} \et_{\be \de} \rp + \ga_{\al \be \ga \de} \rp\\
&=& \fr{1}{4} L^{i j \al \be} R_{ij \al \be} - \fr{1}{8} L^{i j \al \be} R_{ij}{}^{\ga \de} \ga_{\al \be \ga \de}\\
\end{eqnarray}
using vector-form algebra and [[Clifford basis product identities]].

Clifform product identities can be inferred from the identities of the two respective algebras.  For example, since 1-forms anti-commute,
\[ \f{A} \ti \f{B} = \ha \lp \f{A} \f{B} + \f{B} \f{A} \rp = \f{B} \ti \f{A} \]
Some useful identities can be computed using the [[frame]]. For example, for any Clifford vector valued 2-form, $\ff{f}$,
\begin{eqnarray}
\ve{e} \ti \ff{f} & = & -\lp \ve{e} \ti \ve{e} \rp \lp \f{e} \cdot \ff{f} \rp + \ve{e} \ti \lp \lp \ve{e} \ti \ff{f} \rp \ti \f{e} \rp\\
\ff{f} & = & \lp \ve{e} \ti \ff{f} \rp \ti \f{e} - \ve{e} \cdot \lp \f{e} \cdot \ff{f} \rp\\
\lp n-2 \rp \ff{f} & = & \ve{e} \ti \lp \f{e} \ti \ff{f} \rp - \f{e} \cdot \lp \ve{e} \cdot \ff{f} \rp
\end{eqnarray} 
(//add identities as needed//)
The [[Coleman-Mandula theorem|http://prola.aps.org/abstract/PR/v159/i5/p1251_1]] states:
<<<
Let G be a connected symmetry group of the S matrix, and let the following five conditions hold: (1) G contains a subgroup locally isomorphic to the Poincare group. (2) For any M>0, there are only a finite number of one-particle states with mass less than M. (3) Elastic scattering amplitudes are analytic functions of s and t, in some neighborhood of the physical region. (4) The S matrix is nontrivial in the sense that any two one-particle momentum eigenstates scatter (into something), except perhaps at isolated values of s. (5) The generators of G, written as integral operators in momentum space, have distributions for their kernels. Then, we show that G is necessarily locally isomorphic to the direct product of an internal symmetry group and the Poincare group.
<<<

The E8 theory proposed in [[An Exceptionally Simple Theory of Everything]] avoids condition (1) of this theorem because $G = E8$ does not containing a subgroup locally isomorphic to the Poincare group. The expected vacuum spacetime of E8 theory is [[de Sitter spacetime]], which has $SO(4,1)$ as symmetry group, which is nearly, but not, the Poincare group. At low energies the deviation from the Poincare group is infinitesimally small, and the Coleman-Mandula theorem applies to a good approximation, with gravity separate from the other symmetries.

Ref:
*R. Percacci
**[[Mixing internal and spacetime transformations: some examples and counterexamples|http://arxiv.org/abs/0803.0303]]
*K. Cahill
**[[On the unification of the gravitational and electronuclear forces|papers/Cahill - On the unification of the gravitational and electronuclear forces.pdf]]
*** Phys. Rev. D 26, 1916 - 1922 (1982).
*T. Love
**The Geometry of Grand Unification
***Int. J. Th. Phys., 801 (1984).
*F. Nesti and R. Percacci
**[[Gravi-Weak Unification|http://arxiv.org/abs/0706.3307]]
*S. Alexander
**[[Isogravity: Toward an Electroweak and Gravitational Unification|http://arxiv.org/abs/0706.4481]]
/***
|Name|CollapseTiddlersPlugin|
|Source|http://www.TiddlyTools.com/#CollapseTiddlersPlugin|
|Version|2.0.0|
|Author|Eric Shulman|
|OriginalAuthor|Bradley Meck - http://gensoft.revhost.net/Collapse.html|
|License|unknown|
|~CoreVersion|2.1|
|Type|plugin|
|Requires|CollapsedTemplate|
|Description|show/hide content of a tiddler while leaving tiddler title visible|
This plugin provides commands to quickly switch a rendered tiddler between its current ViewTemplate display and a minimal display (title and toolbar) defined by a separate CollapsedTemplate.
!!!Usage
<<<
In [[ToolbarCommands::ViewToolbar|ToolbarCommands]], add:
{{{
collapseTiddler collapseOthers
}}}
you can also embed the following macros in tiddler content:
*{{{<<collapseAll>>}}} - adds 'collapse all' command that applies CollapsedTemplate to each displayed tiddler
*{{{<<expandAll>>}}} - adds 'expand all' command that re-applies ViewTemplate (or equivalent custom template) to each displayed tiddler
*{{{<<foldFirst>>}}} - immediately apply CollapsedTemplate to a given tiddler, as soon as it is displayed.
<<<
!!!Revisions
<<<
2009.05.04 [2.0.0] standardized documentation and added version #
2008.10.05 collapseAll() and expandAll(): added "return false" to button handlers to prevent IE page transition
2008.03.06 refactored all code for size reduction, readability, and I18N/L10N-readiness.  Also added 'folded' flag to tiddler elements (for use by other plugins that need to know if tiddler is folded (e.g., [[SinglePageModePlugin]]
2007.10.11 moved [[FoldFirst]] inline script and converted to {{{<<foldFirst>>}}} macro
2007.12.09 suspend/resume SinglePageMode (SPM/TPM/BPM) when folding/unfolding tiddlers
2007.05.06 add "return false" at the end of each command handler to prevent IE 'page transition' problem.
2007.03.30 add a shadow definition for CollapsedTemplate.  Tweak ViewTemplate shadow so "fold/unfold" and "focus" toolbar items automatically appear when using default templates.  Remove error check for "CollapsedTemplate" existence, since shadow version will now always work as a fallback.
2006.02.24 added fallback to "CollapsedTemplate" if "WebCollapsedTemplate" is not found
2006.02.06 added check for 'readOnly' flag to use alternative "WebCollapsedTemplate"
<<<
!!!Code
***/
//{{{
version.extensions.CollapseTiddlersPlugin= {major: 2, minor: 0, revision: 0, date: new Date(2009,5,4)};

config.commands.collapseTiddler = {
	text: "-",
	tooltip: "Collapse this note",
	collapsedTemplate: "CollapsedTemplate",
	webCollapsedTemplate: "WebCollapsedTemplate",
	handler: function(event,src,title) {
		var e = story.findContainingTiddler(src); if (!e) return false;
		// don't fold tiddlers that are being edited!
		if(story.isDirty(e.getAttribute("tiddler"))) return false;
		var t=config.commands.collapseTiddler.getCollapsedTemplate();
		config.commands.collapseTiddler.saveTemplate(e);
		config.commands.collapseTiddler.display(title,t);
		e.setAttribute("folded","true");
		return false;
	},
	getCollapsedTemplate: function() {
		if (readOnly&&store.tiddlerExists(this.webCollapsedTemplate))
			return this.webCollapsedTemplate;
		else
			return this.collapsedTemplate
	},
	saveTemplate: function(e) {
		if (e.getAttribute("savedTemplate")==undefined)
			e.setAttribute("savedTemplate",e.getAttribute("template"));

	},
	// fold/unfold tiddler with suspend/resume of single/top/bottom-of-page mode
	display: function(title,t) {
		var opt=config.options;
		var saveSPM=opt.chkSinglePageMode; opt.chkSinglePageMode=false;
		var saveTPM=opt.chkTopOfPageMode; opt.chkTopOfPageMode=false;
		var saveBPM=opt.chkBottomOfPageMode; opt.chkBottomOfPageMode=false;
		story.displayTiddler(null,title,t);
		opt.chkBottomOfPageMode=saveBPM;
		opt.chkTopOfPageMode=saveTPM;
		opt.chkSinglePageMode=saveSPM;
	}
}

config.commands.expandTiddler = {
	text: " | ",
	tooltip: "Expand this note",
	handler: function(event,src,title) {
		var e = story.findContainingTiddler(src); if (!e) return false;
		var t = e.getAttribute("savedTemplate");
		config.commands.collapseTiddler.display(title,t);
		e.setAttribute("folded","false");
		return false;
	}
}

config.macros.collapseAll = {
	text: "-",
	tooltip: "Collapse all notes",
	handler: function(place,macroName,params,wikifier,paramString,tiddler){
		createTiddlyButton(place,this.text,this.tooltip,function(){
			story.forEachTiddler(function(title,tiddler){
				if(story.isDirty(title)) return;
				var t=config.commands.collapseTiddler.getCollapsedTemplate();


				config.commands.collapseTiddler.saveTemplate(tiddler);
				config.commands.collapseTiddler.display(title,t);
				tiddler.folded=true;
			});
			return false;
		})
	}
}

config.macros.expandAll = {
	text: " | ",
	tooltip: "Expand all notes",
	handler: function(place,macroName,params,wikifier,paramString,tiddler){
		createTiddlyButton(place,this.text,this.tooltip,function(){
			story.forEachTiddler(function(title,tiddler){
				var t=config.commands.collapseTiddler.getCollapsedTemplate();
				if(tiddler.getAttribute("template")!=t) return; // re-display only if collapsed
				var t=tiddler.getAttribute("savedTemplate");
				config.commands.collapseTiddler.display(title,t);
				tiddler.folded=false;
			});
			return false;
		})
	}
}

config.commands.collapseOthers = {
	text: "\xD8",
	tooltip: "Expand this note and collapse all others",
	handler: function(event,src,title) {
		var e = story.findContainingTiddler(src); if (!e) return false;
		story.forEachTiddler(function(title,tiddler) {
			if(story.isDirty(title)) return;
			var t=config.commands.collapseTiddler.getCollapsedTemplate();
			if (e==tiddler) t=e.getAttribute("savedTemplate");
			config.commands.collapseTiddler.saveTemplate(tiddler);
			config.commands.collapseTiddler.display(title,t);
			tiddler.folded=(e!=tiddler);
		})
		return false;
	}
}

// {{{<<foldFirst>>}}} macro forces tiddler to be folded when *initially* displayed.
// Subsequent re-render does NOT re-fold tiddler, but closing/re-opening tiddler DOES cause it to fold first again.
config.macros.foldFirst = {
	handler: function(place,macroName,params,wikifier,paramString,tiddler){
		var e=story.findContainingTiddler(place);
		if (e.getAttribute("foldedFirst")=="true") return; // already been folded once
		var title=e.getAttribute("tiddler")
		var t=config.commands.collapseTiddler.getCollapsedTemplate();
		config.commands.collapseTiddler.saveTemplate(e);
		config.commands.collapseTiddler.display(title,t);
		e.setAttribute("folded","true");
		e.setAttribute("foldedFirst","true"); // only when tiddler is first rendered
		return false;
	}
}
//}}}
<!--{{{-->
<div>
<span class='toolbar' macro='toolbar +editTiddler expandTiddler collapseOthers closeOthers -closeTiddler'></span>
<span class='title' macro='view title'></span>
</div>
<!--}}}-->
This site is powered by [[TiddlyWiki|http://www.tiddlywiki.com]] <<version>>
!Install these plugins:
*[[InlineJavascriptPlugin]]
**used for the [[DisplayControl]]
**and for [[HideTags]] (used for slides)
*[[MathJaxPlugin]]
**this processes LaTeX.
**insert custom LaTeX command abbreviations into plugin.
**install MathJax TeX otf fonts locally, to ~/Library/Fonts
*[[CollapsePlugin]]
**add symbols
**[[CollapsedTemplate]]
*[[RearrangeTiddlersPlugin]]
*[[ListTaggedPlugin]]
**used for folder/tag listings
*[[AllTagsExceptPlugin]]
**use advanced checkbox to see system tags
*[[CopyTiddlerPlugin]]
**add symbol
*[[DisableWikiLinksPlugin]]
**remove checkboxes and set to always disable
*[[FaviconPlugin]]
*[[ReferencesPlugin]]
*[[RecentPlugin]]
**set to show last 3

Check to make sure didn't install any<<tag plugin>>and forget to list it here. Try using the [[PluginManager]].

!Change these tiddlers to configure operation and appearance:
*These control the content of several boxes:
**[[SiteTitle]]
**[[SiteSubtitle]]
**[[WindowTitle]]
**[[SiteUrl]]
**[[DefaultTiddlers]]
**[[MainMenu]]
**[[SideBarOptions]]
**[[OptionsPanel]]
**[[AdvancedOptions]] -- this is auto-generated now
**[[SideBarTabs]]
***[[TabContents]]
***[[TabTimeline]] - Good as is
***[[TabAll]] - Good as is. Except... this has the text built in. Change in [[SystemConfig]].
***[[TabTags]] - Good as is
**[[DisplayControl]]
*These are css layout templates:
**[[PageTemplate]]
**[[ViewTemplate]]
**[[EditTemplate]]
**[[CollapsedTemplate]]
*This is for slides
**[[HideTags]]
*And these change the system and css options:
**[[SystemConfig]]
**[[StyleSheet]]
***Trouble with [[MyColors]] conflicting with [[ColorPalette]]?
**[[StyleSheetPrint]]
The default config files are invisible and listed as [[ShadowTiddlers]]. These:
*[[StyleSheetLayout]]
*[[StyleSheetColors]]
are augmented and overriden by the [[StyleSheet]].  If they change in the future, with updates, the old version content will likely have to be added to the new [[StyleSheet]].  

!Evil raw html/javascript TW source code tweakage
*edit cookie options, since setting them in [[SystemConfig]] overrides user cookies
*switch line order in {{{config.macros.search.handler}}} for search button after search field
*comment out a couple of displayMessage s?
*comment out tag prompt line in {{{config.macros.tags.handler}}}?

Put TiddlySaver.jar in dg directory and .java.policy in home directory so can save locally

Check to make sure all regularly visible notes are de-tiddlered.

Then, save a "bare.html" copy, without folders or editing tips, and a "minimal.html" copy, with them. Then try to [[ImportTiddlers]].
http://arxiv.org/abs/gr-qc/0603062
*Concise treatment of Hamiltonian formulation of GR with a conformal factor.
*uses metric instead of frame
<<tiddler HideTags>><html>
<table class="gtable">

<tr>
<td>

<table class="gtable">
<tr><td> &nbsp;</td></tr>
<tr><td>&nbsp; </td></tr>
<tr><td>&nbsp;</td></tr>
<tr><td>&nbsp;</td></tr>
<tr><td>&nbsp;</td></tr>
<tr><td>&nbsp;</td></tr>
<tr><td>
<img SRC="talks/Cate2010/Connection field.png" width=300>
</tr></td>
</table>

</td>
<td>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>
<td>

<table class="gtable">
<tr><td> &nbsp;</td></tr>
<tr><td>&nbsp; </td></tr>
<tr><td>&nbsp;</td></tr>
<tr><td>&nbsp;</td></tr>
<tr><td>&nbsp;</td></tr>
<tr><td>&nbsp;</td></tr>
<tr><td>
<img SRC="talks/Cate2010/EM field.png" width=300>
</tr></td>
</table>

</td>
</tr>
</table>
</html>
[[Consequences of Propagating Torsion in Connection-Dynamic Theories of Gravity|papers/9403058.pdf]]
Authors: Sean M. Carroll, George B. Field

We discuss the possibility of constraining theories of gravity in which the connection is a fundamental variable by searching for observational consequences of the torsion degrees of freedom. In a wide class of models, the only modes of the torsion tensor which interact with matter are either a massive scalar or a massive spin-1 boson. Focusing on the scalar version, we study constraints on the two-dimensional parameter space characterizing the theory. For reasonable choices of these parameters the torsion decays quickly into matter fields, and no long-range fields are generated which could be discovered by ground-based or astrophysical experiments. 
/***
|Name|CopyTiddlerPlugin|
|Source|http://www.TiddlyTools.com/#CopyTiddlerPlugin|
|Version|3.2.6|
|Author|Eric Shulman|
|License|http://www.TiddlyTools.com/#LegalStatements|
|~CoreVersion|2.3|
|Type|plugin|
|Description|Quickly create a copy of any existing tiddler|
!!!Usage
<<<
The plugin automatically updates the default (shadow) ToolbarCommands definitions to insert the ''copyTiddler'' command, which will appear as ''copy'' when a tiddler is rendered.  If you are already using customized toolbar definitions, you will need to manually add the ''copyTiddler'' toolbar command to your existing ToolbarCommands tiddler, e.g.:
{{{
|EditToolbar|... copyTiddler ... |
}}}
When the ''copy'' command is selected, a new tiddler is created containing an exact copy of the current text/tags/fields, using a title of "{{{TiddlerName (n)}}}", where ''(n)'' is the next available number (starting with 1, of course).  If you copy while //editing// a tiddler, the current values displayed in the editor are used (including any changes you may have already made to those values), and the new tiddler is immediately opened for editing.

The plugin also provides a macro that allows you to embed a ''copy'' command directly in specific tiddler content:
{{{
<<copyTiddler TidderName label:"..." prompt:"...">>
}}}
where
* ''TiddlerName'' (optional)<br>specifies the //source// tiddler to be copied.  If omitted, the current containing tiddler (if any) will be copied.
* ''label:"..."'' (optional)<br>specifies text to use for the embedded link (default="copy TiddlerName")
* ''prompt:"..."'' (optional)<br>specifies mouseover 'tooltip' help text for link
//Note: to use non-default label/prompt values with the current containing tiddler, use "" for the TiddlerName//
<<<
!!!Configuration
<<<
<<option chkCopyTiddlerDate>> use date/time from existing tiddler (otherwise, use current date/time)
{{{<<option chkCopyTiddlerDate>>}}}
<<<
!!!Revisions
<<<
2010.11.30 3.2.6 use story.getTiddler()
2009.06.08 3.2.5 added option to use timestamp from source tiddler
2009.03.09 3.2.4 fixed IE-specific syntax error
2009.03.02 3.2.3 refactored code (again) to restore use of config.commands.copyTiddler.* custom settings
2009.02.13 3.2.2 in click(), fix calls to displayTiddler() to use current tiddlerElem and use getTiddlerText() to permit copying of shadow tiddler content
2009.01.30 3.2.1 fixed handling for copying field values when in edit mode
2009.01.23 3.2.0 refactored code and added {{{<<copyTiddler TiddlerName>>}}} macro
2008.12.18 3.1.4 corrected code for finding next (n) value when 'sparse' handling is in effect
2008.11.14 3.1.3 added optional 'sparse' setting (avoids 'filling in' missing numbers that may have been previously deleted)
2008.11.14 3.1.2 added optional 'zeroPad' setting
2008.11.14 3.1.1 moved hard-coded '(n)' regex into 'suffixPattern' object property so it can be customized
2008.09.26 3.1.0 changed new title generation to use '(n)' suffix instead of 'Copy of' prefix
2008.05.20 3.0.3 in handler, when copying from VIEW mode, create duplicate array from existing tags array before saving new tiddler.
2007.12.19 3.0.2 in handler, when copying from VIEW mode, duplicate custom fields before saving new tiddler.
2007.09.26 3.0.1 in handler, use findContainingTiddler(src) to get tiddlerElem (and title).  Allows 'copy' command to find correct tiddler when transcluded using {{{<<tiddler>>}}} macro or enhanced toolbar inclusion (see [[CoreTweaks]])
2007.06.28 3.0.0 complete re-write to handle custom fields and alternative view/edit templates
2007.05.17 2.1.2 use store.getTiddlerText() to retrieve tiddler content, so that SHADOW tiddlers can be copied correctly when in VIEW mode
2007.04.01 2.1.1 in copyTiddler.handler(), fix check for editor fields by ensuring that found field actually has edit=='text' attribute
2007.02.05 2.1.0 in copyTiddler.handler(), if editor fields (textfield and/or tagsfield) can't be found (i.e., tiddler is in VIEW mode, not EDIT mode), then get text/tags values from stored tiddler instead of active editor fields.  Allows use of COPY toolbar directly from VIEW mode
2006.12.12 2.0.0 completely rewritten so plugin just creates a new tiddler EDITOR with a copy of the current tiddler EDITOR contents, instead of creating the new tiddler in the STORE by copying the current tiddler values from the STORE.
2005.xx.xx 1.0.0 original version by Tim Morgan
<<<
!!!Code
***/
//{{{
version.extensions.CopyTiddlerPlugin= {major: 3, minor: 2, revision: 6, date: new Date(2010,11,30)};

// automatically tweak shadow EditTemplate to add 'copyTiddler' toolbar command (following 'cancelTiddler')
config.shadowTiddlers.ToolbarCommands=config.shadowTiddlers.ToolbarCommands.replace(/cancelTiddler/,'cancelTiddler copyTiddler');

if (config.options.chkCopyTiddlerDate===undefined) config.options.chkCopyTiddlerDate=false;

config.commands.copyTiddler = {
	text: '\xA9',
	hideReadOnly: true,
	tooltip: 'Make a copy of this note',
	notitle: 'this note',
	prefix: '',
	suffixText: ' (%0)',
	suffixPattern: / \(([0-9]+)\)$/,
	zeroPad: 0,
	sparse: false,
	handler: function(event,src,title)
		{ return config.commands.copyTiddler.click(src,event); },
	click: function(here,ev) {
		var tiddlerElem=story.findContainingTiddler(here);
		var template=tiddlerElem?tiddlerElem.getAttribute('template'):null;
		var title=here.getAttribute('from');
		if (!title || !title.length) {
			if (!tiddlerElem) return false;
			else title=tiddlerElem.getAttribute('tiddler');
		}
		var root=title.replace(this.suffixPattern,''); // title without suffix
		// find last matching title
		var last=title;
		if (this.sparse) { // don't fill-in holes... really find LAST matching title
			var tids=store.getTiddlers('title','excludeLists');
			for (var t=0; t<tids.length; t++) if (tids[t].title.startsWith(root)) last=tids[t].title;
		}
		// get next number (increment from last matching title)
		var n=1; var match=this.suffixPattern.exec(last); if (match) n=parseInt(match[1])+1;
		var newTitle=this.prefix+root+this.suffixText.format([String.zeroPad(n,this.zeroPad)]);
		// if not sparse mode, find the next hole to fill in...
		while (store.tiddlerExists(newTitle)||story.getTiddler(newTitle))
			{ n++; newTitle=this.prefix+root+this.suffixText.format([String.zeroPad(n,this.zeroPad)]); }
		if (!story.isDirty(title)) { // if tiddler is not being EDITED
			// duplicate stored tiddler (if any)
			var text=store.getTiddlerText(title,'');
			var who=config.options.txtUserName;
			var when=new Date();
			var newtags=[]; var newfields={};
			var tid=store.getTiddler(title); if (tid) {
				if (config.options.chkCopyTiddlerDate) var when=tid.modified;
				for (var t=0; t<tid.tags.length; t++) newtags.push(tid.tags[t]);
				store.forEachField(tid,function(t,f,v){newfields[f]=v;},true);
			}
	                store.saveTiddler(newTitle,newTitle,text,who,when,newtags,newfields,true);
			story.displayTiddler(tiddlerElem,newTitle,template);
		} else {
			story.displayTiddler(tiddlerElem,newTitle,template);
			var fields=config.commands.copyTiddler.gatherFields(tiddlerElem); // get current editor fields
			var newTiddlerElem=story.getTiddler(newTitle);
			for (var f=0; f<fields.length; f++) {  // set fields in new editor
				if (fields[f].name=='title') fields[f].value=newTitle; // rename title in new tiddler
				var fieldElem=config.commands.copyTiddler.findField(newTiddlerElem,fields[f].name);
				if (fieldElem) {
					if (fieldElem.getAttribute('type')=='checkbox')
						fieldElem.checked=fields[f].value;
					else 
						fieldElem.value=fields[f].value;
				}
			}
		}
		story.focusTiddler(newTitle,'title');
		return false;
	},
	findField: function(tiddlerElem,field) {
		var inputs=tiddlerElem.getElementsByTagName('input');
		for (var i=0; i<inputs.length; i++) {
			if (inputs[i].getAttribute('type')=='checkbox' && inputs[i].field == field) return inputs[i];
			if (inputs[i].getAttribute('type')=='text' && inputs[i].getAttribute('edit') == field) return inputs[i];
		}
		var tas=tiddlerElem.getElementsByTagName('textarea');
		for (var i=0; i<tas.length; i++) if (tas[i].getAttribute('edit') == field) return tas[i];
		var sels=tiddlerElem.getElementsByTagName('select');
		for (var i=0; i<sels.length; i++) if (sels[i].getAttribute('edit') == field) return sels[i];
		return null;
	},
	gatherFields: function(tiddlerElem) { // get field names and values from current tiddler editor
		var fields=[];
		// get checkboxes and edit fields
		var inputs=tiddlerElem.getElementsByTagName('input');
		for (var i=0; i<inputs.length; i++) {
			if (inputs[i].getAttribute('type')=='checkbox')
				if (inputs[i].field) fields.push({name:inputs[i].field,value:inputs[i].checked});
			if (inputs[i].getAttribute('type')=='text')
				if (inputs[i].getAttribute('edit')) fields.push({name:inputs[i].getAttribute('edit'),value:inputs[i].value});
		}
		// get textareas (multi-line edit fields)
		var tas=tiddlerElem.getElementsByTagName('textarea');
		for (var i=0; i<tas.length; i++)
			if (tas[i].getAttribute('edit')) fields.push({name:tas[i].getAttribute('edit'),value:tas[i].value});
		// get selection lists (droplist or listbox)
		var sels=tiddlerElem.getElementsByTagName('select');
		for (var i=0; i<sels.length; i++)
			if (sels[i].getAttribute('edit')) fields.push({name:sels[i].getAttribute('edit'),value:sels[i].value});
		return fields;
	}
};
//}}}
// // MACRO DEFINITION
//{{{
config.macros.copyTiddler = {
	label: 'copy',
	prompt: 'Make a copy of %0',
	handler: function(place,macroName,params,wikifier,paramString,tiddler) {
		var title=params.shift();
		params=paramString.parseParams('anon',null,true,false,false);
		var label	=getParam(params,'label',this.label+(title?' '+title:''));
		var prompt	=getParam(params,'prompt',this.prompt).format([title||this.notitle]);
		var b=createTiddlyButton(place,label,prompt,
			function(ev){return config.commands.copyTiddler.click(this,ev)});
		b.setAttribute('from',title||'');
	}
};
//}}}
<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
<embed src="talks/TED08/images/Coral_reef_s620.JPG" width="827" height="620"></embed></center></html>@@
<<tiddler HideTags>>$$
\begin{array}{rcll}
\ff{F} \!\!&\!\!=\!\!&\!\! \f{d} \f{H} + \f{H} \f{H}
\;\;\;\;\;\;\;\;\;\;\;\;\; \f{H} = \ha \f{\om} + \fr{1}{4}\f{e}\ph + \f{B} + \f{W}
\\

\!\!&\!\!=\!\!&\!\! \Big( \ha ( \f{d} \f{\om} + \ha \f{\om} \f{\om} ) + \fr{1}{16} M^2 \f{e} \f{e} \Big)_{\p{(}}
\!&\!\! \leftarrow \text{spacetime} \; \ga_{\mu\nu} \\

&&\!\!\! + \Big( \fr{1}{4} \big( \f{d} \f{e} + \ha [ \f{\om}, \f{e} ] \big) \ph - \fr{1}{4} \f{e} \big( \f{d} \ph + [ \f{B} \!+\! \f{W}, \ph ] \big) \Big)_{\p{(}}
\!&\!\! \leftarrow \text{mixed} \; \ga_{\mu\ph} \\

&&\!\!\! + \Big( \f{d} \f{B} + \f{d} \f{W} + \f{W} \f{W} \Big)_{\p{\big(}}
\!&\!\! \leftarrow \text{higher} \; \ga_{\ph\ps} \\

\!\!&\!\!=\!\!&\!\! \ha \big( \ff{R} + \fr{1}{8} M^2 \f{e} \f{e} \big)
+ \fr{1}{4} \big( \ff{T} \ph - \f{e} \f{D} \ph \big)
+ \big( \ff{F_B} + \ff{F_W} \big) \\

\!\!&\!\!=\!\!&\!\! \ff{F_s} + \ff{F_m} +  \ff{F_h}
\end{array}
$$
Modified BF action over 4D base [[manifold]]:
\begin{eqnarray}
S &=& \int \big< \ff{B} \, \ff{F} + \Ph(\f{H},\ff{B}) \big>
= \int \big< \ff{B} \, \ff{F} - {\scriptsize \frac{1}{4}} \ff{B_s} \ff{B_s} \ga + \ff{B_m} \ff{*B_m} + \ff{B_h} \ff{*B_h} \big> \\
&=& \int \big< \ff{F_s} \, \ff{F_s} \ga^- + {\scriptsize \frac{1}{4}} \ff{F_m} \ff{*F_m} + {\scriptsize \frac{1}{4}} \ff{F_h} \ff{*F_h} \big>
\end{eqnarray}
New paper. How to go from a higher dimensional gauge theory, with Chern Simons or Born Infeld action, to Einstein gravity in 4D:
*[[D=4 Einstein gravity from higher D CS and BI gravity and an alternative to dimensional reduction|papers/0703034.pdf]]
[[Welcome]]
The kinetic [[Lagrangian]] term for a [[Dirac spinor]] field in curved [[spacetime]] is
$$
L = \Psi^\dagger \ga_0 \ve{e} \lp \f{\pa} + \ha \f{\om} \rp \Psi 
$$
//(that's not necessarily real...but maybe it is, up to a divergence term?)// Using the [[chiral]] representation for the [[Cl(1,3)]] [[Dirac matrices]], the [[spacetime frame]] and [[spacetime spin connection]] break up to give
\begin{eqnarray}
L &=& \lb \Psi_L^\dagger \;\; \Psi_R^\dagger \rb
\lb \begin{array}{cc}
0 & 1 \\
1 & 0
\end{array} \rb
\lb \begin{array}{cc}
0 & \ve{e}_R \\
\ve{e}_L & 0
\end{array} \rb
\lp \f{\pa} + 
\ha
\lb \begin{array}{cc}
\f{\om}{}_L & 0 \\
0 & \f{\om}{}_R
\end{array} \rb
\rp
\lb \begin{array}{c}
\Psi_L \\
\Psi_R
\end{array} \rb
 \\
&=& \Psi_L^\dagger \ve{e}_L \lp \f{\pa} + \ha \f{\om}{}_L \rp \Psi_L + \Psi_R^\dagger \ve{e}_R\lp \f{\pa} + \ha \f{\om}{}_R \rp \Psi_R
\end{eqnarray}
<<tiddler HideTags>>
\begin{eqnarray}
L_D &=& \bar{\ps} \ga^\mu \lp e_\mu\rp^i \lp
\pa_i 
+ \fr{1}{4} \om_i^{\p{i}\nu\rh} \ga_{\nu\rh}
+ G_i^{\p{i}A} T_A
\rp \ps
+ \bar{\ps} \ph \ps
\end{eqnarray}

| $\; ( e_\mu )^i \; $ |gravitational [[frame]] components (//tetrad, vierbein//)|
| $\; \ga_\mu \; $ |[[Clifford basis vectors]] for [[Cl(1,3)]] |
| $\; \ga_{\mu\nu} = \ga_\mu \ga_\nu \; $ |[[Clifford bivectors|Clifford basis elements]] |
| $\; \om_i^{\p{i}\nu\rh} \; $ |gravitational [[spin connection]] components |
| $\; T_A \; $ |[[Lie algebra]] generators |
| $\; G_i^{\p{i}A} \; $ |[[Yang-Mills gauge field|principal bundle]] components (//connection//) |
| $\; \ph \; $ |Higgs scalar field multiplet |
$$

\begin{array}{rcl}
{\rm Clifford \; algebra} \!\!&\!\! \longleftrightarrow \!\!&\!\! {\rm Lie \; algebra}^{\phantom{(}} \\
\searrow \!\!\!\!\!\! \nwarrow \!\!&\!\! \!\!&\!\! \swarrow \!\!\!\!\!\! \nearrow \\
& {\rm Matrices} &
\end{array}
$$
<<tiddler HideTags>>$$\begin{array}{rcl}
S_D \!\!&\!\!=\!\!&\!\! \int \nf{d^4x} |e| \left\{ \bar{\ps} \ga^\mu \lp e_\mu\rp^i \big( \pa_i + {\small \frac{1}{4}} \om_i^{\p{i}\nu\rh} \ga_{\nu\rh} + A_i^{\p{i}B} T_B \big) \ps + \bar{\ps} \ph \ps \right\} \\
\!\!&\!\!=\!\!&\!\! \int \nf{e} \left\{ \bar{\ps} \ve{e} \big( \f{\pa} + {\small \frac{1}{2}} \f{\om} + \f{A} \big) \ud{\ps} + \bar{\ps} \ph \ud{\ps} \right\}
\end{array}$$

| $\; \f{e} = \f{dx^i} (e_i)^\mu \ga_\mu \;$ |$\in \f{Cl}^1(1,3)$ |gravitational [[frame]] (//tetrad, vierbein//) |
| $\; \ve{e} = \ga^\mu (e_\mu)^i \ve{\pa_i} \;$ |$\in \ve{Cl}{}^1(1,3)$ |inverse [[frame]] |
| $\; \f{\om} = \f{dx^i} \ha \om_i^{\p{i}\mu\nu} \ga_{\mu\nu} \;$ |$\in \f{Cl}^2(1,3)$ |[[spin connection]] |
| $\; \f{A} = \f{dx^i}A_i^{\p{i}B} T_B \;$ |$\in G_{SM} = \f{su}(2)_L + \f{u}(1)_Y + \f{su}(3)$ |[[gauge fields|principal bundle]] |
| $\; \ph \; $ |$\in GL(N,\mathbb{C}) \leftarrow \mathbb{C}^2 = 2_L $ |Higgs scalar field multiplet |
| $\; \ud{\ps} \; $ |$\in 2 \!\times\! (2_L\!+\!2_R) \!\times\! (1\!+\!3) $ |Grassmann valued [[Dirac spinor]] field multiplet |
| $\; \bar{\ps} = \ud{\ps}^\dagger \ga_0  \; $ |$\in 2 \!\times\! (2_R\!+\!2_L) \!\times\! (1\!+\!\bar{3})  $ |conjugate spinor multiplet |
<<tiddler HideTags>>





$$
S_\ps = \int \nf{e} \left\{ \bar{\ps} \ga^\mu \lp e_\mu\rp^i \big(
\pa_i 
+ {\tiny \frac{1}{2}} \om_i^{\p{i}\nu\rh} {\tiny \frac{1}{2}} \ga_{\nu\rh}
+ W_i^{\p{i}\pi} T^W_\pi
+ B_i T^Y
+ g_i^{\p{i}A} T^g_A
\big) \ud{\ps}
+ \bar{\ps} \ph \ud{\ps} \right\}
$$
<<tiddler HideTags>>

$$
0 = \lp \ga^\mu \pa_\mu + i m \rp \ps
$$



$$
\ga^\mu \pa_\mu =
\lb \begin{array}{cccc}
0 & 0 & \pa_0+\pa_3 & \pa_1-i\pa_2 \\
0 & 0 & \pa_1+i\pa_2 & \pa_0-\pa_3 \\
\pa_0-\pa_3 & -\pa_1+i\pa_2 & 0 & 0 \\
-\pa_1-i\pa_2 & \pa_0+\pa_3 & 0 & 0
\end{array} \rb
$$

$$
\ps = 
\lb \matrix{
e_L^\wedge \\ e_L^\vee \\ e_R^\wedge \\ e_R^\vee
} \rb
\in 4^\mathbb{C}
$$
<<tiddler HideTags>>
$$\begin{array}{rcl}
0 \!\!&\!\!=\!\!&\!\! \ga^a \lp e_a \rp^\mu \lp \pa_\mu + {\textstyle \fr{1}{4}} \om_\mu^{\p{\mu}ab} \ga_{ab} \rp \ps + i m \, \ps \\[.5em]
 \!\!&\!\!=\!\!&\!\! \ve{e} \lp \f{d} + \ha \f{\om} \rp \ps + i m \, \ps
\end{array}
$$

| $\; \ga_a \; $ |[[Clifford basis vectors]] for [[Cl(1,3)]], rep in $\mathbb{C}(4)$ |
| $\; \ga_{ab} = \ga_a \ga_b \;\;\; a \ne b \; $ |[[Clifford basis bivectors|Clifford basis elements]] of $Cl^2(1,3) = spin(1,3) \;\;$ |
| $\; ( e_a )^\mu \; $ |[[orthonormal basis vector|frame]] components (//vierbein//) |
| $\; \om_\mu^{\p{\mu}ab} \; $ |[[spin connection]] components |
| $\; \ps \; $ |[[Dirac spinor]], $4^\mathbb{C}_S$ |

| $\; \f{e} = \f{dx^\mu} (e_\mu)^a \ga_a \;$ |$\in Cl^1(1,3)$ |gravitational [[frame]] |
| $\; \ve{e} = \ga^a (e_a)^\mu \ve{\pa_\mu} \;$ |$\in Cl^1(1,3)$ |inverse [[frame]], $\ve{e} \f{e} = 4 \;\;$ |
| $\; \f{\om} = \f{dx^\mu} \ha \om_\mu^{\p{\mu}ab} \ga_{ab} \;$ |$\in Cl^2(1,3) = spin(1,3)$ |[[spin connection]] |
<<tiddler HideTags>>
$$\begin{array}{rcl}
0 \!\!&\!\!=\!\!&\!\! \ga^a \lp e_a \rp^\mu \lp \pa_\mu + {\textstyle \fr{1}{4}} \om_\mu^{\p{\mu}ab} \ga_{ab}  + A_\mu^{\p{\mu}B} T_B \rp \ps + \ph \, \ps \\[.5em]
 \!\!&\!\!=\!\!&\!\! \ve{e} \lp \f{d} + \ha \f{\om} +\f{A} \rp \ps + \ph \, \ps
\end{array}
$$


| $\; T_B \; $ |[[Lie algebra]] basis elements (//generators//),  $\;\; \in \; u(1) \oplus su(2) \oplus su(3) \;\;$ |
| $\; \f{A} = \f{dx^\mu} A_\mu^{\p{\mu}B} T_B \; $ |Yang-Mills [[gauge field|principal bundle]] (//connection//) |
| $\; \ps \; $ |spinor multiplet, $\;\; \in \; 2 \!\otimes\! (2_L\!\oplus\!2_R) \!\otimes\! (1\!\oplus\!3) \;\;\;\;\; (\otimes 3)$  |
| $\; \ph \; $ |Higgs scalar field multiplet (linear operator on $\ps$) |

$$
\begin{array}{rcl}
{\rm Clifford \; algebra} \!\!&\!\! \longleftrightarrow \!\!&\!\! {\rm Lie \; algebra}^{\phantom{(}} & \longleftrightarrow \;\; {\rm Lie \; group} \;\;  \longleftrightarrow \;\; {\rm Geometry}\\
\searrow \!\!\!\!\!\! \nwarrow \!\!&\!\! \!\!&\!\! \swarrow \!\!\!\!\!\! \nearrow & \\
& {\rm Matrices} & &
\end{array}
$$
<<tiddler HideTags>>

$$\begin{array}{rcl}
0 \!\!&\!\!=\!\!&\!\! \ve{e} \lp \f{d} + \ha \f{\om} + \f{A} \rp \ps + \ph \, \ps \\
\!\!&\!\!=\!\!&\!\! \ve{e} \lp \f{d} + \ha \f{\om} + \fr{1}{4} \f{e} \ph + \f{A} \rp \ps \\
\!\!&\!\!=\!\!&\!\! \ve{e} \lp \f{d} + \f{H} \rp \ps \\
\!\!&\!\!=\!\!&\!\! \ve{e} \f{D} \, \ps = D \!\!\!\! / \; \ps
\end{array}
$$


| $\; \f{e} = \f{dx^\mu} (e_\mu)^a \ga_a \;$ |$\in Cl^1(1,3)$ |gravitational [[frame]] |
| $\; \ve{e} = \ga^a (e_a)^\mu \ve{\pa_\mu} \;$ |$\in Cl^1(1,3)$ |inverse [[frame]], $\ve{e} \f{e} = 4$ |
| $\; \f{\om} = \f{dx^\mu} \ha \om_\mu^{\p{\mu}ab} \ga_{ab} \;$ |$\in Cl^2(1,3) = spin(1,3)$ |[[spin connection]] |
| $\; \f{A} = \f{dx^\mu}A_\mu^{\p{\mu}B} T_B \;$ |$\in G_{SM} = su(2)_L \oplus u(1)_Y \oplus su(3) \;\;$ |[[gauge connection|principal bundle]] |
| $\; \ps \; $ |$\in 2 \!\otimes\! (2_L\!\oplus\!2_R) \!\otimes\! (1\!\oplus\!3) \;\;\;\;\; (\otimes 3)$ |spinor field multiplet |
| $\; \f{H} = \ha \f{\om} + \fr{1}{4} \f{e} \ph + \f{A} \; $ |$\in \; ?$ |''unified bosonic connection'' |
| $\; \f{e} \ph \;$ |$\in \; ?$ |''frame-Higgs'' |
The ''Dirac matrices'' provide a $4\times4$ [[Clifford matrix representation]] of [[Cl(1,3)]] or [[Cl(3,1)]]. There are several standard choices, built from the [[Kronecker product]] of [[Pauli matrices]]:

The ([[chiral]]) ''Weyl representation'' of the Dirac matrices of Cl(1,3) is:
\begin{eqnarray}
\ga_0 &=& \si^P_1 \otimes 1 \\
\ga_1 &=& -i \si^P_2 \otimes \si^P_1 \\
\ga_2 &=& -i \si^P_2 \otimes \si^P_2 \\
\ga_3 &=& -i \si^P_2 \otimes \si^P_3
\end{eqnarray}
giving a complex rep for ''Cl(1,3) vectors'',
\begin{eqnarray}
v &=& v^\mu \ga_\mu =
\lb \begin{array}{cc}
0 & v_R \\
v_L & 0
\end{array} \rb
=
\lb \begin{array}{cc}
0 & v^0 - v^\va \si^P_\va \\
v^0 + v^\va \si^P_\va & 0
\end{array} \rb
\\
&=& 
\lb \begin{array}{cccc}
0 & 0 & v^0-v^3 & -v^1+iv^2 \\
0 & 0 & -v^1-iv^2 & v^0+v^3 \\
v^0+v^3 & v^1-iv^2 & 0 & 0 \\
v^1+iv^2 & v^0-v^3 & 0 & 0
\end{array} \rb
\end{eqnarray}
and [[spacetime pseudoscalar|Cl(1,3)]], $\ga = i \si^P_3 \otimes 1$. The $v_{L/R}$ are ''left and right chiral vector parts'' -- $2\times2$ Hermitian matrices projected out by the [[left/right chirality projector]]. (//They satisfy...//) Note that a vector is completely determined by one of its chiral parts.

''Dirac representation'' of CL(1,3),
\begin{eqnarray}
\ga_0 &=& \si^P_3 \otimes 1 \\
\ga_\va &=& -i \si^P_2 \otimes \si^P_\va
\end{eqnarray}

(real) ''Majorana representation'' of Cl(3,1),
\begin{eqnarray}
\ga_0 &=& i \si^P_1 \otimes \si^P_2 \\
\ga_1 &=& 1 \otimes \si^P_1 \\
\ga_2 &=& \si^P_2 \otimes \si^P_2 \\
\ga_3 &=& 1 \otimes \si^P_3
\end{eqnarray}
Multiplying these matrices by $i$ switches them between representations of Cl(1,3) and Cl(3,1).

The different matrix representations may be related by similarity transformations. For example, the Majorana rep is given in terms of the Weyl rep by $\ga^M_\mu = U \ga^W_\mu U^\dagger$, with (recalculate this)
$$
U = \ha
\lb \begin{array}{cccc}
1 & i & 1 & -i \\
i & 1 & i & -1 \\
1 & -i & -1 & -i \\
-i & 1 & i & 1
\end{array} \rb
$$

Ref:
http://en.wikipedia.org/wiki/Dirac_matrices
If $\Psi$ is a [[spinor]] field and $\f{A} \in \f{\rm Lie}(G)$ a [[principal bundle]] connection in a representation matched to the spinor, the [[covariant derivative]] of the spinor field is
$$
\f{\na} \Psi = \lp \f{d} + \f{A} \rp \Psi
$$
Note that $\f{A}$ includes the [[spin connection]], $\f{\om}$ (as the connection for the [[Clifford vector bundle]] subbundle of the full principal bundle) and usually other parts, which will be written as $\f{G}$, so
$$
\f{A} = \ha \f{\om} + \f{G}
$$
If we write the [[frame]] over the base manifold as $\ve{e} = \ga^\mu \ve{e_\mu}$, the ''Dirac operator'' acting on the spinor is defined as
$$
\na \Psi = \ve{e} \f{\na} \Psi = \ga^\mu \lp e_\mu \rp^i \lp \pa_i + \fr{1}{4} \om_i{}^{\nu \rh} \ga_{\nu \rh} + G_i{}^B T_B \rp \Psi
$$
using the [[vector-form algebra]].
A ''Dirac [[spinor]]'', $\Psi$, of the [[spacetime]] [[Cl(1,3)]] [[Clifford algebra]] may be written, using the [[Weyl representation|Dirac matrices]], as a sum of ''left [[chiral]]'' and ''right chiral'' parts,
$$
\Psi = \Psi_L + \Psi_R =
\lb \begin{array}{c}
\ps_L \\
0
\end {array} \rb
+
\lb \begin{array}{c}
0 \\
\ps_R
\end {array} \rb
=
\lb \begin{array}{c}
\ps_L \\
\ps_R
\end {array} \rb
=
\lb \begin{array}{c}
\ps_L^\wedge \\
\ps_L^\vee \\
\ps_R^\wedge \\
\ps_R^\vee
\end {array} \rb
$$
These parts are the ''left handed Weyl spinor'',
$$
\ps_L = \lb \begin{array}{c}
\ps_L^\wedge \\
\ps_L^\vee
\end {array} \rb
$$
and ''right handed Weyl spinor'', $\ps_R$ -- each represented by a column of 2 complex (or complex [[Grassmann|Grassmann number]]) numbers -- the ''spin up'' and ''spin down'' components. These weyl spinors may be projected out,
$$
\Psi_{L/R} = P_{L/R} \Psi
$$
by the [[left/right chirality projector]],
$$
P_{L/R} = \ha \lp 1 \mp i \ga \rp
$$
(In this equation, the four component column with two zero entries is equated to a two component column.)
/***
|Name|DisableWikiLinksPlugin|
|Source|http://www.TiddlyTools.com/#DisableWikiLinksPlugin|
|Version|1.6.0|
|Author|Eric Shulman|
|License|http://www.TiddlyTools.com/#LegalStatements|
|~CoreVersion|2.1|
|Type|plugin|
|Description|selectively disable TiddlyWiki's automatic ~WikiWord linking behavior|
This plugin allows you to disable TiddlyWiki's automatic ~WikiWord linking behavior, so that WikiWords embedded in tiddler content will be rendered as regular text, instead of being automatically converted to tiddler links.  To create a tiddler link when automatic linking is disabled, you must enclose the link text within {{{[[...]]}}}.
!!!!!Usage
<<<
You can block automatic WikiWord linking behavior for any specific tiddler by ''tagging it with<<tag excludeWikiWords>>'' (see configuration below) or, check a plugin option to disable automatic WikiWord links to non-existing tiddler titles, while still linking WikiWords that correspond to existing tiddlers titles or shadow tiddler titles.  You can also block specific selected WikiWords from being automatically linked by listing them in [[DisableWikiLinksList]] (see configuration below), separated by whitespace.  This tiddler is optional and, when present, causes the listed words to always be excluded, even if automatic linking of other WikiWords is being permitted.  

Note: WikiWords contained in default ''shadow'' tiddlers will be automatically linked unless you select an additional checkbox option lets you disable these automatic links as well, though this is not recommended, since it can make it more difficult to access some TiddlyWiki standard default content (such as AdvancedOptions or SideBarTabs)
<<<
!!!!!Configuration

G disabled these so that WikiLinks would never happen.

<<<
<<option chkDisableWikiLinks>> Disable ALL automatic WikiWord tiddler links
<<option chkAllowLinksFromShadowTiddlers>> ... except for WikiWords //contained in// shadow tiddlers
<<option chkDisableNonExistingWikiLinks>> Disable automatic WikiWord links for non-existing tiddlers
Disable automatic WikiWord links for words listed in: <<option txtDisableWikiLinksList>>
Disable automatic WikiWord links for tiddlers tagged with: <<option txtDisableWikiLinksTag>>
<<<
!!!!!Revisions
<<<
2008.07.22 [1.6.0] hijack tiddler changed() method to filter disabled wiki words from internal links[] array (so they won't appear in the missing tiddlers list)
2007.06.09 [1.5.0] added configurable txtDisableWikiLinksTag (default value: "excludeWikiWords") to allows selective disabling of automatic WikiWord links for any tiddler tagged with that value.
2006.12.31 [1.4.0] in formatter, test for chkDisableNonExistingWikiLinks
2006.12.09 [1.3.0] in formatter, test for excluded wiki words specified in DisableWikiLinksList
2006.12.09 [1.2.2] fix logic in autoLinkWikiWords() (was allowing links TO shadow tiddlers, even when chkDisableWikiLinks is TRUE).  
2006.12.09 [1.2.1] revised logic for handling links in shadow content
2006.12.08 [1.2.0] added hijack of Tiddler.prototype.autoLinkWikiWords so regular (non-bracketed) WikiWords won't be added to the missing list
2006.05.24 [1.1.0] added option to NOT bypass automatic wikiword links when displaying default shadow content (default is to auto-link shadow content)
2006.02.05 [1.0.1] wrapped wikifier hijack in init function to eliminate globals and avoid FireFox 1.5.0.1 crash bug when referencing globals
2005.12.09 [1.0.0] initial release
<<<
!!!!!Code
***/
//{{{
version.extensions.DisableWikiLinksPlugin= {major: 1, minor: 6, revision: 0, date: new Date(2008,7,22)};

// G hard coded this
config.options.chkDisableNonExistingWikiLinks=true;
config.options.chkDisableWikiLinks=true;
config.options.chkAllowLinksFromShadowTiddlers=false;

if (config.options.txtDisableWikiLinksList==undefined) config.options.txtDisableWikiLinksList="DisableWikiLinksList";
if (config.options.txtDisableWikiLinksTag==undefined) config.options.txtDisableWikiLinksTag="excludeWikiWords";

// find the formatter for wikiLink and replace handler with 'pass-thru' rendering
initDisableWikiLinksFormatter();
function initDisableWikiLinksFormatter() {
	for (var i=0; i<config.formatters.length && config.formatters[i].name!="wikiLink"; i++);
	config.formatters[i].coreHandler=config.formatters[i].handler;
	config.formatters[i].handler=function(w) {
		// supress any leading "~" (if present)
		var skip=(w.matchText.substr(0,1)==config.textPrimitives.unWikiLink)?1:0;
		var title=w.matchText.substr(skip);
		var exists=store.tiddlerExists(title);
		var inShadow=w.tiddler && store.isShadowTiddler(w.tiddler.title);
		// check for excluded Tiddler
		if (w.tiddler && w.tiddler.isTagged(config.options.txtDisableWikiLinksTag))
			{ w.outputText(w.output,w.matchStart+skip,w.nextMatch); return; }
		// check for specific excluded wiki words
		var t=store.getTiddlerText(config.options.txtDisableWikiLinksList);
		if (t && t.length && t.indexOf(w.matchText)!=-1)
			{ w.outputText(w.output,w.matchStart+skip,w.nextMatch); return; }
		// if not disabling links from shadows (default setting)
		if (config.options.chkAllowLinksFromShadowTiddlers && inShadow)
			return this.coreHandler(w);
		// check for non-existing non-shadow tiddler
		if (config.options.chkDisableNonExistingWikiLinks && !exists)
			{ w.outputText(w.output,w.matchStart+skip,w.nextMatch); return; }
		// if not enabled, just do standard WikiWord link formatting
		if (!config.options.chkDisableWikiLinks)
			return this.coreHandler(w);
		// just return text without linking
		w.outputText(w.output,w.matchStart+skip,w.nextMatch)
	}
}

Tiddler.prototype.coreAutoLinkWikiWords = Tiddler.prototype.autoLinkWikiWords;
Tiddler.prototype.autoLinkWikiWords = function()
{
	// if all automatic links are not disabled, just return results from core function
	if (!config.options.chkDisableWikiLinks)
		return this.coreAutoLinkWikiWords.apply(this,arguments);
	return false;
}

Tiddler.prototype.disableWikiLinks_changed = Tiddler.prototype.changed;
Tiddler.prototype.changed = function()
{
	this.disableWikiLinks_changed.apply(this,arguments);
	// remove excluded wiki words from links array
	var t=store.getTiddlerText(config.options.txtDisableWikiLinksList,"").readBracketedList();
	if (t.length) for (var i=0; i<t.length; i++)
		if (this.links.contains(t[i]))
			this.links.splice(this.links.indexOf(t[i]),1);
};
//}}}
<<tiddler HideTags>>What is done:
*All [[gauge fields|connection]], [[gravity|spacetime]], and Higgs in ''one'' [[connection]], with fermions as [[BRST ghosts|BRST technique]].

To do:
*Will particle assignments work with [[E8]]? (Get the CKMPMNS matrix?)
*Why is the action what it is? (How does symmetry breaking happen?)
*Is a four dimensional base [[manifold]] emergent?
*How does this theory get quantized? (LQG methods should apply.)
**Natural explanation for QM as a bonus?

What this theory will mean, if it all works:
*Gravitational [[frame]] and Higgs are intimately related.
*Naturally combines standard model with gravity -- so it's a [[T.O.E.|theory of everything]]
**(It's also a U.F.T., but I don't like to call it that.)
*Our universe is a very pretty shape!

@@display:block;text-align:center;Gar@Lisi.com
http://deferentialgeometry.org $\p{{}_{(}}$@@
&nbsp;<script label="O" title="toggle sidebar">
   var sb=document.getElementById('sidebar');
   var da=document.getElementById('displayArea');
   if (sb.style.display == 'none') {
      da.style.marginLeft = '18.5em';
      sb.style.display = 'block';}
   else {
      da.style.marginLeft = '0em';
      sb.style.display = 'none';}
</script>&nbsp;<script label="O" title="toggle title">
   var h=document.getElementById('head');
   if (h.style.height == '1.5em') {
      h.style.height = '5.8em';}
   else {
      h.style.height = '1.5em';}
</script>&nbsp;
The rank $6$ exceptional group, ''E6'', is a real, [[simple]], compact, connected [[Lie group|Lie group]]. It may be described by [[exponentiating|exponentiation]] its $78$ dimensional [[Lie algebra]], [[e6]].
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/RW12/images/E6.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math">E6 = spin(10) \,\oplus\, u(1)_{PQ} \,\,\oplus\,\, 16^\mathbb{C}_{S^+}</SPAN>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/RW12/images/E6.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math">E6 = spin(10) \,\oplus\, u(1)_{PQ} \,\,\oplus\,\, 16^\mathbb{C}_{S^+}</SPAN>
</td></tr>
</table>
</center></html>
The rank $8$ exceptional group, ''E8'', is the largest of the real, [[simple]], compact, connected [[Lie groups]] -- and is often regarded as the most beautiful. It may be described by [[exponentiating|exponentiation]] its $248$ dimensional [[Lie algebra]], [[e8]].
<<tiddler HideTags>>Build a real form of complex [[E8]] by using $Cl^2(1,7)=so(1,7)$ instead of $Cl^2(8)=so(8)$. Then ''E8 T.O.E. connection'' is:
$$
\udf{A} = \f{H} + \f{G} + \ud{\Ps}{}_I + \ud{\Ps}{}_{II} + \ud{\Ps}{}_{III} = 
$$
$$
\text{something like}_{\p{\big(}}
$$
$$
{\small
\begin{array}{c}

\!\!\! \lb \begin{array}{cccc}
\frac{1}{2} \f{\om_L} \!+\! i \f{W^3} \!&\! i \f{W^1} \!+\! \f{W^2} \!&\! - \! \frac{1}{4} \f{e_R} \ph_0^* \!& \frac{1}{4} \f{e_R} \ph_+ \! \\

i \f{W^1} \!-\! \f{W^2} \!&\! \frac{1}{2} \f{\om_L} \!-\! i \f{W^3} \!&\! \p{-} \frac{1}{4} \f{e_R} \ph_+^* \!& \frac{1}{4} \f{e_R} \ph_0 \! \\

-\frac{1}{4} \f{e_L} \ph_0 & \frac{1}{4} \f{e_L} \ph_+ & \!\!\!\! \frac{1}{2} \f{\om_R} \!+\! i \f{B} \!\! \!& & \! \\

\p{-}\frac{1}{4} \f{e_L} \ph_+^* & \frac{1}{4} \f{e_L} \ph_0^* & &\! \!\! \frac{1}{2} \f{\om_R} \!-\! i \f{B} \!\!\!\!\!
\end{array} \rb

\!\!+\!\!
\lb \begin{array}{cccc}
i \f{B} \!\! & & & \\
&\!\!\! \frac{-i}{3} \! \f{B} \!+\! i \f{G^{3+8}} \!\!\!&\!\!\! i\f{G^1} \!-\! \f{G^2} \!\!\!&\!\!\! i\f{G^4} \!-\! \f{G^5} \\
&\!\!\! i\f{G^1} \!+\! \f{G^2} \!\!\!&\!\!\! \frac{-i}{3} \! \f{B} \!-\! i \f{G^{3+8}} \!\!\!&\!\!\! i\f{G^6} \!-\! \f{G^7} \\
&\!\!\! i\f{G^4} \!+\! \f{G^5} \!\!\!&\!\!\! i\f{G^6} \!+\! \f{G^7} \!\!\!&\!\!\! \frac{-i}{3} \! \f{B} \!-\!\! \frac{2i}{\sqrt{3}}\f{G^8}
\end{array} \rb
\\
\; \\
+
\lb \begin{array}{cccc}
\ud{\nu}{}^e_L & \ud{u}{}_L^r & \ud{u}{}_L^g & \ud{u}_L^b \\
\ud{e}{}_L & \ud{d}{}_L^r & \ud{d}{}_L^g & \ud{d}{}_L^b \\
\ud{\nu}{}^e_R & \ud{u}{}_R^r & \ud{u}{}_R^g & \ud{u}{}_R^b \\
\ud{e}{}_R & \ud{d}{}_R^r & \ud{d}{}_R^g & \ud{d}{}_R^b
\end{array} \rb
\;+\;
\lb \begin{array}{cccc}
\ud{\nu}{}^\mu_L & \ud{c}{}_L^r & \ud{c}{}_L^g & \ud{c}_L^b \\
\ud{\mu}{}_L & \ud{s}{}_L^r & \ud{s}{}_L^g & \ud{s}{}_L^b \\
\ud{\nu}{}^\mu_R & \ud{c}{}_R^r & \ud{c}{}_R^g & \ud{c}{}_R^b \\
\ud{\mu}{}_R & \ud{s}{}_R^r & \ud{s}{}_R^g & \ud{s}{}_R^b
\end{array} \rb
\;+\;
\lb \begin{array}{cccc}
\ud{\nu}{}^\ta_L & \ud{t}{}_L^r & \ud{t}{}_L^g & \ud{t}_L^b \\
\ud{\ta}{}_L & \ud{b}{}_L^r & \ud{b}{}_L^g & \ud{b}{}_L^b \\
\ud{\nu}{}^\ta_R & \ud{t}{}_R^r & \ud{t}{}_R^g & \ud{t}{}_R^b \\
\ud{\ta}{}_R & \ud{b}{}_R^r & \ud{b}{}_R^g & \ud{b}{}_R^b
\end{array} \rb_{\p{(}}
\end{array}
}
$$
<<tiddler HideTags>>$$
\begin{array}{rcl}
S_D \!\!&\!\!=\!\!&\!\! \int \nf{d^4x} |e| \big\{ \bar{\ps} \ga^\mu (e_\mu)^i \big( \pa_i +\fr{1}{4} \om_i^{\p{i}\nu\rho} \ga_{\nu\rho} + G_i^{\p{i}B} T_B \big) \ud{\ps} + \bar{\ps} \ph \ud{\ps} \big\} \\
        \!\!&\!\!=\!\!&\!\! \int \nf{d^4x} |e| \big\{ \bar{\ps} \ga^\mu (e_\mu)^i \big( \pa_i + \fr{1}{4} \om_i^{\p{i}\nu\rho} \ga_{\nu\rho} + \ha G_i^{\p{i}\ps\ch} \ga_{\ps\ch} + \fr{1}{4} (e_i)^\nu \ph^\ps \ga_{\nu\ps} \big) \ud{\ps} \big\} \\
\!\!&\!\!=\!\!&\!\! \int \nf{e} \big\{ \bar{\ps} \ve{e} \big( \f{d} + \f{H} \big) \ud{\ps} \big\}
= \int \fr{1}{4!} \bar{\ps} \f{e} \f{e} \f{e} \ep \f{D} \ud{\ps}
= \int \nf{e} \bar{\ps} D \!\!\!\!/\; \ud{\ps}
\end{array}
$$
Gravity: $\;\;\;\; \f{\om} = \ha \f{dx^i} \, \om_i^{\p{i}\mu\nu} \ga_{\mu\nu}  \;\; \in Cl(3,1)^2 = spin(3,1)
\;\;\;\;\;\;\;\;\; \f{e} = \f{dx^i} (e_i)^\mu \ga_\mu \;\; \in Cl(3,1)^1 = 4 \vp{A_{\big(}}$
GUT: $\;\;\;\; \f{G} \;\; \in \, su(2)_L+u(1)_Y+su(3)$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \subset su(2)_L+su(2)_R+su(4) = spin(4)+spin(6) \;\; \subset spin(10) \vp{A_{\big(}}$
Fermions: $\;\;\;\; \ud{\ps} \;\; \in 2 \!\times\! (2_L \!+\! 2_R) \!\times\! (1+3) = 32^\mathbb{C} = 64^\mathbb{R} \;\;\; (\times 3){}^{\p{\big(}}$
Higgs: $\;\;\;\; \ph=\ph^\ps \ga_\ps \;\; \in Cl(4)^1 = 4 = \mathbb{C}^2 \;$ or $\; Cl(N)^1 = N{}^{\p{\big(}}$
Connection: $\;\;\;\; \f{H} = \ha \f{\om} + \fr{1}{4} \f{e} \ph + \f{G} \;\; \in spin(3,1) + 4 \!\times\! 10 + spin(10) \;\; \subset spin(3,11) {}^{\p{\Big(}}$
Curvature: $\;\;\;\; \ff{F} = \f{d} \f{H} + \f{H} \f{H} 
= \ha (\ff{R} - \fr{1}{8} \f{e}\f{e} \ph^2) + \fr{1}{4} (\f{T} \ph - \f{e} \f{D} \ph) + \ff{F}^G{}^{\p{\big(}}$
Superconnection: $\;\;\;\; \udf{A} = \f{H} + \ud{\ps} \;\; \in spin(3,11) + 64_S^{+\mathbb{R}}   {}^{\p{\big(}}$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\, \subset spin(4,12) + 128_S^{+\mathbb{R}} = E8(-24)$
Supercurvature: $\;\;\;\; \udff{F} = \f{d} \udf{A} + \udf{A} \udf{A} = \ff{F} + \f{D} \ud{\ps} + \ud{\ps} \ud{\ps}{}^{\p{\big(}}$
$$
S = \int \big< \ff{\bar{B}} \udff{F} - {\textstyle \fr{\pi G}{4}} \ff{B} \ep \ff{B} + {\small g^2} \ff{B'} \ff{*B'} \big> \sim \int \big< \nf{e} \bar{\ps} D \!\!\!\!/\; \ud{\ps} + {\textstyle \fr{1}{16\pi G}} \nf{e} \big( R - {\textstyle \fr{3}{2}} \ph^2 \big) \ph^2 + {\textstyle \fr{1}{4g^2}} \f{D} \ph \fff{*D} \ph + {\textstyle \fr{1}{4g^2}} \ff{F}^G \ff{*F}^G \big> 
$$
<<tiddler HideTags>>@@display:block;text-align:center;
<html><center>
<img src="talks/StAnth09/images/bubblechamber3.png" width="585" height="440">
</center></html>
$\p{{}_{\small (}^{(}}$
[[Garrett Lisi]]&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Joint Mathematics Meetings 1/15/10@@
*Quantization
**Coupling constants run.
***Large $\La$ compatible with UV fixed point.
**Just a connection -- amenable to LQG, spin foams, etc.
*Understand triality-generation relationship better
**Possible collapse or mixing to graviweak $SL(2,\mathbb{C})$.
**The role of $\f{w}+\f{x}\Ph$ and symmetry breaking.
**Getting the CKMPMNS matrix would be nice.
*Why is the action what it is?
**Pulling $\f{e}$ out and putting it into $\ff{F} \ff{*F}$ and $\fff{\od{B}}$ seems weird.
***Why $\f{e}\ph$ simple?
***Four dimensional base manifold emergent?

What this theory will mean, if it all works:
*Combines standard model with gravity -- with LQG, it's a T.o.E.
*Our universe is very pretty.

@@display:block;text-align:center; http://deferentialgeometry.org &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Garrett Lisi@@
<<tiddler HideTags>>
Everything in an $E8$ principal bundle connection,
$$
\udf{A} \in \udf{e8}
$$
Periodic table of interactions (Feynman vertices) from curvature,
$$
\udff{F} = \f{d} \udf{A} + {\scriptsize \frac{1}{2}} \big[ \udf{A}, \udf{A} \big]
$$
described by the $E8$ root polytope. Three generations through triality,
$$
T \, e = \mu \qquad T \, \mu = \ta \qquad T \, \ta = e
$$
Pati-Salam $SU(2)_L \times SU(2)_R \times SU(4)$ GUT and MM gravity together,
$$
S = \int \big< \ff{\od{B}} \udff{F}
+ {\scriptsize \frac{\pi}{4}} \ff{B}{}_G \ff{B}{}_G \ga + \ff{B'} \ff{*B'} \big>
$$
No free parameters -- masses from Higgs VEV's,
$$
g_1 = \sqrt{\fr{3}{5}} \qquad g_2=1 \qquad g_3=1 \qquad \La=\fr{3}{4}\ph^2 \qquad \ph_0 , \ph_1, \Ph \dots 
$$
Everything is pure geometry, and it's very beautiful.
<<tiddler HideTags>>
<<tiddler HideTags>>Superconnection:
$$
\begin{array}{rcl}
\udf{A} \!\!&\!\!=\!\!&\!\! \big( \ha \f{\om} + \fr{1}{4} \f{e} \ph + \f{G} \big) + \ud{\ps} \\
\!\!&\!\! \in \!\!&\!\! H + K \\
\!\!&\!\! \subset \!\!&\!\! \big( spin(3,1) + 4 \!\times\! (2 \!+\! \bar{2}) + su(2)_L + u(1)_Y + su(3) \big) + 2 \!\times\! (2_L\!+\!2_R) \!\times\! (1\!+\!3) \\
\!\!&\!\! \subset \!\!&\!\! \big( spin(3,1) + 4 \!\times\! 10 + spin(10) \big) + 2 \!\times\! 16_S^{+\mathbb{C}} \\
\!\!&\!\! \subset \!\!&\!\! spin(3,11) + 64_S^{+\mathbb{R}} \subset spin(4,12) + 128_S^{+\mathbb{R}} \subset E8
\end{array}
$$
Curvature:
$$
\udff{F} = \f{d} \udf{A} + \udf{A} \udf{A} = \ff{F}^H + \f{D} \ud{\ps} + \ud{\ps} \ud{\ps}
$$
Action:
$$
S = \int \left< \big( \fff{\od{B}} + \ff{B} \big) \udff{F} + \nf{V}(B^H) \right>
$$

Generations?$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\mbox{Axions?} \;\;\;\; spin(1,1)_{PQ}, \;\; \th \ff{F} \ff{F},\;\; \big< \bar{\ps} \f{e} \th \f{e} \th \f{e} \th \ep \f{D} \ud{\ps} \big> \mbox{ ?}
$

Geometric interpretation of the superconnection?$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\mbox{BRST?} \;\;\;\; \ud{\de} \f{K} = - \f{D} \ud{\ps} \;\;\;\;\;\; \mbox{TQFT?}\vp{A_{\big(}}
$
Precise symmetry breaking mechanism?$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\nf{V}(B^H) = \ff{B} \ff{\vv{\Ph}} \ff{B} + \dots \mbox{ ?}
$

Quantization?$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\mbox{Asymptoticly safe R.G. flow of } \La, G, g, \dots \mbox{?    Spinfoams?}
$
<<tiddler HideTags>>$$
\begin{array}{rcl}
S_D \!\!&\!\!=\!\!&\!\! \int \nf{d^4x} |e| \big\{ \bar{\ps} \ga^\mu (e_\mu)^i \big( \pa_i +\fr{1}{4} \om_i^{\p{i}\nu\rho} \ga_{\nu\rho} + G_i^{\p{i}B} T_B \big) \ud{\ps} + \bar{\ps} \ph \ud{\ps} \big\} \\
        \!\!&\!\!=\!\!&\!\! \int \nf{d^4x} |e| \big\{ \bar{\ps} \ga^\mu (e_\mu)^i \big( \pa_i + \fr{1}{4} \om_i^{\p{i}\nu\rho} \ga_{\nu\rho} + \ha G_i^{\p{i}\ps\ch} \ga_{\ps\ch} + \fr{1}{4} (e_i)^\nu \ph^\ps \ga_{\nu\ps} \big) \ud{\ps} \big\} \\
\!\!&\!\!=\!\!&\!\! \int \nf{e} \big\{ \bar{\ps} \ve{e} \big( \f{d} + \f{H} \big) \ud{\ps} \big\}
= \int \fr{1}{4!} \bar{\ps} \f{e} \f{e} \f{e} \ep \f{D} \ud{\ps}
= \int \nf{e} \bar{\ps} D \!\!\!\!/\; \ud{\ps}
\end{array}
$$
Gravity: $\;\;\;\; \f{\om} = \ha \f{dx^i} \, \om_i^{\p{i}\mu\nu} \ga_{\mu\nu}  \;\; \in Cl(3,1)^2 = spin(3,1)
\;\;\;\;\;\;\;\;\; \f{e} = \f{dx^i} (e_i)^\mu \ga_\mu \;\; \in Cl(3,1)^1 = 4 \vp{A_{\big(}}$
GUT: $\;\;\;\; \f{G} \;\; \in \, su(2)_L+u(1)_Y+su(3)$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \subset su(2)_L+su(2)_R+su(4) = spin(4)+spin(6) \;\; \subset spin(10) \vp{A_{\big(}}$
Fermions: $\;\;\;\; \ud{\ps} \;\; \in 2 \!\times\! (2_L \!+\! 2_R) \!\times\! (1+3) = 32^\mathbb{C} = 64^\mathbb{R} \;\;\; (\times 3){}^{\p{\big(}}$
Higgs: $\;\;\;\; \ph=\ph^\ps \ga_\ps \;\; \in Cl(4)^1 = 4 = \mathbb{C}^2 \;$ or $\; Cl(N)^1 = N{}^{\p{\big(}}$
Connection: $\;\;\;\; \f{H} = \ha \f{\om} + \fr{1}{4} \f{e} \ph + \f{G} \;\; \in spin(3,1) + 4 \!\times\! 10 + spin(10) \;\; \subset spin(3,11) {}^{\p{\Big(}}$
Curvature: $\;\;\;\; \ff{F} = \f{d} \f{H} + \f{H} \f{H} 
= \ha (\ff{R} - \fr{1}{8} \f{e}\f{e} \ph^2) + \fr{1}{4} (\f{T} \ph - \f{e} \f{D} \ph) + \ff{F}^G{}^{\p{\big(}}$
Superconnection: $\;\;\;\; \udf{A} = \f{H} + \ud{\ps} \;\; \in spin(3,11) + 64_S^{+\mathbb{R}}   {}^{\p{\big(}}$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\, \subset spin(4,12) + 128_S^{+\mathbb{R}} = E8(-24)$
Supercurvature: $\;\;\;\; \udff{F} = \f{d} \udf{A} + \udf{A} \udf{A} = \ff{F} + \f{D} \ud{\ps} + \ud{\ps} \ud{\ps}{}^{\p{\big(}}$
$$
S = \int \big< \ff{\bar{B}} \udff{F} - {\textstyle \fr{\pi G}{4}} \ff{B} \ep \ff{B} + {\small g^2} \ff{B'} \ff{*B'} \big> \sim \int \big< \nf{e} \bar{\ps} D \!\!\!\!/\; \ud{\ps} + {\textstyle \fr{1}{16\pi G}} \nf{e} \big( R - {\textstyle \fr{3}{2}} \ph^2 \big) \ph^2 + {\textstyle \fr{1}{4g^2}} \f{D} \ph \fff{*D} \ph + {\textstyle \fr{1}{4g^2}} \ff{F}^G \ff{*F}^G \big> 
$$
<<tiddler HideTags>>$$
\begin{array}{rcl}
S_D \!\!&\!\!=\!\!&\!\! \int \nf{d^4x} |e| \big\{ \bar{\ps} \ga^\mu (e_\mu)^i \big( \pa_i +\fr{1}{4} \om_i^{\p{i}\nu\rho} \ga_{\nu\rho} + G_i^{\p{i}B} T_B \big) \ud{\ps} + \bar{\ps} \ph \ud{\ps} \big\} \\
        \!\!&\!\!=\!\!&\!\! \int \nf{d^4x} |e| \big\{ \bar{\ps} \ga^\mu (e_\mu)^i \big( \pa_i + \fr{1}{4} \om_i^{\p{i}\nu\rho} \ga_{\nu\rho} + \ha G_i^{\p{i}\ps\ch} \ga_{\ps\ch} + \fr{1}{4} (e_i)^\nu \ph^\ps \ga_{\nu\ps} \big) \ud{\ps} \big\} \\
\!\!&\!\!=\!\!&\!\! \int \nf{e} \big\{ \bar{\ps} \ve{e} \big( \f{d} + \f{H} \big) \ud{\ps} \big\}
= \int \fr{1}{4!} \bar{\ps} \f{e} \f{e} \f{e} \ep \f{D} \ud{\ps}
= \int \nf{e} \bar{\ps} D \!\!\!\!/\; \ud{\ps}
\end{array}
$$
Gravity: $\;\;\;\; \f{\om} = \ha \f{dx^i} \, \om_i^{\p{i}\mu\nu} \ga_{\mu\nu}  \;\; \in Cl(3,1)^2 = spin(3,1)
\;\;\;\;\;\;\;\;\; \f{e} = \f{dx^i} (e_i)^\mu \ga_\mu \;\; \in Cl(3,1)^1 = 4 \vp{A_{\big(}}$
GUT: $\;\;\;\; \f{G} \;\; \in \, su(2)_L+u(1)_Y+su(3)$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \subset su(2)_L+su(2)_R+su(4) = spin(4)+spin(6) \;\; \subset spin(10) \vp{A_{\big(}}$
Fermions: $\;\;\;\; \ud{\ps} \;\; \in 2 \!\times\! (2_L \!+\! 2_R) \!\times\! (1+3) = 32^\mathbb{C} = 64^\mathbb{R} \;\;\; (\times 3){}^{\p{\big(}}$
Higgs: $\;\;\;\; \ph=\ph^\ps \ga_\ps \;\; \in Cl(4)^1 = 4 = \mathbb{C}^2 \;$ or $\; Cl(N)^1 = N{}^{\p{\big(}}$
Connection: $\;\;\;\; \f{H} = \ha \f{\om} + \fr{1}{4} \f{e} \ph + \f{G} \;\; \in spin(3,1) + 4 \!\times\! 10 + spin(10) \;\; \subset spin(3,11) {}^{\p{\Big(}}$
Curvature: $\;\;\;\; \ff{F} = \f{d} \f{H} + \f{H} \f{H} 
= \ha (\ff{R} - \fr{1}{8} \f{e}\f{e} \ph^2) + \fr{1}{4} (\f{T} \ph - \f{e} \f{D} \ph) + \ff{F}^G{}^{\p{\big(}}$
Superconnection: $\;\;\;\; \udf{A} = \f{H} + \ud{\ps} \;\; \in spin(3,11) + 64_S^{+\mathbb{R}}   {}^{\p{\big(}}$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\, \subset spin(4,12) + 128_S^{+\mathbb{R}} = E8(-24)$
Supercurvature: $\;\;\;\; \udff{F} = \f{d} \udf{A} + \udf{A} \udf{A} = \ff{F} + \f{D} \ud{\ps} + \ud{\ps} \ud{\ps}{}^{\p{\big(}}$
$$
S = \int \big< \ff{\bar{B}} \udff{F} - {\textstyle \fr{\pi G}{4}} \ff{B} \ep \ff{B} + {\small g^2} \ff{B'} \ff{*B'} \big> \sim \int \big< \nf{e} \bar{\ps} D \!\!\!\!/\; \ud{\ps} + {\textstyle \fr{1}{16\pi G}} \nf{e} \big( R - {\textstyle \fr{3}{2}} \ph^2 \big) \ph^2 + {\textstyle \fr{1}{4g^2}} \f{D} \ph \fff{*D} \ph + {\textstyle \fr{1}{4g^2}} \ff{F}^G \ff{*F}^G \big> 
$$
<<tiddler HideTags>>$$
\begin{array}{rcl}
S_D \!\!&\!\!=\!\!&\!\! \int \nf{d^4x} |e| \big\{ \bar{\ps} \ga^a (e_a)^\mu \big( \pa_\mu +\fr{1}{4} \om_\mu^{\p{\mu}bc} \ga_{bc} + A_\mu^{\p{\mu}B} T_B \big) \ud{\ps} + \bar{\ps} \ph \ud{\ps} \big\} \\
        \!\!&\!\!=\!\!&\!\! \int \nf{d^4x} |e| \big\{ \bar{\ps} \Ga^a (e_a)^\mu \big( \pa_\mu + \fr{1}{4} \om_\mu^{\p{\mu}bc} \Ga_{bc} + \ha A_\mu^{\p{\mu}xy} \Ga_{xy} + \fr{1}{4} (e_\mu)^b \ph^x \Ga_{bx} \big) \ud{\ps} \big\} \\
\!\!&\!\!=\!\!&\!\! \int \nf{e} \big\{ \bar{\ps} \ve{e} \big( \f{d} + \f{H} \big) \ud{\ps} \big\}
= \int \fr{1}{4!} \bar{\ps} \f{e} \f{e} \f{e} \ep \f{D} \ud{\ps}
= \int \nf{e} \bar{\ps} D \!\!\!\!/\; \ud{\ps}
\end{array}
$$
Gravity: $\;\;\;\; \f{\om} = \ha \f{dx^\mu} \, \om_\mu^{\p{\mu}ab} \ga_{ab}  \;\; \in Cl^2(1,3) = spin(1,3)
\;\;\;\;\;\;\;\;\;\;\;\; \f{e} = \f{dx^\mu} (e_\mu)^a \ga_a \;\; \in Cl^1(1,3) = 4 \vp{A_{\big(}}$
GUT: $\;\;\;\; \f{A} \;\; \in \, su(2)_L \,\oplus\, u(1)_Y \,\oplus\, su(3)$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \subset su(2)_L \,\oplus\, su(2)_R \,\oplus\, su(4) = spin(4) \,\oplus\, spin(6) \;\; \subset spin(10) \vp{A_{\big(}}$
Fermions: $\;\;\;\; \ud{\ps} \;\; \in 2 \otimes (2_L \oplus 2_R) \otimes (1 \oplus 3) = 32^\mathbb{C} = 64^\mathbb{R} \;\;\; (\times 3){}^{\p{\big(}}$
Higgs: $\;\;\;\; \ph=\ph^x \Ga_x \;\; \in Cl^1(4) = 4 \;$ or $\; Cl^1(10) = 10{}^{\p{\big(}}$
Connection: $\;\;\;\; \f{H} = \ha \f{\om} + \fr{1}{4} \f{e} \ph + \f{G} \;\; \in spin(1,3) \,\oplus\, 4 \!\otimes\! 10 \,\oplus\, spin(10) \;\; \subset spin(11,3) {}^{\p{\Big(}}$
Curvature: $\;\;\;\; \ff{F} = \f{d} \f{H} + \f{H} \f{H} 
= \ha (\ff{R} - \fr{1}{8} \f{e}\f{e} \ph^2) + \fr{1}{4} (\f{T} \ph - \f{e} \f{D} \ph) + \ff{F}^A{}^{\p{\big(}}$
Superconnection: $\;\;\;\; \udf{A} = \f{H} + \ud{\ps} \;\; \in spin(11,3) \,\oplus\, 64^\mathbb{R}_{S+}   {}^{\p{\big(}}$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\, \subset spin(12,4) \,\oplus\, 128^\mathbb{R}_{S+} = E_{8(-24)}$
Supercurvature: $\;\;\;\; \udff{F} = \f{d} \udf{A} + \udf{A} \udf{A} = \ff{F} + \f{D} \ud{\ps} + \ud{\ps} \ud{\ps}{}^{\p{\big(}}$
$$
S = \int \big< \ff{\bar{B}} \udff{F} - {\textstyle \fr{\pi G}{4}} \ff{B} \ep \ff{B} + {\small g^2} \ff{B'} \ff{*B'} \big> \sim \int \big< \nf{e} \bar{\ps} D \!\!\!\!/\; \ud{\ps} + {\textstyle \fr{1}{16\pi G}} \nf{e} \big( R - {\textstyle \fr{3}{2}} \ph^2 \big) \ph^2 + {\textstyle \fr{1}{4g^2}} \f{D} \ph \fff{*D} \ph + {\textstyle \fr{1}{4g^2}} \ff{F}^A \ff{*F}^A \big> 
$$
<<tiddler HideTags>>$$
\begin{array}{rcl}
S_D \!\!&\!\!=\!\!&\!\! \int \nf{d^4x} |e| \big\{ \bar{\ps} \ga^a (e_a)^\mu \big( \pa_\mu +\fr{1}{4} \om_\mu^{\p{\mu}bc} \ga_{bc} + A_\mu^{\p{\mu}B} T_B \big) \ud{\ps} + \bar{\ps} \ph \ud{\ps} \big\} \\
        \!\!&\!\!=\!\!&\!\! \int \nf{d^4x} |e| \big\{ \bar{\ps} \Ga^a (e_a)^\mu \big( \pa_\mu + \fr{1}{4} \om_\mu^{\p{\mu}bc} \Ga_{bc} + \ha A_\mu^{\p{\mu}xy} \Ga_{xy} + \fr{1}{4} (e_\mu)^b \ph^x \Ga_{bx} \big) \ud{\ps} \big\} \\
\!\!&\!\!=\!\!&\!\! \int \nf{e} \big\{ \bar{\ps} \ve{e} \big( \f{d} + \f{H} \big) \ud{\ps} \big\}
= \int \fr{1}{4!} \bar{\ps} \f{e} \f{e} \f{e} \ep \f{D} \ud{\ps}
= \int \nf{e} \bar{\ps} D \!\!\!\!/\; \ud{\ps}
\end{array}
$$
Gravity: $\;\;\;\; \f{\om} = \ha \f{dx^\mu} \, \om_\mu^{\p{\mu}ab} \ga_{ab}  \;\; \in Cl^2(1,3) = spin(1,3)
\;\;\;\;\;\;\;\;\;\;\;\; \f{e} = \f{dx^\mu} (e_\mu)^a \ga_a \;\; \in Cl^1(1,3) = 4 \vp{A_{\big(}}$
GUT: $\;\;\;\; \f{A} \;\; \in \, su(2)_L \,\oplus\, u(1)_Y \,\oplus\, su(3)$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \subset su(2)_L \,\oplus\, su(2)_R \,\oplus\, su(4) = spin(4) \,\oplus\, spin(6) \;\; \subset spin(10) \vp{A_{\big(}}$
Fermions: $\;\;\;\; \ud{\ps} \;\; \in 2 \otimes (2_L \oplus 2_R) \otimes (1 \oplus 3) = 32^\mathbb{C} = 64^\mathbb{R} \;\;\; (\times 3){}^{\p{\big(}}$
Higgs: $\;\;\;\; \ph=\ph^x \Ga_x \;\; \in Cl^1(4) = 4 \;$ or $\; Cl^1(10) = 10{}^{\p{\big(}}$
Connection: $\;\;\;\; \f{H} = \ha \f{\om} + \fr{1}{4} \f{e} \ph + \f{G} \;\; \in spin(1,3) \,\oplus\, 4 \!\otimes\! 10 \,\oplus\, spin(10) \;\; \subset spin(11,3) {}^{\p{\Big(}}$
Curvature: $\;\;\;\; \ff{F} = \f{d} \f{H} + \f{H} \f{H} 
= \ha (\ff{R} - \fr{1}{8} \f{e}\f{e} \ph^2) + \fr{1}{4} (\f{T} \ph - \f{e} \f{D} \ph) + \ff{F}^A{}^{\p{\big(}}$
Superconnection: $\;\;\;\; \udf{A} = \f{H} + \ud{\ps} \;\; \in spin(11,3) \,\oplus\, 64^\mathbb{R}_{S+}   {}^{\p{\big(}}$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\, \subset spin(12,4) \,\oplus\, 128^\mathbb{R}_{S+} = E_{8(-24)}$
Supercurvature: $\;\;\;\; \udff{F} = \f{d} \udf{A} + \udf{A} \udf{A} = \ff{F} + \f{D} \ud{\ps} + \ud{\ps} \ud{\ps}{}^{\p{\big(}}$
$$
S = \int \big< \ff{\bar{B}} \udff{F} - {\textstyle \fr{\pi G}{4}} \ff{B} \ep \ff{B} + {\small g^2} \ff{B'} \ff{*B'} \big> \sim \int \big< \nf{e} \bar{\ps} D \!\!\!\!/\; \ud{\ps} + {\textstyle \fr{1}{16\pi G}} \nf{e} \big( R - {\textstyle \fr{3}{2}} \ph^2 \big) \ph^2 + {\textstyle \fr{1}{4g^2}} \f{D} \ph \fff{*D} \ph + {\textstyle \fr{1}{4g^2}} \ff{F}^A \ff{*F}^A \big> 
$$
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/RW12/images/GraviGUT E8.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math">E_{8(-24)} = spin(12,4) \,\,\oplus\,\, 128^\mathbb{R}_{S+}</SPAN>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/RW12/images/E8.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math"> </SPAN>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>><html><center><iframe src="talks/Mindshare11/anim/E8toE8.html" width="540" height="540" frameborder="0"></iframe>
</center></html>
$$
\udf{A} = \f{H}{}_1 + \f{H}{}_2 + \ud{\Ps}{}_{I} + \ud{\Ps}{}_{II} + \ud{\Ps}{}_{III} \quad \in \;\; \udf{e8} \vp{|_{\big(}}
$$
$$
\begin{array}{rclcl}
\f{H}{}_1 \!\!&\!\!=\!\!&\!\! {\scriptsize \frac{1}{2}} \f{\om} + {\scriptsize \frac{1}{4}} \f{e}\ph + \f{W} + \f{B}{}_1 & \in & \f{so}(7,1) \\[-.1em]
&& \f{\om} & \in & \f{so}(3,1) \\[-.1em]
&& \f{e} \ph = (\f{e}{}_1+\f{e}{}_2+\f{e}{}_3+\f{e}{}_4)\times(\ph_{+/0}+\ph_{-/1}) & \in & \f{4} \times (2+\bar{2}) \\
&& \f{W} + \f{B}{}_1 & \in & \f{su}(2) + \f{su}(2) \\
\f{H}{}_2 \!\!&\!\!=\!\!&\!\! \f{w} + \f{B}{}_2 + \f{x} \Ph + \f{g} & \in & \f{so}(8) \\[-.2em]
&& \f{w} + \f{B}{}_2 & \in & \f{u}(1) + \f{u}(1) \\[-.3em]
&& \f{x} \Ph = (\f{x}{}_{1}+\f{x}{}_{2}+\f{x}{}_{3})\times(\Ph^{r/g/b} + {\Ph}{}^{\bar{r}/\bar{g}/\bar{b}}) & \in & \f{3} \times (3+\bar{3}) \\[-.1em]
&& \f{g} & \in & \f{su}(3) \\
\ud{\Psi}{}_{I} \!\!&\!\!=\!\!&\!\! \ud{\nu}{}_e + \ud{e} + \ud{u} + \ud{d} & \in & 8_{S+} \!\times 8_{S+} \\
\ud{\Psi}{}_{II} \!\!&\!\!=\!\!&\!\! \ud{\nu}{}_\mu + \ud{\mu} + \ud{c} + \ud{s} & \in & 8_{V} \times 8_{V} \\ 
\ud{\Psi}{}_{III} \!\!&\!\!=\!\!&\!\! \ud{\nu}{}_\ta + \ud{\ta} + \ud{t} + \ud{b} & \in & 8_{S-} \!\times 8_{S-} \\
\end{array}
$$
<<tiddler HideTags>>
$$
\udff{F} = \f{d} \udf{A} + \udf{A} \udf{A}
= \ff{F}{}_1+\ff{F}{}_2+ \f{D} \big( \ud{\Ps}{}_{I} + \ud{\Ps}{}_{II} + \ud{\Ps}{}_{III} \big)  \quad \in \;\; \udff{e8} \vp{|_{\Big(}}
$$
$$
\begin{array}{rlcl}
\ff{F}{}_1 \!\!\!\!&=
\ha \big( \ff{R} - \fr{1}{8} \f{e} \f{e} \ph^2 \big)
+ \fr{1}{4} \big( \ff{T} \ph - \f{e} \f{D} \ph \big)
+ \big( \ff{F}{}_{B_1} + \ff{F}{}_W \big)  & \in & \f{so}(7,1) \\[.1em]
& \ff{R} = \f{d} \f{\om} + \ha \f{\om} \f{\om} & \in & \f{so}(3,1) \\[.1em]
& \ff{T} \ph \!-\! \f{e} \f{D} \ph = \big( \f{d} \f{e} \!+\! \ha [ \f{\om}, \f{e} ] \big) \ph - \f{e} \big( \f{d} \ph \!+\! [ \f{B}{}_1 \!+\! \f{W}, \ph ] \big) & \in & \f{4} \times (2+\bar{2}) \\[.2em]
& \ff{F}{}_{B_1} + \ff{F}{}_W = (\f{d} \f{B}{}_1 + \f{B}{}_1 \f{B}{}_1) + (\f{d} \f{W} + \f{W} \f{W}) & \in & \f{su}(2) \!+\! \f{su}(2) \\[.4em]
\ff{F}{}_2 \!\!\!\!&=
\big( \ff{F}{}_{w} + \ff{F}{}_{B_2} + \f{x}\Ph\f{x}\Ph \big)
+ \big( (\f{D} \f{x}) \Ph - \f{x} \f{D} \Ph \big)
+\ff{F}{}_{g}
& \in & \f{so}(8) \\[.1em]
& \ff{F}{}_{w} + \ff{F}{}_{B_2} = \f{d} \f{w} + \f{d} \f{B}{}_2 & \in & \f{u}(1) + \f{u}(1) \\[.1em]
& (\f{D} \f{x}) \Ph \!-\! \f{x} \f{D} \Ph \!=\! 
\big( \f{d} \f{x} \!+\! [ \f{w} \!+\! \f{B}{}_2, \! \f{x} ] \big) \Ph \!-\! \f{x} \big( \f{d} \Ph \!+\! [ \f{g}, \! \Ph ] \big) \!\!\!
 & \in & \f{3} \times (3+\bar{3}) \\[0em]
& \ff{F}{}_{g} = \f{d} \f{g} + \f{g} \f{g} & \in & \f{su}(3)
\end{array}
$$
$$
\f{D} \ud{\Psi} = \big( \f{d} + {\scriptsize \frac{1}{2}} \f{\om} + {\scriptsize \frac{1}{4}} \f{e}\ph \big) \ud{\Ps}
+ \f{W} \ud{\Ps}{}_L + \f{B}{}_1 \ud{\Ps}{}_R - \ud{\Ps} \big( \f{w} + \f{B}{}_2 + \f{x} \Ph \big) - \ud{\Ps}{}_q \, \f{g}
\vp{|^{\Big(}}
$$
<<tiddler HideTags>>
<<tiddler HideTags>>Build new ${\rm Lie}(E8)$ generators from old ones:
$$
\begin{array}{rclclcll}
H_{\al\be} \!\!&\!=\!&\!\! \ga^{\lp16\rp+}_{\al\be} \!\!&\!\!=\!&\!\! \ga^{(8)+}_{\al\be} \otimes 1 \!\!&\!\!\in\!&\!\! so(8)^+ \otimes 1
\!\!&\!=\, so(8)^H \\
G_{\al\be} \!\!&\!=\!&\!\! \ga^{\lp16\rp+}_{\lp\al+8\rp\lp\be+8\rp} \!\!&\!\!=\!&\!\! P^{\lp8\rp}_+ \otimes \ga^{(8)}_{\al\be} \!\!&\!\!\in\!&\!\! 1 \otimes so(8) 
\!\!&\!=\, so(8)^G \\
\Ps^I_{\al\be} \!\!&\!=\!&\!\! \ga^{\lp16\rp+}_{\al\lp\be+8\rp} \!\!&\!\!=\!&\! \ga^{(8)+}_\al \otimes \ga^{(8)}_\be \!\!&\!\!\in\!&\!\! v^{(8)+} \otimes v^{(8)}
\!\!&\!=\, S^I \\
\Ps^{II}_{ab} \!\!&\!=\!&\!\! Q^+_{16\lp a-1\rp+b} \!\!&\!\!=\!&\!\! q^+_a \otimes q^+_b \!\!&\!\!\in\!&\!\! S^{(8)+} \otimes S^{(8)+}
\!\!&\!=\, S^{II} \\
\Ps^{III}_{ab} \!\!&\!=\!&\!\! Q^+_{16\lp a-1\rp+b+8} \!\!&\!\!=\!&\!\! q^+_a \otimes q^-_b \!\!&\!\!\in\!&\!\! S^{(8)+} \otimes S^{(8)-}
\!\!&\!=\, S^{III}
\end{array}
$$

With these basis generators, the ${\rm Lie}(E8)$ elements are:
\begin{eqnarray}
E &=& H + G + \Ps_I + \Ps_{II} + \Ps_{III} \\
&=& \ha h^{\al\be} H_{\al\be} + \ha g^{\al\be} G_{\al\be} + \ps_I^{\al\be} \Ps^I_{\al\be} + \ps_{II}^{ab} \Ps^{II}_{ab} + \ps_{III}^{ab} \Ps^{III}_{ab} \\
&\in& so(8)^H + so(8)^G + S^I + S^{II} + S^{III}_{\p{(}}
\end{eqnarray}

<<tiddler HideTags>>@@display:block;text-align:center;[img[images/png/e8 periodic table.png]]@@
//"E8 is perhaps the most beautiful structure in all of mathematics, but it's very complex."// &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; -- Hermann Nicolai
<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
<embed src="talks/Perimeter07/anim/e8tour.mov" width="602" height="602" controller="false" autoplay="false" loop="false"></embed>
<!-- <embed src="talks/Perimeter07/anim/e8tour (om up)/p1.png" width="608" height="609"></embed> -->
</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@
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<embed src="talks/Perimeter07/anim/e8tour/p1.png" width="608" height="609"></embed>
</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@
<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
<embed src="talks/Perimeter07/anim/e8tour/p21.png" width="608" height="609"></embed>
</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@
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<embed src="talks/Perimeter07/anim/e8tour/p101.png" width="608" height="609"></embed>
</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@
<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
<embed src="talks/Perimeter07/anim/e8tour/p181.png" width="608" height="609"></embed>
</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@
<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
<embed src="talks/Perimeter07/anim/e8tour/p201.png" width="608" height="609"></embed>
</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@
<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
<embed src="talks/Perimeter07/anim/e8tour/p236.png" width="608" height="609"></embed>
</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@
<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
<embed src="talks/Perimeter07/anim/e8tour/p243.png" width="608" height="609"></embed>
</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@
<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
<embed src="talks/Perimeter07/anim/e8tour/p256.png" width="608" height="609"></embed>
</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@
<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
<embed src="talks/Perimeter07/anim/e8tour/p280.png" width="608" height="609"></embed>
</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@
<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
<embed src="talks/Perimeter07/anim/e8tour/p320.png" width="608" height="609"></embed>
</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@
<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
<embed src="talks/Perimeter07/anim/e8tour/p361.png" width="608" height="609"></embed>
</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@
<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
<embed src="talks/Perimeter07/anim/e8tour/p391.png" width="608" height="609"></embed>
</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@
<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
<embed src="talks/Perimeter07/anim/e8tour/p410.png" width="608" height="609"></embed>
</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@
<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
<embed src="talks/Perimeter07/anim/e8tour/p422.png" width="608" height="609"></embed>
</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@
<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
<embed src="talks/Perimeter07/anim/e8tour/p430.png" width="608" height="609"></embed>
</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@
<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
<embed src="talks/Perimeter07/anim/e8tour/p482.png" width="608" height="609"></embed>
</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@
<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
<embed src="talks/Perimeter07/anim/e8tour/p562.png" width="608" height="609"></embed>
</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@
<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
<embed src="talks/Perimeter07/anim/e8tour/p642.png" width="608" height="609"></embed>
</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@
<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
<embed src="talks/Perimeter07/anim/e8tour/p662.png" width="608" height="609"></embed>
</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@
<<tiddler HideTags>>@@display:block;text-align:center;${\rm Lie}(E8)$ has $(248-8)=240$ roots in 8D space -- vertices of $P4_{2,1}$:$\p{{}_{\big(}}$
<html><center><embed src="talks/FQXi07/video/e8anim.mov" width="510" height="510" controller="false" autoplay="false" loop="false"></embed></center></html>$E8$ T.O.E.: Each vertex corresponds to an elementary particle.$\p{{}{\Big(}^{(}}$@@ 
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/RW12/images/Triality.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math"> </SPAN>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>>The ${\rm Lie}(E8)$ brackets between elements in the various parts:
$$
\begin{array}{cc}
\begin{array}{rcl}
\big[ H_1, H_2 \big] \!\!&\!=\!&\!\! H_1 H_2 - H_2 H_1 \\
\big[ G_1, G_2 \big] \!\!&\!=\!&\!\! G_1 G_2 - G_2 G_1 \\
&&\\
\big[ H, \Ps_I \big] \!\!&\!=\!&\!\! H \, \Ps_I \\
\big[ H, \Ps_{II} \big] \!\!&\!=\!&\!\! H^+ \, \Ps_{II} \\
\big[ H, \Ps_{III} \big] \!\!&\!=\!&\!\! H^+ \, \Ps_{III} \\
&&\\
\big[ G, \Ps_I \big] \!\!&\!=\!&\!\! \Ps_I \, G \\
\big[ G, \Ps_{II} \big] \!\!&\!=\!&\!\! - \Ps_{II} \, G^+ \\
\big[ G, \Ps_{III} \big] \!\!&\!=\!&\!\! - \Ps_{III} \, G^-
\end{array}
&
\begin{array}{rcl}
\big[ \Ps^1_I, \Ps^2_I \big] \!\!&\!=\!&\!\! -2 \big( \Ps^1_I \, {\Ps^2_I}^T \big)_H \\
&& -2 \big( {\Ps^1_I}^T \Ps^2_I \big)_{G_{\p{(}}}  \\
\\
\big[ \Ps^1_{II}, \Ps^2_{II} \big] \!\!&\!=\!&\!\! - \big( \Ps^1_{II} \Ga^+ {\Ps^2_{II}}^T \big)_H \\
&&\!\! - \big( {\Ps^1_{II}}^T \Ga^+ \Ps^2_{II} \big)_{G_{\p{(}}} \\
\big[ \Ps^1_{III}, \Ps^2_{III} \big] \!\!&\!=\!&\!\! - \big( \Ps^1_{II} \Ga^+ {\Ps^2_{II}}^T \big)_H \\
&&\!\! - \big( {\Ps^1_{II}}^T \Ga^- \Ps^2_{II} \big)_G \\
&&\\
\big[ \Ps_I, \Ps_{II} \big] \!\!&\!=\!&\!\! - \big( \Ps_I \Ga^{++} \Ps_{II} \big)_{III} \\
\big[ \Ps_I, \Ps_{III} \big] \!\!&\!=\!&\!\! - \big( \Ps_I \Ga^{+-} \Ps_{III} \big)_{II} \\
\big[ \Ps_{II}, \Ps_{III} \big] \!\!&\!=\!&\!\! - \big( \Ps_{II} \Ga^{++} \Ps_{III} \big)_I
\end{array}
\end{array}
$$
Note: $H$ acts on $\Ps$'s from the left and $G$ acts from the right.$^{\p{\big(}}_{\p{(}}$
<<tiddler HideTags>>$$\begin{array}{rcl}
\udf{A} = \f{H} + \ud{\ps} \;\; \!\!&\!\!\in\!\!&\!\! spin(11,3) \,\oplus\, 64^\mathbb{R}_{S+} \\
\!\!&\!\!\subset\!\!&\!\! spin(11,3) \,\oplus\, 64^\mathbb{R}_{S+} \,\oplus\, 64^\mathbb{R}_{S-} \,\oplus\, 14_V \,\oplus\, 14_V \,\oplus\, spin(1,1) \\
\!\!&\!\!=\!\!&\!\! spin(12,4) \,\oplus\, 128^\mathbb{R}_{S+} \\
\!\!&\!\!=\!\!&\!\!  E_{8(-24)} \\
\end{array}
$$
''$E_{8(-24)}$ structure''
$$\begin{array}{rcl}
\f{H} = \ha \f{\om} + \fr{1}{4}\f{e}\ph + \f{A} = \ha \f{H}^{xy} \Ga'_{xy} \!\!&\!\! \in \!\!&\!\! spin(12,4) \\
\ud{\ps} = \ud{\ps}^\ph Q'_\ph \!\!&\!\! \in \!\!&\!\! 128^\mathbb{R}_{S+}
\end{array}
$$
$$
\begin{array}{rcl}
[\Ga'_{wx}, \Ga'_{yz}] \!\!&\!\!=\!\!&\!\! 2 \eta_{xy} \Ga'_{wz} - 2 \eta_{xz} \Ga'_{wy} + 2 \eta_{wz} \Ga'_{xy} - 2 \eta_{wy} \Ga'_{xz}  \\
[\Ga'_{xy}, Q'_\ph] \!\!&\!\! = \!\!&\!\! - [Q'_\ph , \Ga'_{xy}] = \Ga'^+_{xy} Q'_\ph = Q'_\ps (\Ga'^+_{xy})^\ps_{\p{\ps} \ph} \\
[Q'_\ph, Q'_\ps] \!\!&\!\! = \!\!&\!\! - \Ga_{xy} (\Ga'^{+xy})_{\ps \ph} =  - \Ga_{xy} \eta^{xw} \eta^{yz} (\Ga'^+_{wz})^\la_{\p{\la} \ph} g_{\la \ps} \\ [.5em]
g_{\la \ps} \!\!&\!\!=\!\!&\!\!  (\Ga'_1 \Ga'_2 \Ga'_3 \Ga'_{16})^+_{\la \ps}
\end{array}
$$

Spinors from E8 geometry,
$$\begin{array}{rcl}
\udff{F} \!\!&\!\! = \!\!&\!\! \ff{F^H} + \f{D} \ud{\ps} + \ud{\ps} \ud{\ps} \\[.5em]
\ve{e} \f{D} \ud{\ps} \!\!&\!\! = \!\!&\!\! \ve{e} \lp \f{d} \ud{\ps} + {\textstyle \ha} [ \f{H}, \ud{\ps} ] \rp
= \ve{e} \lp \f{d} + \f{H} \rp \ud{\ps} = D \!\!\!\! / \ud{\ps}
\end{array}
$$
<<tiddler HideTags>>$$\begin{array}{rcl}
\udf{A} = \f{H} + \ud{\ps} \;\; \!\!&\!\!\in\!\!&\!\! \lp spin(1,3) \,\oplus\, 4 \!\otimes\! (2 \!\,\oplus\,\! \bar{2}) \,\oplus\, su(2)_L \,\oplus\, u(1)_Y \,\oplus\, su(3) \rp \,\oplus\, \lp 2 \!\otimes\! (2_L\!\,\oplus\,\!2_R) \!\otimes\! (1\!\,\oplus\,\!3) \rp \\
\!\!&\!\!\subset\!\!&\!\! spin(11,3) \,\oplus\, 64^\mathbb{R}_{S+} \\
\!\!&\!\!\subset\!\!&\!\! spin(11,3) \,\oplus\, ( 64^\mathbb{R}_{S+} \,\oplus\, 64^\mathbb{R}_{S-}) \,\oplus\, (14_V \,\oplus\, 14_V) \,\oplus\, spin(1,1) \\
\!\!&\!\!=\!\!&\!\! spin(12,4) \,\oplus\, 128^\mathbb{R}_{S+} =  E_{8(-24)}
\end{array}
$$
''$E_{8(-24)}$ structure''
$$
\Ga'_{xy} \in spin(12,4) \;\;\;\;\;\;\;\; Q'_\ph  \in  128^\mathbb{R}_{S+}
$$
$$\begin{array}{rcl}
[\Ga'_{wx}, \Ga'_{yz}] \!\!&\!\!=\!\!&\!\! 2 \eta_{xy} \Ga'_{wz} - 2 \eta_{xz} \Ga'_{wy} + 2 \eta_{wz} \Ga'_{xy} - 2 \eta_{wy} \Ga'_{xz}  \\
[\Ga'_{xy}, Q'_\ph] \!\!&\!\! = \!\!&\!\! - [Q'_\ph , \Ga'_{xy}] = \Ga'^+_{xy} Q'_\ph = Q'_\ps (\Ga'^+_{xy})^\ps_{\p{\ps} \ph} \\
[Q'_\ph, Q'_\ps] \!\!&\!\! = \!\!&\!\! - \Ga_{xy} (\Ga'^{+xy})_{\ps \ph} =  - \Ga_{xy} \eta^{xw} \eta^{yz} (\Ga'^+_{wz})^\la_{\p{\la} \ph} g_{\la \ps} \\ [.5em]
(Q'_\la, Q'_\ps) \!\!&\!\!=\!\!&\!\! g_{\la \ps} = (\Ga'_1 \Ga'_2 \Ga'_3 \Ga'_{16})^+_{\la \ps}
\end{array}
$$

Spinors from E8 geometry,
$$
\f{H} = {\textstyle \ha} \f{H}{}^{xy} \Ga'_{xy}  \;\;\;\;\;\;\;\; \ud{\ps} = \ud{\ps}{}^{\ph} Q'_\ph
$$
$$
\ve{e} \f{D} \ud{\ps} = \ve{e} \lp \f{d} \ud{\ps} + {\textstyle \ha} [ \f{H}, \ud{\ps} ] \rp = \ve{e} \lp \f{d} + \f{H} \rp \ud{\ps}
$$
<<tiddler HideTags>>
Via the [[Pati-Salam]] GUT:

$$
\begin{array}{rcl}
\udf{A} = \f{H} + \ud{\ps} \;\; \!\!&\!\!\in\!\!&\!\! \lp spin(1,3) \,\oplus\, 4 \!\otimes\! (2 \!\,\oplus\,\! \bar{2}) \,\oplus\, su(2)_L \,\oplus\, u(1)_Y \,\oplus\, su(3) \rp \,\oplus\, \lp 2 \!\otimes\! (2_L\!\,\oplus\,\!2_R) \!\otimes\! (1\!\,\oplus\,\!3) \rp \!\otimes\! 2 \\[.5em]
\!\!&\!\!\subset\!\!&\!\! \lp spin(1,3) \,\oplus\, 4 \!\otimes\! (2 \!\,\oplus\,\! \bar{2}) \,\oplus\, \lp su(2)_L \,\oplus\, su(2)_R \,\oplus\, su(4) \rp \rp \,\oplus\, \lp 2 \!\otimes\! (2_L\!\,\oplus\,\!2_R) \!\otimes\! 4 \rp \!\otimes\! 2 \\[.5em]
\!\!&\!\!=\!\!&\!\! \lp spin(1,3) \,\oplus\, 4 \!\otimes\! 4 \,\oplus\, \lp spin(4) \,\oplus\, spin(6) \rp \rp \,\oplus\, \lp 2 \!\otimes\! 4 \!\otimes\! 4 \!\otimes\! 2 \rp \\[.5em]
\!\!&\!\!=\!\!&\!\! \lp spin(1,7) \,\oplus\, spin(6) \rp \,\oplus\, \lp 8 \!\otimes\! 4 \!\otimes\! 2 \rp \\[.5em]
\!\!&\!\!\subset\!\!&\!\! \lp spin(1,7) \,\oplus\, spin(7,1) \rp \,\oplus\, \lp 8 \!\otimes\! 8 \rp \\[.5em]
\!\!&\!\!\subset\!\!&\!\! \lp spin(1,7) \,\oplus\, spin(7,1) \rp \,\oplus\, \lp 8 \!\otimes\! 8 \rp \,\oplus\, \lp 8 \!\otimes\! 8 \rp \,\oplus\, \lp 8 \!\otimes\! 8 \rp \\[.5em]
\!\!&\!\!=\!\!&\!\! spin(8,8) \,\oplus\, 128^\mathbb{R}_{S+} \\[.5em]
\!\!&\!\!=\!\!&\!\! E_{8(8)}
\end{array}
$$
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The ''Ehresmann Cartan connection'', $\f{\ve{\cal C}}$, is an [[Ehresmann principal bundle connection]] over the total space, $E_G$, of an [[Ehresmann Cartan geometry]]. In coordinates ($x$ over $M$ patches, $x_s$ over $G/H$ patches, and $y$ over $H$ patches) adapted to the reference sections, the Ehresmann Cartan connection may be written locally as
\begin{eqnarray}
\f{\ve{\cal C}}(x, x_s, y) &=& \f{C^J}(x) \, \ve{\xi^L_J}(x_s, y) + \f{\ve{\cal I}} \\
&=& \f{C^J}(x) \, L^I{}_J(x_s, y) \, \ve{\xi^R_I}(x_s, y) + \f{\ve{\cal I}}  
\end{eqnarray}
in which $\ve{\xi^L_J}$ and $\ve{\xi^R_J} \sim T_J$ are the [[left and right action vector fields|Lie group geometry]] for the fibers, $G_x$, the [[left-right rotator]] is
$$
L^I{}_J(x_s, y) = \ve{\xi^L_J} \f{\xi_R^I} = \lp T^I, g^-(x_s,y) \, T_J \, g(x_s,y) \rp
$$
the [[Ehresmann-Maurer-Cartan form|Maurer-Cartan form]] (the [[identity projection|vector projection]] along the fibers) is
$$
\f{\ve{\cal I}}(x_s,y) = \f{\xi_R^J} \ve{\xi^R_J} = \f{dx_s^a} \ve{\pa^s_a} + \f{dy^p} \ve{\pa_p} = \f{\ve{{\cal I}_{G/H}}} + \f{\ve{{\cal I}_H}}
$$
and $\f{C^J}(x)$ are the components of the [[Cartan connection|Cartan geometry]] over $M$. The Ehresmann Cartan connection is a projection, $\f{\ve{\cal C}} \f{\ve{\cal C}} = \f{\ve{\cal C}}$, is [[right invariant]], $R_g^*\f{\ve{\cal C}} = \f{\ve{\cal C}}$, and has a natural [[spectral decomposition|spectral decomposition of the Ehresmann principal bundle connection]]. It may also be written as a [[Lieform]] over the total space by [[contracting|vector-form algebra]] it with the [[Maurer-Cartan form]], $\f{\cal I}(x_s,y) = \f{\xi_R^J} T_J = g^- \f{d} g$, over the total space to get the ''Ehresmann Cartan connection form'',
\begin{eqnarray}
\f{\cal C}(x,x_s,y) &=& \f{\ve{\cal C}} \f{\cal I} = \f{C^J}(x) \ve{\xi^L_J} \f{\xi_R^I} T_I + \f{\ve{\cal I}} \f{\cal I} \\
&=& \lp \f{C^J} L^I{}_J(x_s,y) + \f{\xi_R^I} \rp T_I \\
&=& g^-(x_s,y) \, \f{C}(x) \, g(x_s,y) + g^-(x_s,y) \, \f{d} \, g(x_s,y) 
\end{eqnarray}
This form [[pulls back|pullback]] along the canonical reference section, $\si_0^G$, to give the Cartan connection,
$$
\si_0^{G*} \f{\cal C} = \f{C}(x)
$$
and satisfies $R_g^* \f{\cal C} = g^- \f{\cal C} g$ under the right action.

The ''[[FuN curvature]] of the Ehresmann Cartan connection'' is
\begin{eqnarray}
\ff{\ve{\cal F}}(x,x_s,y) &=& - \ha \lb \f{\ve{\cal C}}, \f{\ve{\cal C}} \rb_L \\
&=& \lp \f{d} \f{C^K} + \ha \f{C^I} \f{C^J} C_{IJ}{}^K \rp \ve{\xi^L_K}(x_s,y)
\end{eqnarray}
which is vector valued in the vertical subspace and right invariant, $R_g^* \ff{\ve{\cal F}} = \ff{\ve{\cal F}}$. The ''FuN curvature form of the Ehresmann Cartan connection'' is a $Lie(G)$ valued 2-form over $E_G$,
\begin{eqnarray}
\ff{\cal F} &=& \ff{\ve{\cal F}} \f{\cal I} =  \lp \f{d} \f{C^K} + \ha \f{C^I} \f{C^J} C_{IJ}{}^K \rp g^-(x_s,y) \, T_K \, g(x_s,y) \\
&=& g^-(x_s,y) \lp \f{d} \f{C} + \ha \lb \f{C}, \f{C} \rb \rp g(x_s,y) \\
&=& g^-(x_s,y) \lp \f{d} \f{C} + \f{C} \f{C} \rp g(x_s,y)
\end{eqnarray}
This form pulls back along the canonical reference section to give the [[Cartan geometry]] curvature,
$$
\si_0^{G*} \ff{\cal F} = \f{d} \f{C} + \f{C} \f{C} = \ff{F}(x)
$$
and satisfies $R_g^* \ff{\cal F} = g^- \ff{\cal F} g$ under the right action.
When $H$ is [[reductive]] in $G$ (which is usually assumed) the [[Cartan connection|Cartan geometry]] splits as
$$
\f{C}(x) = \f{e}(x) + \f{A}(x) = \f{e^A} K_A + \f{A^P} H_P
$$
the [[Ehresmann Cartan connection]] can be made to follow this split,
$$
\begin{eqnarray}
\f{\ve{\cal C}}(x,x_s,y) &=& \f{C^J}(x) \, L^I{}_J(x_s, y) \, \ve{\xi^R_I}(x_s, y) + \f{\ve{\cal I}} \\
&=& \f{\ve{\cal E}} + \f{\ve{\cal A}}
\end{eqnarray}
$$
with the ''Ehresmann Cartan frame'' and ''Ehresmann Cartan H-connection'' defined over patches of the total space, $E_G$, of the [[Ehresmann Cartan geometry]] as:
$$
\begin{eqnarray}
\f{\ve{\cal E}}(x, x_s, y) &=& \f{e^A}(x) \, (L^h)^I{}_K(y) \, (L^r)^K{}_A(x_s) \, \ve{\xi^R_I}(x_s, y) \\
\f{\ve{\cal A}}(x, x_s, y) &=& \f{A^P}(x) \, (L^h)^I{}_K(y) \, (L^r)^K{}_P(x_s) \, \ve{\xi^R_I}(x_s, y) + \f{\ve{\cal I}}
\end{eqnarray}
$$
with the [[left-right rotator]] and [[Killing vector fields|Lie group geometry]] split over the [[reductive Lie group geometry]]. The [[Ehresmann Cartan connection form|Ehresmann Cartan connection]], $\f{\cal C} = \f{\ve{\cal C}} \f{\cal I}$, also splits,
$$
\begin{eqnarray}
\f{\cal C}(x,x_s,y) &=& g^-(x_s,y) \, \f{C}(x) \, g(x_s,y) + g^-(x_s,y) \, \f{d} \, g(x_s,y) \\
&=& \f{\cal E} + \f{\cal A}
\end{eqnarray}
$$
with the ''Ehresmann Cartan frame form'' and ''Ehresmann Cartan H-connection form'' defined over $E_G$ as:
$$
\begin{eqnarray}
\f{\cal E}(x, x_s, y) &=& \f{\ve{\cal E}} \f{\cal I} = \f{{\cal E}^I} T_I = g^- \, \f{e} \, g(x_s,y) \in \f{\rm Lie}(G) \\
\f{\cal A}(x, x_s, y) &=& \f{\ve{\cal A}} \f{\cal I} = \f{{\cal A}^I} T_I = g^- \, \f{A} \, g(x_s,y) + g^- \, \f{d} \, g(x_s,y) \in \f{\rm Lie}(G)
\end{eqnarray}
$$
with $g(x_s,y) = r(x_s) \, h(y)$.

This splitting is not a natural thing to do for the Ehresmann Cartan connection or connection form, for which the gauge group is $G$; however, it makes more sense when the Ehresmann Cartan conenction form is pulled back to the [[Cartan homogeneous space bundle]] or [[Cartan H-bundle]]. 
An ''Ehresmann [[Cartan geometry]]'' modeled on an $n_K$ dimensional [[homogeneous space]], $S=G/H$, is described by an [[Ehresmann principal bundle connection]] (the [[Ehresmann Cartan connection]]), $\f{\ve{\cal A}} = \f{\ve{\cal C}}$, over a $(n_M + n_G)$ dimensional total space, $E_G \sim M \times G$, built from an $n_M = n_S$ dimensional base, $M$, and $n_G$ dimensional fiber, $F = G$. This fiber of an ''Ehresmann Cartan geometry'' has a subgroup, $H \subset G$, so the bundle produces two [[associated]] bundles, the [[Cartan H-bundle]], $E_H \sim M \times H$, and the [[Cartan homogeneous space bundle]], $E_S \sim M \times S$. The Ehresmann Cartan connection gives the ''Ehresmann Cartan connection form'', $\f{\cal C} = \f{\ve{\cal C}} \f{\cal I} \in \f{\rm Lie}(G)$, which gives the associated [[Cartan H-bundle connection form|Cartan H-bundle]], $\f{{\cal C}_H}$, over $E_H$ and [[Cartan homogeneous space connection form|Cartan homogeneous space bundle]], $\f{{\cal C}_S}$, over $E_S$. A ''generalized Ehresmann Cartan geometry'' has $n_M \neq n_S$.

There is a convenient set of local coordinates for the total space. The $n_M$ coordinates, $x^a$, cover patches of the base manifold, $M$, the $n_H$ coordinates, $y^p$, correspond to elements $h(y) \in H \subset G$, and the remaining $n_S$ homogeneous space coordinates, $x_s^a$, correspond to $x_s \in S$. So the combined coordinates, $(x_s, y)$, cover patches of $G$ and the total combined coordinates, $(x,x_s,y)$, cover patches of $E_G$ -- so a point of $E_G$ may be written as
$$
p \sim (x,x_s,y) \sim (x,x_s,h(y)) \sim (x,g(x_s,y))
$$
The chosen [[coset representative section|homogeneous space]], $r : S \to G$, allows points of $G$ to be specified in terms of points of $G/H$ and $H$ via the right action, 
$$
g(x_s, y) = R_{h(y)} r(x_s) = r(x_s) \, h(y)
$$
The ''Cartan geometry [[reference section|Ehresmann gauge transformation]]'', $\si^G : M \to E_G$, is then determined by the reference section, $\si^H : M \to E_H$, of the Cartan H-bundle and the reference section, $\si^S : M \to E_S$, of the Cartan homogeneous space bundle. With $\si^H(x) = {\big (} x,h(y_\si(x)) {\big )}$ and $\si^S(x) = (x, x_{s\si}(x))$ we have:
$$
\si^G(x) = {\big (} x, r(x_{s\si}(x)) \, h(y_\si(x)) {\big )} 
$$
The [[canonical reference section|Ehresmann principal bundle connection]], $\si_0^G(x) = (x,1) \sim (x,0,0)$, of $E_G$ corresponds to the canonical reference section, $\si_0^H(x) = (x,1) \sim (x,0)$, of $E_H$ and the zero point reference section, $\si_0^S(x) = (x, 0)$, of $E_S$. 

The Ehresmann Cartan geometry total space, $E_G$, is not only a bundle over $M$ -- it is also a bundle over $E_H$ and over $E_S$. The fundamental bundle maps, $\pi^G_H : E_G \to E_H$ and $\pi^G_S : E_G \to E_S$, are given by $\pi^G_H(x,x_s,y)=(x,y)$ and $\pi^G_S(x,x_s,y) = (x,x_s)$. There are also reference sections, $\si'^S : E_H \to E_G$ and $\si'^H : E_S \to E_G$, over these bases, determined by the reference sections over their partner bundle, $\si'^S(x,y)=(x,x_{s\si}(x),y)$ and $\si'^H(x,x_s)=(x,x_s,y_\si(x))$. The complete web of bundle maps is summarized by:
$$
\begin{array}{ccc}
E_G & \matrix{\lower8mu {\overset{\si'^S}{\longleftarrow}}\\ \raise8mu {\underset{\pi^G_H}{\longrightarrow}}} & E_H\\
{}^{\pi^G_S} \! {\big \downarrow} {\big \uparrow} \! {}_{\si'^H} & {}_{\pi_G} \! \! \! \! \searrow \! \! \nwarrow \! \! \! \! {}^{\si^G} & {}^{\pi_H} \! {\big \downarrow} {\big \uparrow} \! {}_{\si^H}\\
E_S & \matrix{\lower8mu {\overset{\si^S}{\longleftarrow}}\\ \raise8mu {\underset{\pi_S}{\longrightarrow}}} & M
\end{array}
$$
and we have
$$
\begin{eqnarray}
\pi_G &=& \pi_H \circ \pi^G_H = \pi_S \circ \pi^G_S \\
\si^G &=& \si'^H \circ \si^S = \si'^S \circ \si^H
\end{eqnarray}
$$

The geometry of an Ehresmann Cartan geometry and its associated bundles is described by the [[Ehresmann Cartan connection]] and its curvature.
The geometry of a [[fiber bundle]] may be described via a [[connection]], $\f{A}$, and [[covariant derivative]], $\f{\na}$, defined over the base manifold, $M$, or alternatively via an ''Ehresmann connection'', $\f{\ve{\cal A}}$, defined over the total space, $E$, of the bundle. This [[vector valued form]] is a [[vector projection]], $\f{\ve{\cal A}}\f{\ve{\cal A}}=\f{\ve{\cal A}}$, that succinctly describes the geometric structure of the bundle, including the [[Lie group]] symmetry. As a projection, it splits the tangent vector space at each point, $p$, of $E$, into range and kernel subspaces,
$$
T_p E = V_r + V_0
$$
The range subspace of $\f{\ve{\cal A}}$ is the ''vertical subspace'', $V_r=V_V$, and the collection of these vertical vector fields over $E$ is an involutive [[distribution]], $\ve{\De_V}=\ve{\De_r}$, of vectors tangent to the fibers of the bundle, $\pi_* \ve{\De_V} = 0$. In this way, the Ehresmann connection determines the fibers of the fiber bundle. The vector fields, $\ve{\xi_A} \sim T_A$, in $\ve{\De_V}$ are the flow fields of the group action on the fibers. They are in involution since
$$
\lb \ve{\xi_A},\ve{\xi_B} \rb_L = C_{AB}{}^C \ve{\xi_C}
$$
The kernel subspace of $\f{\ve{\cal A}}$ is the ''horizontal subspace'', $V_0=V_H$, and the collection of these horizontal vector fields over $E$ form a ''horizontal distribution'', $\ve{\De_H}=\ve{\De_0}$, that may or may not be in involution (more on that further down). 

The Ehresmann connection respects the symmetry of the structure group. If we take the group action on manifold points to be a right action $p \mapsto R_g p = pg$, the Ehresmann connection at different points along fibers are related by the [[pullback]],
$$
R_g^* \f{\ve{\cal A}} (pg) = \f{\ve{\cal A}} (p)
$$
In this way the Ehresmann connection is related to the [[Ehresmann-Maurer-Cartan form|Maurer-Cartan form]] on the fiber. Note that the 1-form and vector parts of the Ehresmann connection are being pulled back -- in coordinates this could well be written as
$$
\lp R_g^* \f{dz^i} \rp {\cal A}_i{}^j (pg) \ve{\pa_j} = \f{dz^i} {\cal A}_i{}^j (p) \lp R_{g*} \ve{\pa_j} \rp
$$
The vertical and horizontal distributions satisfy
\begin{eqnarray}
\ve{\De_V} \f{\ve{\cal A}} &=& \ve{\De_V} \\
\ve{\De_H} \f{\ve{\cal A}} &=& 0
\end{eqnarray}
and so must also respect the symmetry of the structure group,
\begin{eqnarray}
\ve{\De_V} (p g) &=& R_{g*} \ve{\De_V} (p) \\
\ve{\De_H} (p g) &=& R_{g*} \ve{\De_H} (p)
\end{eqnarray}
so knowing the distributions at any point $p$ in $E$ implies the distributions at any other point on the fiber containing $p$.

Iff the [[FuN curvature]] of the Ehresmann connection vanishes,
$$
\ff{\ve{{\cal F}}} = - \ha \lb \f{\ve{\cal A}},\f{\ve{\cal A}} \rb_L = 0
$$
the horizontal distribution is also in involution and may be integrated to get ''horizontal section''s. 

Refs:
*http://philsci-archive.pitt.edu/archive/00002133/01/geometrie.pdf
*http://www.mat.univie.ac.at/~michor/gaubook.pdf
*http://www.mat.univie.ac.at/~michor/listpubl.html
*http://www.emis.de/monographs/KSM/index.html
For any [[vector valued form]] field, $\nf{\ve{\cal K}}$, on the total space of a [[fiber bundle]], a natural grade 1 [[derivation]] is provided by the [[FuN derivative]] with respect to the [[Ehresmann connection]], defining the ''Ehresmann covariant derivative'',
$$
\f{\cal D} \nf{\ve{\cal K}} = - {\cal L}_{\f{\ve{\cal A}}} \nf{\ve{\cal K}}
$$
Once a choice of [[gauge|Ehresmann gauge transformation]] is made, the Ehresmann connection may be expressed in local coordinates as
$$
\f{\ve{\cal A}}(x,y) = \f{A^B}(x) \ve{\xi_B}(y) + \f{\ve{W}}(y)
$$
If the vector valued form field is right invariant over the total space and may be written as
$$
\nf{\ve{\cal K}} = \nf{K^B}(x) \ve{\xi_B}(y)
$$
then, using a couple of [[FuN identities]], its Ehresmann covariant derivative is
\begin{eqnarray}
\f{\cal D} \nf{\ve{\cal K}} &=& - \lb \f{\ve{\cal A}}, \nf{\ve{\cal K}} \rb_L = - \lb \f{A^B}(x) \ve{\xi_B}(y), \nf{K^C}(x) \ve{\xi_C}(y) \rb_L - \lb \f{\ve{W}}(y), \nf{K^C}(x) \ve{\xi_C}(y) \rb_L \\
&=& - \f{A^B} \nf{K^C} \lb \ve{\xi_B}, \ve{\xi_C} \rb_L - \f{A^B} \lp {\cal L}_{\ve{\xi_B}} \nf{K^C} \rp \ve{\xi_C}
+ \lp {\cal L}_{\ve{\xi_C}} \f{A^B} \rp \nf{K^C} \ve{\xi_B}
+ \lp \f{d} \f{A^B} \rp \ve{\xi^B} \nf{K^C} \ve{\xi_C}
+ \lp \ve{\xi^C} \f{A^B} \rp \lp \f{d} \nf{K^C} \rp \ve{\xi_B} \\
&-& \lp \f{\ve{W}} \f{\pa} \rp \nf{K^C} \ve{\xi_C} + \lp -1 \rp^k \lp \nf{K^C} \ve{\xi_C} \f{\pa} \rp \f{\ve{W}} + \lp \f{\pa} \f{\ve{W}} \rp \nf{K^C} \ve{\xi_C} + \lp \f{\pa} \nf{K^C} \ve{\xi_C} \rp \f{\ve{W}} \\
&=& \lp \f{d} \nf{K^C} \rp \ve{\xi_C} - \f{A^B} \nf{K^C} \lb \ve{\xi_B}, \ve{\xi_C} \rb_L
\end{eqnarray}
An [[Ehresmann connection]] may be described in local coordinates by choosing a ''reference [[section|fiber bundle]]'', $\si_0$, that maps from some base manifold, $M$, to the total space, $E$. If coordinates $x^a$ are used in a local patch over $M$, and coordinates $y^p$ are used in patches over a typical fiber, these coordinates can be chosen so $y=0$ on the reference section, and the Ehresmann connection can be written locally over $E$ as
$$
\f{\ve{\cal A}}(x,y) = \f{dx^a} A_a{}^B(x) \xi_B{}^p(y) \ve{\pa_p} + \f{dy^p} \ve{\pa_p} = \f{A^B} \ve{\xi_B} + \f{\ve{\cal I}}
$$
In which $\ve{\xi_B}$ are the [[right (or left) invariant vector fields|Lie group geometry]] on the fibers. Another section ([[gauge|gauge transformation]]), $\si:M \rightarrow E$, may be chosen by flowing the original section along a [[diffeomorphism]] along the fibers, $\ph(x,y) = (x,y_\ph(x,y))$, to $\si = \ph \circ \si_0$. The new section is described in the original coordinates by $y^p_\si(x)$. Since the Ehresmann connection is valued in $TE$ it can't be [[pulled back|pullback]] along the section; however, the [[vector projection onto a section]],
$$
\f{\ve{P_\si}} = \f{dx^a} \ve{\pa_a} + \f{dx^a} \fr{\pa y_\si^p}{\pa x^a} \ve{\pa_p}
$$
can be used to project to the TE valued 1-form on the section,
$$
\f{\ve{{\cal A}_\si}} = \f{\ve{P_\si}} \f{\ve{\cal A}} = \f{dx^a} \lp A_a{}^B(x) \xi_B{}^p(y_\si) + \fr{\pa y_\si^p}{\pa x^a} \rp \ve{\pa_p}
$$
The Ehresmann connection everywhere in the total space is determined by the connection components, $A_a{}^B(x)$, on a chosen section. Changing to the connection on a different section is called a passive [[gauge transformation]].

An alternative way of effecting a gauge transformation is to flow the Ehresmann connection by the diffeomorphism, $\f{\ve{\cal A'}} = \phi^*\f{\ve{\cal A}}$, along the fibers while projecting it onto the original section, $\si_0$. This is called an ''active gauge transformation'', and gives the same result,
$$
\f{\ve{{\cal A'}_{\si_0}}} = \f{\ve{P_{\si_0}}} \f{\ve{\cal A'}} = \f{\ve{P_{\si_0}}} \ph^* \f{\ve{\cal A}} = \f{\ve{P_{\ph \circ \si_0}}} \f{\ve{\cal A}} = \f{\ve{P_\si}} \f{\ve{\cal A}} = \f{\ve{\cal A_\si}}
$$

Ref:
*[[Geometrical aspects of local gauge symmetry|papers/Geometrical aspects of local gauge symmetry.pdf]]
It was enlightening to consider the [[Ehresmann principal bundle connection]] as a construction in the entire space, $E$, of a [[principal bundle]] with base space, $M=S$. It is equally enlightening to consider the ''Ehresmann homogeneous space geometry'' as the analogous construction in the [[Lie group geometry]], $G$, with a base space that is a [[homogeneous space geometry]], $S = G/H$.

Points of the [[homogeneous space]], $x = [r(x)] \in M = G/H$, are mapped to $G$, by the homogeneous reference section, $r: S \to G$, and any point (element) in $G$, as a function of coordinates $x$ of $M$ and $y$ of $H$, may be specified by
$$
g(z) = g(x,y) = r(x) h(y) \in G
$$
in which $h(y) \in H$ operates on $r(x) \in G$ via the [[right action|group]]. In these coordinates, adapted to the reference section, the reference section is the [[submanifold]] corresponding to $y=0$. The [[Maurer-Cartan form]], $\f{\cal I} = \f{{\cal I}^J} T_J$, over $G$, is
$$
\begin{eqnarray}
\f{\cal I}(z) &=& g^- \f{d} g = h^- \lp r^- \f{d} r \rp h + h^- \f{d} h \\
&=& h^- \f{I} h + h^- \f{d} h \\
&=& \f{{\cal E}_S} + \f{{\cal A}_S}
\end{eqnarray}
$$
When $H$ is [[reductive]] in $G$ (which is assumed) the [[Maurer-Cartan frame|homogeneous space geometry]],
$$
\f{I}(x) = r^- \f{d} r = \f{e_S} + \f{A_S}
$$
splits into the homogeneous space frame, $\f{e_S}(x) = \f{e_S^A} K_A \in \f{\rm Lie}(G/H)$, and homogeneous H-connection, $\f{A_S}(x) = \f{A_S^P} H_P \in \f{\rm Lie}(H)$. These correspond to the ''Ehresmann homogeneous space frame form'' and ''Ehresmann homogeneous H-connection form'' over $G$,
$$
\begin{eqnarray}
\f{{\cal E}_S} = \f{{\cal E}_S^A} K_A &=& h^- \f{e_S} h \in \f{\rm Lie}(G/H) \\
\f{{\cal A}_S} = \f{{\cal A}_S^P} H_P &=& h^- \f{A_S} h + h^- \f{d} h \in \f{\rm Lie}(H)
\end{eqnarray}
$$
Their 1-form components are computed using the [[Killing form]],
$$
\begin{eqnarray}
\f{{\cal E}_S^A}(z) &=& \f{e_S^B} \lp K^A, h^- K_B h \rp = \lp L^h \rp^A{}_B \, \f{e_S^B}(x) \\
\f{{\cal A}_S^P}(z) &=& \f{A_S^Q} \lp H^P, h^- H_Q h \rp + \lp H^P, h^- \f{d} h \rp = \lp L^h \rp^P{}_Q \, \f{A_S^Q}(x) + \f{{\cal I}_H^P}(y)
\end{eqnarray}
$$
with the appearance of the [[left-right rotator]]s, $\lp L^h \rp^I{}_J(y)$, and the Maurer-Cartan form, $\f{{\cal I}_H}$, for $H$. These are the same as the frame components, $\f{e^A}(z) = \f{{\cal E}_S^A}(z)$ and $\f{e^P}(z) = \f{{\cal A}_S^P}(z)$, for a [[reductive Lie group geometry]]. Using the correspondence between the Lie algebra generators and the left invariant vector fields of the Lie group geometry, $T_I \sim \ve{\xi^R_I}$, allows us to write the [[Ehresmann-Maurer-Cartan vector valued form|Maurer-Cartan form]] as
$$
\begin{eqnarray}
\f{\ve{\cal I}} &=& \f{\ve{{\cal E}_S}} + \f{\ve{{\cal A}_S}} \\
&=& \f{{\cal E}_S^A} \ve{\xi^R_A} + \f{{\cal A}_S^P} \ve{\xi^R_P} \\
&=& \f{e_S^B} \lp L^h \rp^A{}_B \, \ve{\xi^R_A} + \f{A_S^Q} \lp L^h \rp^P{}_Q \, \ve{\xi^R_P} + \f{\ve{{\cal I}_H}} \\
&=& \f{I^J} \lp L^r \rp_J{}^K \, \ve{\xi^L_K} + \f{\ve{{\cal I}_H}}
\end{eqnarray}
$$
with the ''Ehresmann homogeneous space frame'', $\f{\ve{{\cal E}_S}}$, and ''Ehresmann homogeneous H-connection'', $\f{\ve{{\cal A}_S}}$, satisfying $\f{\ve{{\cal E}_S}} \f{\cal I} = \f{{\cal E}_S}$ and $\f{\ve{{\cal A}_S}} \f{\cal I} = \f{{\cal A}_S}$ using the Maurer-Cartan form, $\f{\cal I} = \f{\xi_R^J}(z) T_J$, over $G$. The Ehresmann homogeneous H-connection, $\f{\ve{{\cal A}_S}} = \f{{\cal A}_S^P} \ve{\xi^R_P}(z)$, is a [[Ehresmann principal bundle connection]] for an H-bundle and satisfies $\f{\ve{{\cal A}_S}} \f{\cal I_H} = \f{{\cal A}_S}$, since $\ve{\xi^R_P}(z) = \ve{\xi^{HR}_P}(y)$ in a reductive Lie group geometry. Another way of looking at an Ehresmann homogeneous space geometry is as a [[Cartan H-bundle]] with $\f{C} = \f{I}$.

Since the Ehresmann-Maurer-Cartan VVF is the identity projection, its [[FuN curvature]] vanishes, $\ff{\ve{\cal F}} = -\ha \lb \f{\ve{\cal I}}, \f{\ve{\cal I}} \rb_L = 0$.
from [[Ehresmann connection]]

equivalent to [[parallel transport]]
[<img[images/png/fiber bundle.png]]A [[principal bundle]] consists of a total space, $E$, built locally from the direct product of a [[Lie group geometry]] (the typical fiber, $F=G$) over a base manifold, $M$. The same [[Lie group]], $G$, is the structure group, acting on the fibers, and hence on the total space, via left action. This group also acts on the fibers, and the total space, via right action. A [[connection]], $\f{A}(x)=\f{A^B}T_B=\f{dx^a}A_a{}^BT_B$, a [[Lieform]] over the base space, describes principal bundle geometry. By choosing a [[reference section|Ehresmann gauge transformation]], $\si_0:M \to E$, this connection may be related to an [[Ehresmann connection]], $\f{\ve{\cal A}}$, over the total space.

There is a convenient set of local coordinates for the total space. The $n_M$ coordinates, $x^a$, with [[spacetime]] [[indices]], cover patches of the base manifold and the $n_G$ coordinates, $y^p$, are from the typical fiber. So a point of $E$ may be described by $p=(x,y)$ or equivalently by $p=(x,g)$ -- where $g(y)$ is the Lie group (fiber) element parameterized by $y$. The fiber bundle projection is then simply $\pi(x,y)=x$. The coordinates are chosen so that $y^p=0$, and hence $g=1$, on the ''canonical reference section'', $\si_0(x)=(x,y_0(x))=(x,0) \sim (x,1)$, which provides the ''canonical local trivialization'', $(x,g) \in E$. With these coordinates each fiber corresponds to a coordinate surface of constant $x$. The $y$ [[coordinate basis vectors]] are in the vertical subspace, $\ve{\pa_p} \in \ve{\De_V}$, but the $x$ coordinate basis vectors are not necessarily in the horizontal subspace, $\ve{\pa_a} \notin \ve{\De_H}$. We will abuse the use of the same label, $x^a$, for the coordinates on the base and some on the total space. Using these coordinates, the ''Ehresmann principal bundle connection'' (a [[vector projection]]) over the total space is
$$
\f{\ve{\cal A}}(x,y) = \f{A}^B(x) \ve{\xi}{}^L_B(y) + \f{\ve{\cal I}}
$$
in which $\ve{\xi^L_B}$ are the [[left action vector fields|Lie group geometry]] for the Lie group geometry, defined by
$$
\ve{\xi^L_B} \f{\pa} g(y) = T_B g
$$
and
$$
\f{\ve{\cal I}} = \f{\xi_L^B}(y) \ve{\xi^L_B}(y) = \f{\xi_R^B}(y) \ve{\xi^R_B}(y) = \f{dy^p} \ve{\pa_p}
$$
is the [[Ehresmann-Maurer-Cartan form|Maurer-Cartan form]] (the [[identity projection|vector projection]] along the fibers). The Ehresmann connection is a projection, $\f{\ve{\cal A}} \f{\ve{\cal A}} = \f{\ve{\cal A}}$, is [[right invariant]], $R_h^*\f{\ve{\cal A}} = \f{\ve{\cal A}}$, and has a natural [[spectral decomposition|spectral decomposition of the Ehresmann principal bundle connection]]. It may also be written as a Lie algebra valued 1-form over the total space by [[contracting|vector-form algebra]] it with the [[Maurer-Cartan form]], $\f{\cal I}(y) = \f{\xi_R^B}(y)T_B = g^- \f{d} g$, over the total space to get the ''Ehresmann connection form'',
\begin{eqnarray}
\f{\cal A}(x,y) &=& \f{\ve{\cal A}} \f{\cal I} = \f{A^B}(x) \ve{\xi^L_B}(y) \f{\xi_R^C}(y) T_C + \f{\ve{\cal I}} \f{\cal I} \\
&=& \lp \f{A^B} L^C{}_B(y) + \f{\xi_R^C} \rp T_C \\
&=& g^-(y) \f{A}(x) g(y) + g^-(y) \f{d^y} g(y) 
\end{eqnarray}
using the defining equation for the [[left-right rotator]],
$$
L^C{}_B(y) = \ve{\xi^L_B}(y) \f{\xi_R^C}(y) = \lp T^C, g^-(y) T_B g(y) \rp 
$$
This form pulls back along the canonical reference section ($y=0$) to give the principal bundle connection,
$$
\si_0^*\f{\cal A} = \f{A}(x)
$$
and satisfies $R_h^* \f{\cal A} = h^- \f{\cal A} h$ under the right action.

The ''[[FuN curvature]] of the Ehresmann principal bundle connection'' is
\begin{eqnarray}
\ff{\ve{\cal F}}(x,y) &=& - \ha \lb \f{\ve{\cal A}}, \f{\ve{\cal A}} \rb_L = \lp 1 - \f{\ve{\cal A}} \rp \lp \f{\pa} \f{\ve{\cal A}} \rp = \lp 1 - \f{\ve{\cal A}} \rp \lp \lp \f{\pa^x} + \f{\pa^y} \rp \f{\ve{\cal A}} \rp \\
&=& \lp 1 - \f{A^B}(x) \ve{\xi^L_B}(y) - \f{dy^p} \ve{\pa_p} \rp 
\lp \lp \f{d^x} \f{A^C} \rp \ve{\xi^L_C}(y) - \f{A^C} \f{\pa^y} \ve{\xi^L_C} \rp \\
&=& \lp \f{d^x} \f{A^C} \rp \ve{\xi^L_C} - \f{A^B} \f{A^C} \ve{\xi^L_B} \f{\pa^y} \ve{\xi^L_C} \\
&=& \lp \f{d^x} \f{A^D} + \ha \f{A^B} \f{A^C} C_{BC}{}^D \rp \ve{\xi^L_D}(y)
\end{eqnarray}
using the [[Lie bracket for left action vector fields|Lie group geometry]],
$$
\ve{\xi^L_{\lb B \rd}} \lp \f{\pa} \ve{\xi^L_{\ld C \rb}} \rp = \ha \lb \ve{\xi^L_B} , \ve{\xi^L_C} \rb_L = - \ha C_{BC}{}^D \ve{\xi^L_D}
$$
This curvature is vector valued in the vertical subspace, and is right invariant, $R_h^* \ff{\ve{\cal F}} = \ff{\ve{\cal F}}$. The ''FuN curvature form of the Ehresmann connection'' is a $Lie(G)$ valued 2-form over $E$,
\begin{eqnarray}
\ff{\cal F} &=& \ff{\ve{\cal F}} \f{\cal I} =  \lp \f{d^x} \f{A^D} + \ha \f{A^B} \f{A^C} C_{BC}{}^D \rp g^-(y) T_D g(y) \\
&=& g^-(y) \lp \f{d^x} \f{A} + \ha \lb \f{A}, \f{A} \rb \rp g(y) \\
&=& g^-(y) \lp \f{d^x} \f{A} + \f{A} \f{A} \rp g(y)
\end{eqnarray}
This form pulls back along the canonical reference section to give the [[principal bundle]] curvature,
$$
\si_0^*\ff{\cal F} = \f{d} \f{A} + \f{A} \f{A} = \ff{F}(x)
$$
and satisfies $R_h^* \ff{\cal F} = h^- \ff{\cal F} h$ under the right action.
The [[Ehresmann covariant derivative]] of a [[vector valued form]] field, $\nf{\ve{\cal K}}$, over the total space of a [[principal bundle]] using an [[Ehresmann principal bundle connection]] is
$$
\f{\cal D} \nf{\ve{\cal K}} = - {\cal L}_{\f{\ve{\cal A}}} \nf{\ve{\cal K}}
$$
using the [[FuN derivative]]. The VVF will usually be right invariant and expressible as
$$
\nf{\ve{\cal K}} = \nf{K^B}(x) \ve{\xi^L_B}(y)
$$
corresponding to the [[Lieform]],
$$
\nf{\cal K} = \nf{\ve{\cal K}} \f{\cal I} = \nf{K^B}(x) g^-(y) T_B g(y) 
$$
obtained with the [[Maurer-Cartan form]], $\f{\cal I}(y) = \f{\xi_R^B} T_B$, and the [[left-right rotator]]. The [[pullback]] of this form along a section, $\si_1=(x,g_1(x))$, gives the Lieform over the base,
$$
\nf{K_1}(x) = \si_1^* \nf{\cal K} = \nf{K^B}(x) g^-_1(x) T_B g_1(x) = g^-_1(x) \nf{K}(x) g_1(x)
$$
in which the form pulled back along the reference section is $\nf{K}(x) = \nf{K^B}(x) T_B$. The Ehresmann covariant derivative of the VVF,
$$
\f{\cal D} \nf{\ve{\cal K}} = \lp \f{d} \nf{K^C} \rp \ve{\xi^L_C} - \f{A^B} \nf{K^C} \lb \ve{\xi^L_B}, \ve{\xi^L_C} \rb_L
= \lp \f{d} \nf{K^D} + \f{A^B} \nf{K^C} C_{BC}{}^D \rp \ve{\xi^L_D}
$$
gives a definition for the ''Ehresmann covariant derivative of a Lieform'',
$$
\f{\cal D} \nf{\cal K} = \lp \f{\cal D} \nf{\ve{\cal K}} \rp \f{\cal I} = \lp \f{d} \nf{K^D} + \f{A^B} \nf{K^C} C_{BC}{}^D \rp g^-(y) T_D g(y)
$$
which pulls back along any chosen section to give the [[covariant derivative|principal bundle]] of $\nf{K}$ on the base, 
\begin{eqnarray}
\lp \f{D_1} \nf{K_1} \rp(x) &=& \si_1^* \lp \f{\cal D} \nf{\cal K} \rp 
= \lp \f{d} \nf{K^D} + \f{A^B} \nf{K^C} C_{BC}{}^D \rp g^-_1(x) T_D g_1(x) \\
&=& g^-_1(x) \lp \f{d} \nf{K} + \lb \f{A}, \nf{K} \rb \rp g_1(x)
= g^-_1(x) \lp \f{\na} \nf{K} \rp g_1(x)
\end{eqnarray}
(This should be equivalent to the usual definition of the Ehresmann covariant derivative of Lieform via
$$
\ve{v_1} \ve{v_2} \dots \ve{v_{k+1}} \f{\cal D} \nf{\cal K} = \ve{v^H_1} \ve{v^H_2} \dots \ve{v^H_{k+1}} \f{d} \nf{\cal K}(x,y)
$$
in which the vectors on the right are horizontal projections of the ones on the left, $\ve{v^H} = \ve{v}(1-\f{\ve{\cal A}})$. //That definition needs to be checked.//)

Similarly, if a VVF is left invariant and may be expressed as
$$
\nf{\ve{\cal K}} = \nf{K^B}(x) \ve{\xi^R_B}(y)
$$
corresponding to the Lieform,
$$
\nf{\cal K} = \nf{\ve{\cal K}} \f{\cal I} = \nf{K^B}(x) T_B 
$$
The [[pullback]] of this form along any section, $\si_1=(x,g_1(x))$, gives the same Lieform over the base,
$$
\nf{K_1}(x) = \si_1^* \nf{\cal K} = \nf{K^B}(x) T_B = \nf{K}(x)
$$
The Ehresmann covariant derivative of this VVF,
$$
\f{\cal D} \nf{\ve{\cal K}} = \lp \f{d} \nf{K^C} \rp \ve{\xi^R_C} - \f{A^B} \nf{K^C} \lb \ve{\xi^L_B}, \ve{\xi^R_C} \rb_L
= \lp \f{d} \nf{K^C} \rp \ve{\xi^R_C}
$$
gives the Lieform,
$$
\f{\cal D} \nf{\cal K} = \lp \f{\cal D} \nf{\ve{\cal K}} \rp \f{\cal I} = \lp \f{d} \nf{K^C} \rp T_C = \f{d} \nf{\cal K}
$$
which pulls back along any chosen section to give the [[exterior derivative]] of $\nf{K}$ on the base, 
$$
\lp \f{D_1} \nf{K_1} \rp = \si_1^* \lp \f{\cal D} \nf{\cal K} \rp
= \f{d} \nf{K}
$$
A passive [[Ehresmann gauge transformation]] for an [[Ehresmann principal bundle connection]] corresponds to changing to a different choice of section along which to pull back the Ehresmann connection form. The choice of reference section is equivalent to the choice of a local trivialization for a [[fiber bundle]]. Once a reference section, $\si_0:M \to E$, and principal bundle connection, $\f{A}$, have been used to build the principal bundle Ehresmann connection, $\f{\ve{\cal A}}$, a different section, $\si_1$, can be introduced and used to pull back a different principal bundle connection, $\f{A'}$ -- this is a [[gauge transformation]]. Using coordinates adapted to the reference section, the Ehresmann connection is
$$
\f{\ve{\cal A}}(x,y) = \f{A^B} \ve{\xi^L_B}(y) + \f{\ve{\cal I}}
$$
and the Ehresmann connection form is
$$
\f{\cal A} = \f{\ve{\cal A}} \f{\cal I} = g^-(y) \f{A}(x) g(y) + g^- \f{d^y} g(y)
$$
using the [[Maurer-Cartan form]], $\f{\cal I}(y) = \f{\xi_R^B} T_B$, and [[left-right rotator]]. The new section, $\si_1 = \ph \circ \si_0$, may be obtained by flowing the reference section by an equivariant vertical [[diffeomorphism]], $\ph(x,y) = (x,y_\ph(x,y))$, satisfying $\ph(ph)=\ph(p)h$ and giving $\si_1(x) = (x,y_1(x)) = (x,y_\ph(x,0))$. This is equivalent to transforming the original section by right (//?//) action by an element of $G$ to $\si_1(x) = \si_0(x) \, g(y_1(x)) = \si_0 \, g_1(x)$. The [[vector projection onto a section]], $\si_1$, is
$$
\f{\ve{P_1}} = \f{dx^a} \ve{\pa_a} + \f{dx^a} \fr{\pa y_1^p}{\pa x^a} \ve{\pa_p}
$$
and is used to project the Ehresmann connection to
$$
\f{\ve{{\cal A}_1}} = \f{\ve{P_1}} \f{\ve{\cal A}} = \f{A^B}(x) \ve{\xi^L_B}(y_1) + \f{dx^a} \fr{\pa y_1^p}{\pa x^a} \ve{\pa_p}
$$
and the Ehresmann connection form to
$$
\f{{\cal A}_1} = \f{\ve{P_1}} \f{\cal A} = \f{\ve{{\cal A}_1}} \f{\cal I} = \f{\ve{P_1}} \f{\ve{\cal A}} \f{\cal I} = g^-(y_1(x)) \f{A}(x) g(y_1(x)) + g^-(y_1(x)) \f{d^x} g(y_1(x)) 
$$
on the section. This, the gauge transformed connection, gives the [[pullback]] of the Ehresmann connection form along the section,
$$
\f{A'}(x) = \si_1^* \f{\cal A} = \si_1^* \f{{\cal A}_1} = g^-_1(x) \f{A} g_1(x) + g^-_1(x) \f{d^x} g_1(x) 
$$
identified as the [[principal bundle gauge transformation|principal bundle]] with $g_1(x)=g^-(x)$.

An alternative way of effecting a gauge transformation is to flow the Ehresmann connection form by the diffeomorphism, $\f{\cal A'} = \ph^*\f{\cal A}$, then pull it back along the original section, $\si_0$. This ''active gauge transformation'' gives the same result,
$$
\si_0^* \f{\cal A'} = \si_0^* \lp \ph^* \f{\cal A} \rp
= \lp \ph \circ \si_0 \rp^* \f{\cal A}
= \si_1^* \f{\cal A} = \f{A'}
$$

Ref:
*[[Geometrical aspects of local gauge symmetry|papers/Geometrical aspects of local gauge symmetry.pdf]]
[[Ehresmann lift]] for an [[Ehresmann principal bundle connection]].
A [[vector bundle]] consists of a total space, $E$, built locally from the direct product of a [[vector space]] (the typical fiber, consisting of elements $v = v^\al b_\al \in V = F$) and a base manifold, $M$. A [[vector bundle connection]], $\f{A}{}_\al{}^\be(x) = \f{dx^i} A_{i \al}{}^\be(x)$, a 1-form over the base space valued in some subgroup of the general linear group, describes the geometry of the vector bundle. By choosing a ''reference [[section|fiber bundle]]'', $\si_0:M \to E$, this connection may be related to an [[Ehresmann connection]], $\f{\ve{\cal A}}$, over the total space.

There is a convenient set of local coordinates for the total space. The $n$ coordinates, $x^a$, with [[spacetime]] [[indices]], are from the base manifold and the $K$ coordinates, $v^\al$, are from the typical fiber (vector space). So a point of $E$ (above some patch of $M$) may be described by $p = (x,v)$. The fiber bundle projection is then simply $\pi(x,v) = x$. The coordinates are chosen so that $v^\al = 0$ on the reference section, a ''canonical local trivialization'', $\si_0(x) = (x,v_0(x)) = (x,0)$. With these coordinates each fiber corresponds to a coordinate surface of constant $x$. The [[coordinate basis vectors]] for the $v^\al$ coordinates, $\ve{\pa_\al} \in \ve{\De_V}$, are in the vertical subspace, but the $x$ coordinate basis vectors are not necessarily in the horizontal subspace, $\ve{\pa_a} \notin \ve{\De_H}$. We will abuse the use of the same label, $x^a$, for the coordinates on the base and some on the total space. Using these coordinates, the linear ''Ehresmann vector bundle connection'' over the total space is
$$
\f{\ve{\cal A}}(x,v) = \lp \f{dx^i} A_{i \al}{}^\be(x) v^\al + \f{dv^\be} \rp \ve{\pa_\be}
$$

In analogy with the [[Maurer-Cartan form]], we build a vector ($V$) valued 1-form,
$$
\f{\cal I} = \f{dv^\be} b_\be
$$
and use it to define the ''Ehresmann vector bundle connection form'',
$$
\f{\cal A} = \f{\ve{\cal A}} \f{\cal I} = \lp \f{A}{}_\al{}^\be v^\al + \f{dv^\be} \rp b_\be
$$
This allows us to define the [[vector bundle covariant derivative|vector bundle connection]] of any section, $\si_1 : M \mapsto E$, $\si_1(x) = (x, v_1(x))$, as the [[pullback]] of $\f{\cal A}$ along the section to M,
$$
\si_1^* \f{\cal A} = \f{A}{}_\al{}^\be v_1^\al(x) b_\be + \f{dx^a} \fr{\pa v_1^\be}{\pa x^a} b_\be = \f{\na} v_1(x)
$$ 

The ''[[FuN curvature]] of the Ehresmann vector bundle connection'' is
\begin{eqnarray}
\ff{\ve{\cal F}}(x,y) &=& - \ha \lb \f{\ve{\cal A}}, \f{\ve{\cal A}} \rb_L = \lp 1 - \f{\ve{\cal A}} \rp \lp \f{\pa} \f{\ve{\cal A}} \rp \\
&=& \lp 1 - \f{A}{}_\al{}^\be v^\al \ve{\pa_\be} - \f{dv^\be} \ve{\pa_\be} \rp 
\lp \lp \f{d^x} \f{A}{}_\ga{}^\de \rp v^\ga \ve{\pa_\de} - \f{A}{}_\ga{}^\de \f{dv^\ga} \ve{\pa_\de} \rp \\
&=& \lp \f{d^x} \f{A}{}_\al{}^\de - \f{A}{}_\al{}^\be \f{A}{}_\be{}^\de \rp v^\al \ve{\pa_\de} \\
&=& \ff{F}{}_\al{}^\de v^\al \ve{\pa_\de}
\end{eqnarray}
in which the [[vector bundle curvature]], $\ff{F}{}_\al{}^\de$, appears.

The [[Ehresmann covariant derivative]] of any [[vector valued form]] over the total space (such as the curvature above) that can be written as
$$
\nf{\ve{\cal K}} = \nf{K}{}_\ga{}^\de(x) v^\ga \ve{\pa_\de}
$$
is defined using the [[FuN derivative]] as
\begin{eqnarray}
\f{\cal D} \nf{\ve{\cal K}} &=& - {\cal L}_{\f{\ve{\cal A}}} \nf{\ve{\cal K}}
= - \f{\ve{\cal A}} \lp \f{\pa} \nf{\ve{\cal K}} \rp + \lp -1 \rp^k \nf{\ve{\cal K}} \lp \f{\pa} \f{\ve{\cal A}} \rp + \f{\pa} \lp \nf{\ve{\cal K}} \f{\ve{\cal A}} \rp \\
&=& - \lp \f{A}{}_\al{}^\be(x) v^\al + \f{dv^\be} \rp \ve{\pa_\be} \lp \lp \f{d^x} \nf{K}{}_\ga{}^\de \rp v^\ga \ve{\pa_\de} + \lp -1 \rp^k \nf{K}{}_\ga{}^\de \f{dv^\ga} \ve{\pa_\de} \rp \\
& & + \lp -1 \rp^k \nf{K}{}_\ga{}^\de v^\ga \ve{\pa_\de} \lp \lp \f{d^x} \f{A}{}_\la{}^\be \rp v^\al \ve{\pa_\be} - \f{A}{}_\al{}^\be \f{dv^\al} \ve{\pa_\be} \rp
+ \f{\pa} \lp \nf{K}{}_\ga{}^\de v^\ga \ve{\pa_\de} \rp \\
&=& \lp \f{d^x} \nf{K}{}_\ga{}^\de - \f{A}{}_\ga{}^\be \nf{K}{}_\be{}^\de + \f{A}{}_\be{}^\de \nf{K}{}_\ga{}^\be \rp v^\ga \ve{\pa_\de} \\
&=& \lp \f{\na} \nf{K}{}_\ga{}^\de \rp v^\ga \ve{\pa_\de}
\end{eqnarray}

An [[Ehresmann gauge transformation]] corresponding to a change in local trivialization, $b_\be \mapsto b'_\be = g_\be{}^\al(x) b_\al$, gives changes in $\f{A}{}_\al{}^\be$ and other coefficients corresponding to a [[vector bundle gauge transformation]].
In a [[spacetime]], equivalent to a [[Cl(1,3)]] or [[Cl(3,1)]] [[Clifford vector bundle]], the [[Clifford-Ricci curvature]], $\f{R}$, [[Clifford curvature scalar]], $R$, [[frame]], $\f{e}$, ''cosmological constant'', $\La$, and ''Clifford energy-momentum tensor'', $\f{T}$, for matter are dynamically related by ''Einstein's equation'',
$$
\f{R} - \ha R \f{e} = \et_{00} ( \La \f{e} - 8 \pi G \f{T} ) 
$$
in which $\et_{00}$ specifies the [[Minkowski metric]] sign convention. In a vacuum, $\f{T} = 0$, Einstein's equation contracted with the coframe, $\ve{e}$, gives
$$
\ve{e} \cdot \lp \f{R} - \ha R \f{e} \rp = R - \ha R n = \et_{00} \La n
$$
requiring the curvature scalar to be constant, $R = - \fr{2n}{n-2} \et_{00} \La = - 4 \et_{00} \La$ (with $n=4$), and giving the ''vacuum Einsten's equation'',
$$
\f{R} = - \fr{2}{n-2} \et_{00} \La \f{e} = - \et_{00} \La \f{e}
$$
Any spacetime that satisfies $\f{R} = \al \f{e}$ for some constant, $\al$, is an ''Einstein space''.

Einstein's equation is derived by extremizing the ''Einstein-Hilbert action'',
$$
S = \int \nf{e} \lp \fr{1}{16 \pi G} \lp R + 2 \et_{00} \La \rp + L_M \rp
$$
with respect to $\ve{e}$, in natural [[units]].
http://arxiv.org/abs/gr-qc/0606062
*looks to be a good reveiw of ECT

In ECT...
The curvature picks up a contribution from torsion, and the Ricci curvature is no longer guaranteed to be symmetric in its indices. This change in the equation of motion allows matter with a spin component to couple to the angular momentum of the gravitational field.

In teleparallel theories of gravity the spin connection is purely torsional ($\f{\nu}=0$, $\f{d} \f{e}=0$, $\f{\ka} \neq 0$) and the [[spacetime]] is, in that sense, flat, with the gravitational field a force represented solely by torsion.

Ref:
[[Huang - Cosmological Solutions with Torsion in a Model of de Sitter Gauge Theory of Gravity|papers/Huang - Cosmological Solutions with Torsion in a Model of de Sitter Gauge Theory of Gravity.pdf]]
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<td><div class="math">
\begin{array}{c}
W =
\lb \begin{array}{cc}
{\small \frac{i}{2}} W^3 & W^+ \\
W^- & {\small - \! \frac{i}{2}} W^3
\end{array} \rb \vp{|_{\Big(}}
\quad
B_1 =
\lb \begin{array}{cc}
{\small \frac{i}{2}} B_1^3 & B_1^+ \\
B_1^- & {\small - \! \frac{i}{2}} B_1^3
\end{array} \rb \vp{|_{\Big(}}
\\
\big[
\lb \begin{array}{cc}
W & \\
& B_1
\end{array} \rb
,
\lb \begin{array}{cc}
 & \ph_B \\
\ph_W &
\end{array} \rb
\big] \vp{|_{\Big(}}
\\
\qquad \qquad \qquad \qquad \quad
\ph_{W/B} =
\lb \begin{array}{cc}
-  \ph_{0/1} & \ph_+ \\
\ph_- & \ph_{1/0}
\end{array} \rb \vp{|_{\Big(}}
\\
\lb \begin{array}{cc}
W & \\
& B_1
\end{array} \rb
\quad
\lb \begin{array}{c}
\nu_{eL} \\ e_L \\ \nu_{eR} \\ e_R
\end{array} \rb
\quad
\lb \begin{array}{c}
u_L \\ d_L \\ u_R \\ d_R
\end{array} \rb \\[.5em]
\big( \fr{\sqrt{3}}{\sqrt{5}} B_1^3 - \fr{\sqrt{2}}{\sqrt{5}} B_2 \big)
= (\fr{\sqrt{3}}{\sqrt{5}}) \ha Y 
 \;\to\; g_1=\fr{\sqrt{3}}{\sqrt{5}}
\end{array}
</div></td>

<td>&nbsp;&nbsp;&nbsp;</td>

<td border=none>
<table class="ptable">
<tr>
<th ALIGN=CENTER COLSPAN="2"><SPAN class="math">SO(4)</SPAN></th>
<th></th>
<th ALIGN=CENTER><SPAN class="math">W^3</SPAN></th>
<th ALIGN=CENTER><SPAN class="math">B_1^3</SPAN></th>
<th></th>
<th ALIGN=CENTER><SPAN class="math">\fr{\sqrt{2}}{\sqrt{3}} B_2</SPAN></th>
<th ALIGN=CENTER><SPAN class="math">\ha Y</SPAN></th>
<th ALIGN=CENTER><SPAN class="math">Q</SPAN></th>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\mcir{#FFFF00} </SPAN></td>
<td ALIGN=CENTER><SPAN class="math">W^+</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">1</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">1</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\mcir{#FFFF00} </SPAN></td>
<td ALIGN=CENTER><SPAN class="math">W^-</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">- 1</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-1</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\mcir{#FFFFFF} </SPAN></td>
<td ALIGN=CENTER><SPAN class="math">B_1^+</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">1</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">1</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">1</SPAN></td>
</tr>
<tr class="butt">
<td ALIGN=CENTER><SPAN class="math">\mcir{#FFFFFF} </SPAN></td>
<td ALIGN=CENTER><SPAN class="math">B_1^-</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">- 1</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-1</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-1</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\msqu{#B2B200} </SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\ph_+</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\ha</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\ha</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\ha</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">1</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\mdia{#F2F200} </SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\ph_-</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-1</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\msqu{#999999}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\ph_0</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\ha</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
</tr>
<tr class="butt">
<td ALIGN=CENTER><SPAN class="math">\mdia{#4D4D4D}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\ph_1</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\fr{1}{2}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\btri{#B2B200}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\nu_{eL}</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\fr{1}{2}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\ha</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\btri{#F2F200}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">e_L</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\fr{1}{2}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-1</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\btri{#999999}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\nu_{eR}</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\ha</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
</tr>
<tr class="butt">
<td ALIGN=CENTER><SPAN class="math">\btri{#D9D9D9}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">e_R</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\ha</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-1</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-1</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">
\trip{\mtri{#BF6000}}{\mtri{#668000}}{\mtri{#8F00B2}}
</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">u_L</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\fr{1}{2}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{6}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\fr{1}{6}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\fr{2}{3}</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">
\trip{\mtri{#F77C00}}{\mtri{#99BF00}}{\mtri{#AD00F7}}
</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">d_L</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{6}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\fr{1}{6}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">
\trip{\mtri{#990000}}{\mtri{#009900}}{\mtri{#0000B2}}
</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">u_R</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{6}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\fr{2}{3}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\fr{2}{3}</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">
\trip{\mtri{#D90000}}{\mtri{#00BF00}}{\mtri{#0000F7}}
</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">d_R</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{6}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{3}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
</table>
</td>

</tr>
</table>
</center>
</html>
<<tiddler HideTags>>
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/IfA11/images/Hypercharge and weak charge.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math">su(2)_L \oplus u(1)_Y \oplus (2_L+2_R) \otimes (1+1)</SPAN>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/RW12/images/Hypercharge and weak charge.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math">su(2)_L \oplus u(1)_Y \oplus (2_L+2_R) \otimes (1+1)</SPAN>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/RW12/images/Hypercharge and weak charge.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math">su(2)_L \oplus u(1)_Y \oplus (2_L+2_R) \otimes (1+1)</SPAN>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/RW12/images/Electroweak breaking.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math">Q = Y  \cos(\th_W)  + W  \sin(\th_W)</SPAN>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="images/png/ewhiggs.png" height="300">
</td></tr> <tr><td>
 <SPAN class="math">\;</SPAN>
</td></tr>
<tr><td>
 <SPAN class="math">\;</SPAN>
</td></tr>
<tr><td>
A fiber twisting around maximal torus inside <SPAN class="math">SU(2)_L \otimes U(1)_Y</SPAN>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>>@@display:block;text-align:center;<html><iframe src="epe/EPE3.html" width=680 height=540 scrolling="no" frameborder="0"></iframe></html>@@
<html>
<center>
<table class="ptable">
<tr>
<th ALIGN=CENTER COLSPAN="3"><SPAN>Forces (bosons)</SPAN></th>
<th><SPAN class="math">\;\;\;\;</SPAN></th>
<th ALIGN=CENTER COLSPAN="3"><SPAN>Matter (fermions)</SPAN></th>
<th><SPAN class="math">\;\;</SPAN></th>
<th ALIGN=CENTER COLSPAN="3"><SPAN>Second generation</SPAN></th>
<th><SPAN class="math">\;\;</SPAN></th>
<th ALIGN=CENTER COLSPAN="3"><SPAN>Third generation</SPAN></th>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\mcir{#000000} </SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\p{{\Big(}^{(}_{\big(}} \ga \p{{\Big(}^{(}_{\big(}} </SPAN></td>
<td ALIGN=CENTER><SPAN>electromagnetism</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\btri{#F2F200} \btri{#D9D9D9}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">e</SPAN></td>
<td ALIGN=CENTER><SPAN>electron</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\mtri{#F2F200} \mtri{#D9D9D9}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\mu</SPAN></td>
<td ALIGN=CENTER><SPAN>muon</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\stri{#F2F200} \stri{#D9D9D9}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\tau</SPAN></td>
<td ALIGN=CENTER><SPAN>tau</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\mcir{#FFFF00} </SPAN></td>
<td ALIGN=CENTER><SPAN class="math">W,Z</SPAN></td>
<td ALIGN=CENTER><SPAN>weak</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\butr{#F2F200} \butr{#D9D9D9}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\bar{e}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\p{{\Big(}^{\Big(}}</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\mutr{#F2F200} \mutr{#D9D9D9}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\bar{\mu}</SPAN></td>
<td ALIGN=CENTER><SPAN></SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\sutr{#F2F200} \sutr{#D9D9D9}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\bar{\tau}</SPAN></td>
<td ALIGN=CENTER><SPAN></SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td ALIGN=CENTER><SPAN class="math">g</SPAN></td>
<td ALIGN=CENTER><SPAN>strong</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\btri{#B2B200} \btri{#999999}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\nu_e</SPAN></td>
<td ALIGN=CENTER><SPAN>electron <br> neutrino</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\mtri{#B2B200} \mtri{#999999}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\nu_\mu</SPAN></td>
<td ALIGN=CENTER><SPAN>muon <br> neutrino</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\stri{#B2B200} \stri{#999999}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\nu_\tau</SPAN></td>
<td ALIGN=CENTER><SPAN>tau <br> neutrino</SPAN></td>
</tr>
<tr class="butt">
<td ALIGN=CENTER><SPAN class="math">\mcir{#59FF59} </SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\p{{\Big(}^{\big(}_{\big(}} \om \p{{\Big(}^{\big(}_{\big(}}</SPAN></td>
<td ALIGN=CENTER><SPAN>gravity</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\butr{#B2B200} \butr{#999999}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\bar{\nu}{}_e</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\p{{\Big(}_(}</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\mutr{#B2B200} \mutr{#999999}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\bar{\nu}{}_\mu</SPAN></td>
<td ALIGN=CENTER><SPAN></SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\sutr{#B2B200} \sutr{#999999}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\bar{\nu}{}_\tau</SPAN></td>
<td ALIGN=CENTER><SPAN></SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">
\begin{array}{c}
\mdia{#F2F200} \, \mdia{#BF6000} \\[-.5em]
\msqu{#B2B200} \, \msqu{#F77C00}
\end{array}
</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\ph</SPAN></td>
<td ALIGN=CENTER><SPAN>Higgs</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">
\begin{array}{c}
\btri{#BF6000} \btri{#990000} \\[-.8em]
\btri{#668000} \btri{#009900} \\[-.8em]
\btri{#8F00B2} \btri{#0000B2}
\end{array}
</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">u</SPAN></td>
<td ALIGN=CENTER><SPAN>up <br> quark</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">
\begin{array}{c}
\mtri{#BF6000} \mtri{#990000} \\[-.8em]
\mtri{#668000} \mtri{#009900} \\[-.8em]
\mtri{#8F00B2} \mtri{#0000B2}
\end{array}
</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">c</SPAN></td>
<td ALIGN=CENTER><SPAN>charm <br> quark</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">
\begin{array}{c}
\stri{#BF6000} \stri{#990000} \\[-.8em]
\stri{#668000} \stri{#009900} \\[-.8em]
\stri{#8F00B2} \stri{#0000B2}
\end{array}
</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">t</SPAN></td>
<td ALIGN=CENTER><SPAN>top <br> quark</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math"></SPAN></td>
<td ALIGN=CENTER><SPAN class="math"></SPAN></td>
<td ALIGN=CENTER><SPAN></SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">
\begin{array}{c}
\butr{#BF6000} \butr{#990000} \\[-.8em]
\butr{#668000} \butr{#009900} \\[-.8em]
\butr{#8F00B2} \butr{#0000B2}
\end{array}
</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\bar{u}</SPAN></td>
<td ALIGN=CENTER><SPAN></SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">
\begin{array}{c}
\mutr{#BF6000} \mutr{#990000} \\[-.8em]
\mutr{#668000} \mutr{#009900} \\[-.8em]
\mutr{#8F00B2} \mutr{#0000B2}
\end{array}
</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\bar{c}</SPAN></td>
<td ALIGN=CENTER><SPAN></SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">
\begin{array}{c}
\sutr{#BF6000} \sutr{#990000} \\[-.8em]
\sutr{#668000} \sutr{#009900} \\[-.8em]
\sutr{#8F00B2} \sutr{#0000B2}
\end{array}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\bar{t}</SPAN></td>
<td ALIGN=CENTER><SPAN></SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math"></SPAN></td>
<td ALIGN=CENTER><SPAN class="math"></SPAN></td>
<td ALIGN=CENTER><SPAN></SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">
\begin{array}{c}
\btri{#F77C00} \btri{#D90000} \\[-.8em]
\btri{#99BF00} \btri{#00BF00} \\[-.8em]
\btri{#AD00F7} \btri{#0000F7}
\end{array}
</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">d</SPAN></td>
<td ALIGN=CENTER><SPAN>down <br> quark</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">
\begin{array}{c}
\mtri{#F77C00} \mtri{#D90000} \\[-.8em]
\mtri{#99BF00} \mtri{#00BF00} \\[-.8em]
\mtri{#AD00F7} \mtri{#0000F7}
\end{array}
</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">s</SPAN></td>
<td ALIGN=CENTER><SPAN>strange <br> quark</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">
\begin{array}{c}
\stri{#F77C00} \stri{#D90000} \\[-.8em]
\stri{#99BF00} \stri{#00BF00} \\[-.8em]
\stri{#AD00F7} \stri{#0000F7}
\end{array}
</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">b</SPAN></td>
<td ALIGN=CENTER><SPAN>bottom <br> quark</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math"></SPAN></td>
<td ALIGN=CENTER><SPAN class="math"></SPAN></td>
<td ALIGN=CENTER><SPAN></SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">
\begin{array}{c}
\butr{#F77C00} \butr{#D90000} \\[-.8em]
\butr{#99BF00} \butr{#00BF00} \\[-.8em]
\butr{#AD00F7} \butr{#0000F7}
\end{array}
</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\bar{d}</SPAN></td>
<td ALIGN=CENTER><SPAN></SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">
\begin{array}{c}
\mutr{#F77C00} \mutr{#D90000} \\[-.8em]
\mutr{#99BF00} \mutr{#00BF00} \\[-.8em]
\mutr{#AD00F7} \mutr{#0000F7}
\end{array}
</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\bar{s}</SPAN></td>
<td ALIGN=CENTER><SPAN></SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">
\begin{array}{c}
\sutr{#F77C00} \sutr{#D90000} \\[-.8em]
\sutr{#99BF00} \sutr{#00BF00} \\[-.8em]
\sutr{#AD00F7} \sutr{#0000F7}
\end{array}
</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\bar{b}</SPAN></td>
<td ALIGN=CENTER><SPAN></SPAN></td>
</tr>
</table>
</center>
</html>
Images can be included by their filename or full URL. It's good practice to include a title to be shown as a tooltip, and when the image isn't available. An image can also link to another note or or a URL
[img[Romanesque broccoli|images/fractalveg.jpg][http://www.flickr.com/photos/jermy/10134618/]]
{{{
[img[Romanesque broccoli|images/fractalveg.jpg]
   [http://www.flickr.com/photos/jermy/10134618/]]
[img[title|filename]]
[img[filename]]
[img[title|filename][link]]
[img[filename][link]]
}}}
[<img[Forest|images/forest.jpg][http://www.flickr.com/photos/jermy/8749660/]][>img[Field|images/field.jpg][http://www.flickr.com/photos/jermy/8749285/]]You can also float images to the left or right: the forest is left aligned with {{{[<img[}}}, and the field is right aligned with {{{[>img[}}}.
@@clear(left):clear(right):display(block):You can use CSS to clear the floats@@
{{{
[<img[Forest|images/forest.jpg][http://www.flickr.com/photos/jermy/8749660/]]
[>img[Field|images/field.jpg][http://www.flickr.com/photos/jermy/8749285/]]
You can also float images to the left or right:
 the forest is left aligned with {{{[<img[}}},
and the field is right aligned with {{{[>img[}}}.
@@clear(left):clear(right):display(block):
You can use CSS to clear the floats@@
}}}
Decent intros using MaxEnt:
http://arxiv.org/abs/cond-mat/0507388
http://arxiv.org/abs/physics/9805024

some Q.A.Wang papers
*http://arxiv.org/abs/cond-mat/0312329
**I don't like his use of ergodicity in defining the long time average equal to the Bayesian expectation value.
**nice: uses fixed average action and MaxEnt to get partition function with action per path instead of energy per state
***I might like using Rovelli's Hamiltonian constraint dynamics better.
*http://arxiv.org/abs/cond-mat/0407515
**seems inferior to previous paper

Good Hawking paper on it:
http://arxiv.org/abs/gr-qc/9501014
*boundary term in depth
*Hamiltonian formulation
*relationship between partition functions for static spacetimes (timelike Killing vector).

related discussion on time and Tomita flow in Rovelli book

Nice new paper including possible action for BF gravity:
*http://arxiv.org/pdf/1103.2971v1
<<tiddler HideTags>>$$
\begin{array}{llcl}

1918, \!\!&\!\! {\rm Weyl} \!\!&\!\! : & \f{A} \in \f{\rm Lie}(G) \p{{}_{\big(}} \\

1954, \!\!&\!\! {\rm Y.M.} \!\!&\!\! : & \f{A} = \f{B} + \f{W} + \f{G}
\;\; \in \; \f{\rm Lie}(G) = \f{su}(1) + \f{su}(2) + \f{su}(3) \p{{}_{\big(}} \\

1967, \!\!&\!\! {\rm F.P.} \!\!&\!\! : & \udf{A} = \f{A} + \ud{g} \p{{}_{\big(}}
\;\; \in \; \udf{\rm Lie}(G) \\

1977, \!\!&\!\! {\rm M.M.} \!\!&\!\! : & \f{A} = \f{\om} + \f{e}
\;\; \in \; \f{\rm Lie}(G) = \f{so}(1,4) \p{{}_{\Big(_(}} \\


2002, \!\!&\!\! {\rm Y.T.} \!\!&\!\! : & \ud{\ps} = \ud{g} \p{{}_{\big(}} \\

2005, \!\!&\!\! {\rm Y.T.} \!\!&\!\! : & \udf{A} = {\small \frac{1}{2}} \f{\om} + {\small \frac{1}{4}} \f{e} \ph + \f{B} + \f{W} + \f{G} + \ud{\nu^e} + \ud{e} + \ud{u} + \ud{d}   \\
 & & & \;\;\;\,\, \in \; \f{\rm Lie}(G) = \f{Cl}(1,7) \p{{}_{\big(}} \\

{\rm now}, \!&\! {\rm Y.T.} \!\!&\!\! : & \udf{A} = {\small \frac{1}{2}} \f{\om} + {\small \frac{1}{4}} \f{e} \ph + \f{B} + \f{W} + \f{G} + \ud{\nu^e} + \ud{e} + \ud{u} + \ud{d}  \\
 & & & \;\;\;\;\;\;\;\; + \ud{\nu^\mu} + \ud{\mu} + \ud{c} + \ud{s}
+ \ud{\nu^\ta} + \ud{\ta} + \ud{t} + \ud{b} \\
 & & & \;\;\;\,\, \in \; \f{\rm Lie}(G) = \f{e8} ? \p{{}_{\big(_(}}
\end{array}
$$
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/RW12/images/GraviGUTSM.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math">spin(1,3) \,\oplus\, 4 \!\otimes\! (2 \!\,\oplus\,\! \bar{2}) \,\oplus\, su(2)_L \,\oplus\, u(1)_Y \,\oplus\, su(3) \,\oplus\, 2 \!\otimes\! (2_L\!\,\oplus\,\!2_R) \!\otimes\! (1\!\,\oplus\,\!3)</SPAN>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/IfA11/images/GraviGUTSM.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math">spin(1,3) \,\oplus\, 4 \!\otimes\! (2 \!\,\oplus\,\! \bar{2}) \,\oplus\, su(2)_L \,\oplus\, u(1)_Y \,\oplus\, su(3) \,\oplus\, 2 \!\otimes\! (2_L\!\,\oplus\,\!2_R) \!\otimes\! (1\!\,\oplus\,\!3)</SPAN>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>>The $14$ Lie algebra elements of the smallest exceptional Lie group, $G2$:
$$
\begin{array}{rcccccccl}
g2 \!\!&\!\!=\!\!&\!\! su(3) \!\!&\!\! + \!\!&\!\! 3 \!\!&\!\! + \!\!&\!\! \bar{3} \!\!&& \\
&&\!\! \f{g} \!\!&\!\! + \!\!&\!\! \ud{q} \!\!&\!\! + \!\!&\!\! \ud{\bar{q}} \!\!&\! \in \! &\! \udf{g2} 
\end{array}
$$
Structure of $G2$ implies Lie bracket equivalent to fundamental action,
$$
[ g,q ] = \big[ g^A T_A,q^B T_B \big] = g \, q =
\lb
\matrix{
\! \fr{i}{2} g^3 \!+\! {\scriptsize \frac{i}{2\sqrt{3}}} g^8 \!\! & g^{r\bar{g}} & g^{r\bar{b}} \\
g^{\bar{r}g} & \!\! {\scriptsize -\!\frac{i}{2}} g^3 \!+\! {\scriptsize \frac{i}{2\sqrt{3}}} g^8 \!\! & g^{g\bar{b}} \\
g^{\bar{r}b} & g^{\bar{g}b} & {\scriptsize -\!\frac{i}{\sqrt{3}}} g^8
}
\rb
\lb \matrix{
q^r \\ q^g \\ q^b
} \rb
$$
corresponding to the strong interactions, such as
<html>
<center>
<table class="gtable">
<tr>
<td>
<div class="math">
\big[ g^{r\bar{g}}, q^g \big] = q^r
</div>
</td>
<td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>
<td ALIGN=CENTER><img SRC="images/png/quark gluon vertex.png" height=160px></td>
</tr>
</table>
</center>
</html>
''Bold''
{{{''Bold''}}}
==Strikethrough==
{{{==Strikethrough==}}}
__Underline__ 
{{{__Underline__}}}
//Italic// 
{{{//Italic//}}}
2^^3^^=8 
{{{2^^3^^=8}}}
a~~ij~~ = -a~~ji~~ 
{{{a~~ij~~ = -a~~ji~~}}}
@@highlight@@ 
{{{@@highlight@@}}}

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@@color:green;green coloured@@
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@@text-shadow:black 3px 3px 8px;font-size:18pt;display:block;margin:1em 1em 1em 1em;border:1px solid black;Access any CSS style@@
{{{@@text-shadow:black 3px 3px 8px;font-size:18pt;display:block;margin:1em 1em 1em 1em;border:1px solid black;Access any CSS style@@}}}
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//For backwards compatibility, the following highlight syntax is also accepted://
@@bgcolor(#ff0000):color(#ffffff):red coloured@@
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The rank $4$ exceptional group, ''F4'', is a real, [[simple]], compact, connected [[Lie group|Lie group]]. It may be described by [[exponentiating|exponentiation]] its $52$ dimensional [[Lie algebra]], [[f4]].  An explicit construction can be found in:

Ref:
*[[Cerchiai - Mapping the geometry of the F4 group|papers/Cerchiai - Mapping the geometry of the F4 group.pdf]]
$$
\begin{array}{rcll}
F4 &:& ( \ha \om_L^3, \ha \om_R^3, W^3, B_1^3 ) \;\;\; \p{(B_2, g^3, g^8)} &\left\{ 
\begin{array}{l}\text{graviweak interactions} \\ \text{three generations}\end{array} \right.
\\
G2 &:& \p{( \ha \om_L^3, \ha \om_R^3, W^3, B_1^3 ) \;\;\;}(B_2, g^3, g^8) &\left\{ 
\begin{array}{l}\text{strong interactions} \\ \text{anti-particles}\end{array} \right.
\\[-.3em]
\rlap{\hbox{@(hr noshade size="1" style="position:relative; left:-2em;
      width:30em; border:0px; border-top:1px solid black")}}\\[-.5em]
E8 &:& ( \ha \om_L^3, \ha \om_R^3, W^3, B_1^3, w, B_2, g^3, g^8) & \, \left\{ \; \text{everything} \right.
\end{array}
$$

Breakdown of E8 to the standard model and gravity:
\begin{eqnarray}
e8 &=& f4 + g2 + 26 \! \times \! 7 \\
&=& so(7,1) + su(3) + (8_{S+}\!+\!8_V+\!8_{S-})\!\times\!(1\!+\!1\!+\!3\!+\!\bar{3}) + 3\!\times\!(3\!+\!\bar{3}) + 2 \\[.4em]
A &=& \big( {\scriptsize \frac{1}{2}} \om + {\scriptsize \frac{1}{4}} e \ph + W + B_1 \big) + g + 3 \! \times \! \Ps + x \Ph + B_2 + w
\end{eqnarray}
Two new quantum numbers and some non-standard particles:
$$
\{ \; w \quad (B_1^3\!+\!B_2) \quad B_1^\pm \quad x_{1/2/3} \Ph^{r/g/b} \quad x_{1/2/3} \Ph^{\bar{r}/\bar{g}/\bar{b}} \; \} \vp{\big(}
$$
<<tiddler HideTags>>
<<tiddler HideTags>>
@@display:block;text-align:center;
<html><center><embed src="images/png/f4.png" width="510" height="510"></embed></center></html>
@@ 
confirmed attendees:
*[[Scott Aaronson|http://www.scottaaronson.com/]], QMI, quantum computing
*[[Fred Adams|http://www.physics.lsa.umich.edu/department/directory/bio.asp?ID=1]]${}^*$, constant change, astrophysics
*[[Anthony Aguirre|http://scipp.ucsc.edu/~aguirre/]], cosmology
*[[Stephon Alexander|http://www.phys.psu.edu/people/display/index.html?person_id=4901]], astrophysics
**Just put out a new paper: Isogravity
**friends with James Bjorken (and everyone else, apparently)
*[[Markus Aspelmeyer|http://homepage.univie.ac.at/Markus.Aspelmeyer/]]${}^*$, QM foundations
*[[Paul A Benioff|http://www.phy.anl.gov/theory/staff/pab.html]], QM foundations, older guy
*[[Caslav Brukner|http://homepage.univie.ac.at/Caslav.Brukner/index.htm]], QM foundations
*[[Dmitry Budker|http://www.fqxi.org/aw-budker2.html]]${}^*$, constant change (experimental)
*[[Gregory Chaitin|http://www.umcs.maine.edu/~chaitin/]], math, complexity, and philosophy of science
*[[Hyung Choi|http://www.zoominfo.com/people/Choi_Hyung_78134925.aspx]], metanexus, QM foundations, science and religion (uh oh)
*[[Louis Crane|http://www.fqxi.org/aw-crane2.html]]${}^*$, QGR, QM histories
*[[Paul Davies|http://cosmos.asu.edu/]], QMI, astrophysics, popular author
*[[John Donoghue|http://www.fqxi.org/aw-donoghue2.html]]${}^*$, emergent symmetry
*[[Richard Easther|http://www.fqxi.org/aw-easther.html]]${}^*$, superstring cosmology
*[[David Ritz Finkelstein|http://www.physics.gatech.edu/people/faculty/dfinkelstein.html#personal]],  older particle physicist
**Lie algebra expert. Proponent of stable Lie algebras.
*[[Rodolfo Gambini|http://www.fqxi.org/aw-pullin2.html]]${}^*$, QM GR
*[[Jaume Garriga|http://www.ffn.ub.es/gcg/personal/jaume.html]], cosmology, branes
*[[Steven Gratton|http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=find+ea+gratton%2C+steven]], cosmology, inflation
*[[Alan Guth|http://web.mit.edu/physics/facultyandstaff/faculty/alan_guth.html]], err, invented inflation
*[[Lucien Hardy|http://www.perimeterinstitute.ca/index.phpindex.php?option=com_content&task=view&id=30&Itemid=7&view_directory=1&pi=1078]], Causaloids
*[[Adrian Kent|http://www.damtp.cam.ac.uk/user/apak/]], QM foundations
*[[Lawrence Krauss|http://www.phys.cwru.edu/~krauss/]], astrophysics and cosmology, popular author, dislikes KK and strings, lectures a LOT
*[[Matthew Leifer|http://www.fqxi.org/aw-leifer2.html]], QM foundations
*[[Eugene Lim|http://pantheon.yale.edu/~eal48/papers.html]]${}^*$, cosmology
*A. [[Garrett Lisi]]${}^*$
*[[Abraham Loeb|http://www.fqxi.org/aw-loeb2.html]]${}^*$, SETI, astronomy
*[[Fotini Markopoulou|http://www.fqxi.org/aw-markopoulou2.html]]${}^*$, quantum graphity
*[[Laura Mersini|http://en.wikipedia.org/wiki/Laura_Mersini]], cosmology
*[[Farzad Nekoogar|http://www.fqxi.org/aw-nekoogar.html]]${}^*$, popularizer of theoretical physics -- [[multiversal journeys|http://www.multiversaljourneys.com/]]
*[[Ken Olum|http://www.fqxi.org/aw-olum2.html]]${}^*$, GR, wants to rule out wormholes and other GR exotics
*[[Maulik Parikh|http://www.fqxi.org/aw-khoury2.html]]${}^*$, GR boundaries, mach's principle, hep-th and strings
*[[Philip Pearle|http://physerver.hamilton.edu/people/]], QM  foundations, older guy
*[[Ekkehard Peik|http://www.fqxi.org/aw-peik2.html]]${}^*$, constant change (experimental)
*[[Simon Saunders|http://www.fqxi.org/aw-saunders2.html]]${}^*$, QM foundations
*Lee Smolin (not going)
*[[Robert Spekkens|http://www.fqxi.org/aw-spekkens2.html]]${}^*$, QM foundations
*[[Max Tegmark|http://web.mit.edu/physics/facultyandstaff/faculty/max_tegmark.html]],  astrophysics, cosmology, trouble maker...
*[[Mark Trodden|http://physics.syr.edu/~trodden/]], cosmology, particle physics -- QFT
*[[Roderich Tumulka|http://www.fqxi.org/aw-tumulka2.html]]${}^*$, Bohmian QM
*[[Jos Uffink|http://www.phys.uu.nl/igg/jos/]], QM foundations
*[[Vitaly Vanchurin|http://cosmos.phy.tufts.edu/~vitaly/]], cosmic strings 
*[[Xiao-Gang Wen|http://www.fqxi.org/aw-wen2.html]]${}^*$, gravity and light emergent from substrate :P
*[[Serge Winitzki|http://www.theorie.physik.uni-muenchen.de/~serge/]], quantum cosmology
**Likes wiki, and likes ToE.
**Inflation expert -- says $R \ph^2$ term would be great, among others.
***Strong constraints on these coefficients.
*[[Toby Wiseman|http://schwinger.harvard.edu/~wiseman/]], string theory
*[[Wojciech Zurek|http://public.lanl.gov/whz/]], cosmology and astrophysics, chaos, QM foundations

${}^*$ grant winners (19)

Press and Foundation people
*[[Graham P Collins|http://www.sff.net/people/GPC/]], Scientific American Magazine
*[[Valerie Jamieson|http://www.scienceinpublic.com/scienceweek/speakers.htm#Valerie%20Jamieson%20background]], New Scientist Magazine
**particle physics background
*[[Wade Davis|http://en.wikipedia.org/wiki/Wade_Davis]], National Geographic Explorer-in-Residence, ethnobiologist
*[[Charles Harper|http://www.templeton.org/about_us/who_we_are/leadership_team/charles_harper/]], Senior Vice-President, John Templeton Foundation
**He's paying, try not to insult him.
*[[Amanda High|http://www.nptrust.org/about_npt/key_staff.asp#high]], Vice President, National Philanthropic Trust
**What's she doing at the FQXi conference
*[[Howard Burton|http://www.perimeterinstitute.ca/index.php?option=com_content&task=view&id=30&Itemid=72&pi=index.php&e=Founding%20Executive%20Director&cat_id=53&cat_table=2]], Executive Director, Perimeter Institute for Theoretical Physics
**Just got ousted from PI position, even though he founded it. Used to be main PI talent scout.
*[[Christopher Liedel|http://executiveeducation.wharton.upenn.edu/fellows/feb_info/roster_detail.cfm?id=KRSM00000024468]], Executive Vice President & Chief Financial Officer, National Geographic Society
*Robert Kuhn,  Kuhn foundation -- makes science documentaries for PBS
/*{{{*/
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n.rel = "shortcut icon"; 
n.href = "favicon.ico"; 
document.getElementsByTagName("head")[0].appendChild(n);
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Choosing the anti-Grassmann 3-form to be $\fff{\od{B}} = \nf{e} \od{\Ps} \ve{e} \,$ gives the massive Dirac action in curved spacetime:
\begin{eqnarray}
S_f &=& \int \big< \fff{\od{B}} \udff{F} \big>
= \int \big< \fff{\od{B}} \f{D} \ud{\Ps} \big> \\
&=& \int \big< \nf{e} \od{\Ps} \ve{e} \big( \f{d} \ud{\Ps} + \f{H}{}_1 \ud{\Ps} - \ud{\Ps} \f{H}{}_2 \big) \big> \\
&=& \int \big< \nf{e} \od{\Ps} \ve{e} \big( ( \f{d} + {\scriptsize \frac{1}{2}} \f{\om} + {\scriptsize \frac{1}{4}} \f{e}\ph + \f{W} + \f{B}{}_1 ) \ud{\Ps}
- \ud{\Ps} ( \f{w} + \f{B}{}_2 + \f{x} \Ph + \f{g} )  \big) \big> \\
&=& \int \nf{d^4 x} \, |e| \, \big< \od{\Ps} \ga^\mu (e_\mu)^i \big( \pa_i \ud{\Ps} + {\scriptsize \frac{1}{4}} \om_i^{\p{i} \mu \nu} \ga_{\mu \nu} \ud{\Ps} + W_i \ud{\Ps} + B_{1i} \ud{\Ps} \\
&& \hphantom{\int \nf{d^4 x} \, |e| \, \big< \od{\Ps} \ga^\mu (e_\mu)^i \big(} + \ud{\Ps} w_i + \ud{\Ps} B_{2i} + \ud{\Ps} x_i \Ph + \ud{\Ps} g_i \big) + \od{\Ps} \, \ph \, \ud{\Ps} \big>
\end{eqnarray}
The $\od{\Ps} \, \ph \, \ud{\Ps}$ is the standard Higgs mass term.$\vp{\Huge(}$
The $\od{\Ps} \ga^\mu \ud{\Ps} x_\mu \Ph$ term... I don't understand yet -- promising for CKM.
<<tiddler HideTags>>
<<tiddler HideTags>><html><center>
<img src="talks/Zuck09/images/Fiber bundle.png" width="336" height="200">
</center></html>
@@display:block;text-align:center;Principal bundle over 4D base.@@
Fermion, $\ud{\ps}$, Higgs, $\ph$, and frame, $\f{e}$, fibers in various representations of the structure group,
$$
Spin(1,3) \times ( SU(2) \times U(1) \times SU(3) ) / Z_6
$$

Connection: $\;\; \f{A} = \f{dx^\mu} A_\mu^{\p{k}B} T_B \;\;\;\;\;\;\;\;\;\;\;\;\;$ Ehresmann connection: $\;\; \f{\ve{\cal A}}(x,y) = \f{A}^B(x) \ve{\xi}_B(y) + \f{\ve{\cal I}}  \;$
 
Curvature: $\;\; \ff{F} =  \f{d} \f{A} + \ha [ \f{A}, \f{A} ] \;\;\;\;\;\;\;\;\;\;\;$ Frolicher-Nijenhuis: $\;\; \ff{\ve{\cal F}} = - \ha [ \f{\ve{\cal A}}, \f{\ve{\cal A}} ] = - \f{\ve{\cal A}} ( \f{\pa} \f{\ve{\cal A}} ) + \f{\pa} ( \f{\ve{\cal A}} \f{\ve{\cal A}} ) \;$
 
Action: $\;\; S(\ff{F}, \f{D} \ud{\ps}, \f{A}, \ud{\ps}, \ph, \f{e}, \dots)^{\p{\big(}} \;$
<<tiddler HideTags>>
<html><center>
<img src="talks/StAnth09/images/Fiber bundle c.png" height="400">
</center></html>

@@display:block;text-align:center;U(1) fiber bundle over 4D base.@@

The ''FuN curvature'' (//''Frolicher-Nijenhuis curvature''//), $\ff{\ve{\cal F}}=\f{dx^i} \f{dx^j} \ha {\cal F}_{ij}{}^k \ve{\pa_k}$, of a [[vector valued form]], $\f{\ve{\cal A}}=\f{dx^i} {\cal A}_i{}^j \ve{\pa_j}$, is its [[FuN bracket|FuN derivative]] with itself,
\begin{eqnarray}
\ff{\ve{\cal F}} &=& - \ha \lb \f{\ve{\cal A}}, \f{\ve{\cal A}} \rb_L
= - \lb \f{\ve{\cal A}}, \f{\pa} \rb \f{\ve{\cal A}} \\
&=& - \f{\ve{\cal A}} \lp \f{\pa} \f{\ve{\cal A}} \rp + \f{\pa} \lp \f{\ve{\cal A}} \f{\ve{\cal A}} \rp \\
&=& - \lp \f{\ve{\cal A}} \f{\pa} \rp \f{\ve{\cal A}} + \lp \f{\pa} \f{\ve{\cal A}} \rp \f{\ve{\cal A}}  
\end{eqnarray}
In components, this is
$$
{\cal F}_{ij}{}^k = - {\cal A}_i{}^m \pa_m {\cal A}_j{}^k + {\cal A}_j{}^m \pa_m {\cal A}_i{}^k + {\cal A}_m{}^k \pa_i {\cal A}_j{}^m - {\cal A}_m{}^k \pa_j A_i{}^m
$$

If, as often happens, a vector valued form is a [[vector projection]], $\f{\ve{\cal P}} = \f{\ve{\cal P}} \f{\ve{\cal P}}$, its FuN curvature is
\begin{eqnarray}
\ff{\ve{\cal F}} &=& \f{\pa} \f{\ve{\cal P}} - \f{\ve{\cal P}} \lp \f{\pa} \f{\ve{\cal P}} \rp \\
&=& \lp 1 - \f{\ve{\cal P}} \rp \lp \f{\pa} \f{\ve{\cal P}} \rp  
\end{eqnarray}
which satisfies $\f{\ve{\cal P}} \ff{\ve{\cal F}}=0$ -- the form part of the FuN curvature is in the kernel (horizontal part) of the projection. If the vector projection is an [[Ehresmann connection]], any two vectors contracted with the FuN curvature give the vertical part of the [[Lie bracket|Lie derivative]] of the horizontal part of the vectors,
$$
\ve{u} \ve{v} \ff{\ve{\cal F}} = - \ha {\lb \ve{u_H} , \ve{v_H} \rb_L}_V = - \ha \lb \ve{u} \lp 1- \f{\ve{\cal P}} \rp , \ve{v} \lp 1- \f{\ve{\cal P}} \rp \rb_L \f{\ve{\cal P}}
$$
//check that//
The ''Frolicher-Nijenhuis Lie derivative'' -- which we refer to as the //''FuN derivative''// -- is a [[natural]] operator generalizing the [[Lie derivative]] to handle [[vector valued form]] fields. The FuN derivative of a vector valued $k$-form field, $\nf{\ve{K}}$, with respect to a vector field, $\ve{v}$, is written terms of [[partial derivative]]s as
\begin{eqnarray}
{\cal L}_{\ve{v}} \nf{\ve{K}} &=& \lim_{t \to 0} \fr{\ph_t^*\nf{\ve{K}} - \nf{\ve{K}}}{t} = \ve{v} \lp \f{\pa} \nf{\ve{K}} \rp + \f{\pa} \lp \ve{v} \nf{\ve{K}} \rp - \lp \nf{\ve{K}} \f{\pa} \rp \ve{v} \\
&=&  \lp \ve{v} \f{\pa} \rp \nf{\ve{K}} +  \lp \f{\pa} \ve{v} \rp \nf{\ve{K}} - \lp \nf{\ve{K}} \f{\pa} \rp \ve{v}
\end{eqnarray}
This defines the ''Frolicher-Nijenhuis bracket'' (//''FuN bracket''//) for these fields, and enforcing antisymmetry defines the FuN derivative of a vector field with respect to a vector valued form.,
$$
{\cal L}_{\nf{\ve{K}}} {\ve{v}} = \lb \nf{\ve{K}} , \ve{v} \rb_L = - \lb \ve{v} , \nf{\ve{K}} \rb_L = - {\cal L}_{\ve{v}} \nf{\ve{K}} 
$$
Similarly, generalizing Cartan's formula for the Lie derivative, the FuN derivative of a differential form is
\begin{eqnarray}
{\cal L}_{\nf{\ve{K}}} \nf{F} &=& \lb \nf{\ve{K}} , \nf{F} \rb_L = \nf{\ve{K}} \lp \f{d} \nf{F} \rp + \lp -1 \rp^k \f{d} \lp \nf{\ve{K}} \nf{F} \rp \\
 &=& \lp \nf{\ve{K}} \f{\pa} \rp \nf{F} + \lp -1 \rp^k \lp \f{\pa} \nf{\ve{K}} \rp \nf{F}
\end{eqnarray}
which also defines the FuN bracket of these objects. The above expression for the FuN bracket, and Cartan's formula, can also be written using the natural [[exterior derivative]] in a [[commutator]] bracket,
$$
\lb \nf{\ve{K}} , \nf{F} \rb_L = \lb \nf{\ve{K}} , \f{d} \rb \nf{F}
$$
thereby demonstrating the naturalness of the FuN derivative acting on forms. These definitions generalize furthest to give the glorious FuN bracket (and FuN derivative) between vector valued $k$ and $l$ forms 
\begin{eqnarray}
\lb \nf{\ve{K}} , \nf{\ve{L}} \rb_L &=& {\cal L}_{\nf{\ve{K}}} \nf{\ve{L}} \\
&=& \nf{\ve{K}} \lp \f{\pa} \nf{\ve{L}} \rp - \lp -1 \rp^{kl} \nf{\ve{L}} \lp \f{\pa} \nf{\ve{K}} \rp + \lp -1 \rp^k \f{\pa} \lp \nf{\ve{K}} \nf{\ve{L}} \rp - \lp -1 \rp^{kl+l} \f{\pa} \lp \nf{\ve{L}} \nf{\ve{K}} \rp \\
&=& \lp \nf{\ve{K}} \f{\pa} \rp \nf{\ve{L}} - \lp -1 \rp^{kl} \lp \nf{\ve{L}} \f{\pa} \rp \nf{\ve{K}} + \lp -1 \rp^k \lp \f{\pa} \nf{\ve{K}} \rp \nf{\ve{L}} - \lp -1 \rp^{kl+l} \lp \f{\pa}\nf{\ve{L}} \rp \nf{\ve{K}} \\
&=& \lb \nf{\ve{K}} , \f{\pa} \rb \nf{\ve{L}} - \lp -1 \rp^{kl} \lb \nf{\ve{L}} , \f{\pa} \rb \nf{\ve{K}}
\end{eqnarray}
which gives all the FuN brackets and Lie derivatives as special cases. This FuN bracket of vector valued $k$ and $l$ forms is a vector valued $(k+l)$-form, and is defined to satisfy
$$
{\cal L}_{\lb \nf{\ve{K}} , \nf{\ve{L}} \rb_L} = \lb {\cal L}_{\nf{\ve{K}}} , {\cal L}_{\nf{\ve{L}}} \rb
$$
when acting on vectors or forms.

The FuN derivative has a number of other nice [[properties|FuN identities]].

//(Most everything here was learned from talking with [[Michael Edwards]] and reading [[Peter Michor]] et al. (Though the above explicit expression is mine, so if it's wrong, blame [[me|Garrett Lisi]]))//
The [[FuN derivative]] with respect to a [[vector valued k-form|vector valued form]], ${\cal L}_{\nf{\ve{K}}}$, is a grade $k$ [[derivation]] that combines with itself and other operators in a number of ways. Like the [[Lie bracket|Lie derivative identities]], it is linear in both arguments.

The FuN Lie bracket may or may not change sign under the exchange of its vector valued $k$-form and vector valued $l$-form arguments,
$$
\lb \nf{\ve{K}}, \nf{\ve{L}} \rb_L = - \lp -1 \rp^{kl} \lb \nf{\ve{L}}, \nf{\ve{K}} \rb_L
$$
As a derivation, the FuN derivative operates on products of forms via the graded Liebniz rule,
$$
{\cal L}_{\nf{\ve{K}}} \lp \nf{F} \nf{G} \rp 
= \lp {\cal L}_{\nf{\ve{K}}} \nf{F} \rp \nf{G} + \lp -1 \rp^{kf} \nf{F} \lp {\cal L}_{\nf{\ve{K}}} \nf{G} \rp
$$
But it is not a derivation over products of VVFs and forms. Using some [[vector valued form identities]] we get
\begin{eqnarray}
{\cal L}_{\nf{\ve{K}}} \lp \nf{\ve{L}} \nf{F} \rp &=& \lp \nf{\ve{K}} \f{\pa} \rp \lp \nf{\ve{L}} \nf{F} \rp + \lp -1 \rp^k \lp \f{\pa} \nf{\ve{K}} \rp \lp \nf{\ve{L}} \nf{F} \rp \\
&=& \lp \lp \nf{\ve{K}} \f{\pa} \rp \nf{\ve{L}} \rp \nf{F}  
+ \lp-1\rp^{k\lp l-1\rp} \lb \nf{\ve{L}} \lp \lp \nf{\ve{K}} \f{\pa} \rp \nf{F} \rp
- \lp \nf{\ve{L}} \lp \nf{\ve{K}} \f{\pa} \rp \rp \nf{F} \rb \\
& &
+ \lp-1\rp^k \lp \lp \f{\pa} \nf{\ve{K}} \rp \nf{\ve{L}} \rp \nf{F}
+ \lp-1\rp^{kl} \nf{\ve{L}} \lp \lp \f{\pa} \nf{\ve{K}} \rp \nf{F} \rp
- \lp-1\rp^{kl} \lp \nf{\ve{L}} \lp \f{\pa} \nf{\ve{K}} \rp \rp \nf{F} \\
&=& 
\lp {\cal L}_{\nf{\ve{K}}} \nf{\ve{L}} \rp \nf{F} 
+ \lp-1\rp^{k\lp l-1\rp} \nf{\ve{L}} \lp {\cal L}_{\nf{\ve{K}}} \nf{F} \rp 
 - \lp-1\rp^{k\lp l-1\rp} {\cal L}_{\nf{\ve{L}}\nf{\ve{K}}} \nf{F}
\end{eqnarray}
and
\begin{eqnarray}
{\cal L}_{\nf{\ve{L}}\nf{\ve{K}}} \nf{F} &=& \lp \lp \nf{\ve{L}} \nf{\ve{K}} \rp \f{\pa} \rp \nf{F} - \lp-1\rp^{\lp l+k\rp} \lp \f{\pa} \lp \nf{\ve{L}} \nf{\ve{K}} \rp \rp \nf{F} \\
&=& \lp \nf{\ve{L}} \lp \nf{\ve{K}} \f{\pa} \rp \rp \nf{F} - \lp-1\rp^k \lp
\lp-1\rp^l \lp \f{\pa} \nf{\ve{L}} \rp \nf{\ve{K}} - \nf{\ve{L}} \lp \f{\pa} \nf{\ve{K}} \rp + \lp \nf{\ve{L}} \f{\pa} \rp \nf{\ve{K}} 
\rp \nf{F} \\
&=& 
\nf{\ve{L}} \lp {\cal L}_{\nf{\ve{K}}} \nf{F} \rp
+ \lp-1\rp^{k\lp l-1\rp} \lp {\cal L}_{\nf{\ve{K}}} \nf{\ve{L}} \rp \nf{F}
- \lp-1\rp^{k\lp l-1\rp} {\cal L}_{\nf{\ve{K}}} \lp \nf{\ve{L}} \nf{F} \rp 
\end{eqnarray}
which are linked by the last lines of each -- they are the same equation (an equation that may be used to define the FuN bracket of two VVF's in terms of the FuN derivatives of forms). A similar identity exists for three VVF's:
$$
{\cal L}_{\nf{\ve{L}}\nf{\ve{K}}} \nf{\ve{M}}
= \nf{\ve{L}} \lp {\cal L}_{\nf{\ve{K}}} \nf{\ve{M}} \rp
+ \lp-1\rp^{k\lp l-1\rp} \lp {\cal L}_{\nf{\ve{K}}} \nf{\ve{L}} \rp \nf{\ve{M}}
- \lp-1\rp^{k\lp l-1\rp} {\cal L}_{\nf{\ve{K}}} \lp \nf{\ve{L}} \nf{\ve{M}} \rp
- \lp-1\rp^{m\lp k+ l-1\rp} \lp {\cal L}_{\nf{\ve{M}}} \nf{\ve{L}} \rp \nf{\ve{K}}
$$

When the two VVF's are written as $\nf{\ve{K}}=\nf{K^A} \ve{X_A}$ and $\nf{\ve{L}}=\nf{L^A} \ve{Y_A}$ their FuN bracket is
\begin{eqnarray}
\lb \nf{\ve{K}}, \nf{\ve{L}} \rb_L &=& \nf{K^A} \nf{L^B} \lb \ve{X_A}, \ve{Y_B} \rb_L + \nf{K^A} \lp {\cal L}_{\ve{X_A}} \nf{L^B} \rp \ve{Y_B}
- \lp {\cal L}_{\ve{Y_B}} \nf{K^A} \rp \nf{L^B} \ve{X_A} \\
&+& \lp -1 \rp^k \lp \f{d} \nf{K^A} \rp \ve{X_A} \nf{L^B} \ve{Y_B}
+ \lp -1 \rp^k \lp \ve{Y_B} \nf{K^A} \rp \lp \f{d} \nf{L^B} \rp \ve{X_A}
\end{eqnarray}
and, for the FuN bracket of a vector valued 1-form with itself,
\begin{eqnarray}
\lb \f{\ve{K}}, \f{\ve{K}} \rb_L &=& \f{K^A} \f{K^B} \lb \ve{X_A}, \ve{X_B} \rb_L + 2 \f{K^A} \lp {\cal L}_{\ve{X_A}} \f{K^B} \rp \ve{X_B}
- 2 \lp \f{d} \f{K^A} \rp \ve{X_A} \f{K^B} \ve{X_B}
\end{eqnarray}

When acting on forms, the FuN derivative commutes with the [[exterior derivative]],
$$
0 = \lb {\cal L}_{\nf{\ve{K}}}, \f{d} \rb = {\cal L}_{\nf{\ve{K}}} \f{d} + \lp -1 \rp^k \f{d} {\cal L}_{\nf{\ve{K}}} 
$$
In fact, the FuN derivative of a form with respect to the [[identity projection|vector projection]] is the exterior derivative,
$$
{\cal L}_{\nf{\ve{I}}} \nf{F} = \f{d} \nf{F}
$$
and of a VVF is zero, ${\cal L}_{\nf{\ve{I}}} \nf{\ve{K}} = 0$.

Acting on itself twice, the FuN bracket satisfies the ''graded Jacobi identity'',
$$
\lb \nf{\ve{K}}, \lb \nf{\ve{L}} , \nf{\ve{M}} \rb_L \rb_L = \lb \lb \nf{\ve{K}}, \nf{\ve{L}} \rb_L, \nf{\ve{M}} \rb_L - \lp -1 \rp^{kl} \lb \nf{\ve{L}}, \lb \nf{\ve{K}} , \nf{\ve{M}} \rb_L \rb_L
$$
The rank $2$ exceptional group, ''G2'', is a real, [[simple]], compact, connected [[Lie group|Lie group]]. It may be described by [[exponentiating|exponentiation]] its $14$ dimensional [[Lie algebra]], [[g2]].
<html>
<center>
<table class="gtable">
<tr border=none>

<td border=none>
<table class="ptable">
<tr>
<th COLSPAN="2"><SPAN class="math">G2+</SPAN></th>
<th></th>
<th><SPAN class="math">x</SPAN></th>
<th><SPAN class="math">y</SPAN></th>
<th><SPAN class="math">z</SPAN></th>
<th></th>
<th><SPAN class="math">g^3</SPAN></th>
<th><SPAN class="math">g^8</SPAN></th>
<th><SPAN class="math">\fr{\sqrt{8}}{\sqrt{3}} B_2</SPAN></th>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{r\bar{g}}</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{r}g}</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{r\bar{b}}</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{r}b}</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{g}b}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">-\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{g\bar{b}}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>

<tr>
<td><SPAN class="math">\btri{#D90000} </SPAN></td>
<td><SPAN class="math">q^r_I</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">\smash{\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#00BF00} </SPAN></td>
<td><SPAN class="math">q^g_I</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#0000F7} </SPAN></td>
<td><SPAN class="math">q^b_I</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\butr{#D90000} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^r_I</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\butr{#00BF00} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^g_I</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">\fr{1}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\butr{#0000F7} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^b_I</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">{\fr{1}{\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">\fr{1}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\rlap{\mutr{#0000F7}}{\mtri{#0000F7}}</SPAN></td>
<td><SPAN class="math">q_{II}</SPAN></td>
<td></td>
<td COLSPAN="3"><SPAN class="math">\mp 1</SPAN></td>
<td></td>
<td>"</td>
<td>"</td>
<td><SPAN class="math">\pm \fr{2}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\rlap{\stri{#0000F7}}{\sutr{#0000F7}}</SPAN></td>
<td><SPAN class="math">q_{III}</SPAN></td>
<td></td>
<td COLSPAN="3"><SPAN class="math">\pm 1 \;\; \pm \! 1</SPAN></td>
<td></td>
<td>"</td>
<td>"</td>
<td><SPAN class="math">\mp \fr{4}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\rlap{\butr{#999999}}{\btri{#999999}}</SPAN></td>
<td><SPAN class="math">l</SPAN></td>
<td></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\pm 1</SPAN></td>
</tr>
</table>
</td>

<td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>

<td>
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<html>
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<table class="gtable">
<tr border=none>

<td border=none>
<table class="ptable">
<tr>
<th COLSPAN="2"><SPAN class="math">G2+</SPAN></th>
<th></th>
<th><SPAN class="math">x</SPAN></th>
<th><SPAN class="math">y</SPAN></th>
<th><SPAN class="math">z</SPAN></th>
<th></th>
<th><SPAN class="math">g^3</SPAN></th>
<th><SPAN class="math">g^8</SPAN></th>
<th><SPAN class="math">\fr{\sqrt{8}}{\sqrt{3}} B_2</SPAN></th>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{r\bar{g}}</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{r}g}</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{r\bar{b}}</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{r}b}</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{g}b}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">-\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{g\bar{b}}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>

<tr>
<td><SPAN class="math">\btri{#D90000} </SPAN></td>
<td><SPAN class="math">q^r_I</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">\smash{\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#00BF00} </SPAN></td>
<td><SPAN class="math">q^g_I</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#0000F7} </SPAN></td>
<td><SPAN class="math">q^b_I</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\butr{#D90000} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^r_I</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\butr{#00BF00} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^g_I</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">\fr{1}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\butr{#0000F7} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^b_I</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">{\fr{1}{\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">\fr{1}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\rlap{\mutr{#0000F7}}{\mtri{#0000F7}}</SPAN></td>
<td><SPAN class="math">q_{II}</SPAN></td>
<td></td>
<td COLSPAN="3"><SPAN class="math">\mp 1</SPAN></td>
<td></td>
<td>"</td>
<td>"</td>
<td><SPAN class="math">\pm \fr{2}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\rlap{\stri{#0000F7}}{\sutr{#0000F7}}</SPAN></td>
<td><SPAN class="math">q_{III}</SPAN></td>
<td></td>
<td COLSPAN="3"><SPAN class="math">\pm 1 \;\; \pm \! 1</SPAN></td>
<td></td>
<td>"</td>
<td>"</td>
<td><SPAN class="math">\mp \fr{4}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\rlap{\butr{#999999}}{\btri{#999999}}</SPAN></td>
<td><SPAN class="math">l</SPAN></td>
<td></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\pm 1</SPAN></td>
</tr>
</table>
</td>

<td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>

<td>
<!-- <embed src="talks/Perimeter07/anim/g2spin.mov" width="452" height="452" controller="false" autoplay="false" loop="false"></embed>  -->
<embed src="talks/Perimeter07/anim/g2spin/p1.png" width="462" height="462"></embed>
</td>

</tr>
</table>
</center>
</html>
<<tiddler HideTags>>
<html>
<center>
<table class="gtable">
<tr border=none>

<td border=none>
<table class="ptable">
<tr>
<th COLSPAN="2"><SPAN class="math">G2+</SPAN></th>
<th></th>
<th><SPAN class="math">x</SPAN></th>
<th><SPAN class="math">y</SPAN></th>
<th><SPAN class="math">z</SPAN></th>
<th></th>
<th><SPAN class="math">g^3</SPAN></th>
<th><SPAN class="math">g^8</SPAN></th>
<th><SPAN class="math">\fr{\sqrt{8}}{\sqrt{3}} B_2</SPAN></th>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{r\bar{g}}</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{r}g}</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{r\bar{b}}</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{r}b}</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{g}b}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">-\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{g\bar{b}}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>

<tr>
<td><SPAN class="math">\btri{#D90000} </SPAN></td>
<td><SPAN class="math">q^r_I</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">\smash{\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#00BF00} </SPAN></td>
<td><SPAN class="math">q^g_I</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#0000F7} </SPAN></td>
<td><SPAN class="math">q^b_I</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\butr{#D90000} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^r_I</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\butr{#00BF00} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^g_I</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">\fr{1}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\butr{#0000F7} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^b_I</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">{\fr{1}{\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">\fr{1}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\rlap{\mutr{#0000F7}}{\mtri{#0000F7}}</SPAN></td>
<td><SPAN class="math">q_{II}</SPAN></td>
<td></td>
<td COLSPAN="3"><SPAN class="math">\mp 1</SPAN></td>
<td></td>
<td>"</td>
<td>"</td>
<td><SPAN class="math">\pm \fr{2}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\rlap{\stri{#0000F7}}{\sutr{#0000F7}}</SPAN></td>
<td><SPAN class="math">q_{III}</SPAN></td>
<td></td>
<td COLSPAN="3"><SPAN class="math">\pm 1 \;\; \pm \! 1</SPAN></td>
<td></td>
<td>"</td>
<td>"</td>
<td><SPAN class="math">\mp \fr{4}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\rlap{\butr{#999999}}{\btri{#999999}}</SPAN></td>
<td><SPAN class="math">l</SPAN></td>
<td></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\pm 1</SPAN></td>
</tr>
</table>
</td>

<td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>

<td>
<!-- <embed src="talks/Perimeter07/anim/g2spin.mov" width="452" height="452" controller="false" autoplay="false" loop="false"></embed>  -->
<embed src="talks/Perimeter07/anim/g2spin/p20.png" width="462" height="462"></embed>
</td>

</tr>
</table>
</center>
</html>
<<tiddler HideTags>>
<html>
<center>
<table class="gtable">
<tr border=none>

<td border=none>
<table class="ptable">
<tr>
<th COLSPAN="2"><SPAN class="math">G2+</SPAN></th>
<th></th>
<th><SPAN class="math">x</SPAN></th>
<th><SPAN class="math">y</SPAN></th>
<th><SPAN class="math">z</SPAN></th>
<th></th>
<th><SPAN class="math">g^3</SPAN></th>
<th><SPAN class="math">g^8</SPAN></th>
<th><SPAN class="math">\fr{\sqrt{8}}{\sqrt{3}} B_2</SPAN></th>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{r\bar{g}}</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{r}g}</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{r\bar{b}}</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{r}b}</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{g}b}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">-\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{g\bar{b}}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>

<tr>
<td><SPAN class="math">\btri{#D90000} </SPAN></td>
<td><SPAN class="math">q^r_I</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">\smash{\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#00BF00} </SPAN></td>
<td><SPAN class="math">q^g_I</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#0000F7} </SPAN></td>
<td><SPAN class="math">q^b_I</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\butr{#D90000} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^r_I</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\butr{#00BF00} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^g_I</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">\fr{1}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\butr{#0000F7} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^b_I</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">{\fr{1}{\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">\fr{1}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\rlap{\mutr{#0000F7}}{\mtri{#0000F7}}</SPAN></td>
<td><SPAN class="math">q_{II}</SPAN></td>
<td></td>
<td COLSPAN="3"><SPAN class="math">\mp 1</SPAN></td>
<td></td>
<td>"</td>
<td>"</td>
<td><SPAN class="math">\pm \fr{2}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\rlap{\stri{#0000F7}}{\sutr{#0000F7}}</SPAN></td>
<td><SPAN class="math">q_{III}</SPAN></td>
<td></td>
<td COLSPAN="3"><SPAN class="math">\pm 1 \;\; \pm \! 1</SPAN></td>
<td></td>
<td>"</td>
<td>"</td>
<td><SPAN class="math">\mp \fr{4}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\rlap{\butr{#999999}}{\btri{#999999}}</SPAN></td>
<td><SPAN class="math">l</SPAN></td>
<td></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\pm 1</SPAN></td>
</tr>
</table>
</td>

<td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>

<td>
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</td>

</tr>
</table>
</center>
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<<tiddler HideTags>>
<html>
<center>
<table class="gtable">
<tr border=none>

<td border=none>
<table class="ptable">
<tr>
<th COLSPAN="2"><SPAN class="math">G2+</SPAN></th>
<th></th>
<th><SPAN class="math">x</SPAN></th>
<th><SPAN class="math">y</SPAN></th>
<th><SPAN class="math">z</SPAN></th>
<th></th>
<th><SPAN class="math">g^3</SPAN></th>
<th><SPAN class="math">g^8</SPAN></th>
<th><SPAN class="math">\fr{\sqrt{8}}{\sqrt{3}} B_2</SPAN></th>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{r\bar{g}}</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{r}g}</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{r\bar{b}}</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{r}b}</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{g}b}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">-\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{g\bar{b}}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>

<tr>
<td><SPAN class="math">\btri{#D90000} </SPAN></td>
<td><SPAN class="math">q^r_I</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">\smash{\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#00BF00} </SPAN></td>
<td><SPAN class="math">q^g_I</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#0000F7} </SPAN></td>
<td><SPAN class="math">q^b_I</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\butr{#D90000} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^r_I</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\butr{#00BF00} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^g_I</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">\fr{1}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\butr{#0000F7} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^b_I</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">{\fr{1}{\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">\fr{1}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\rlap{\mutr{#0000F7}}{\mtri{#0000F7}}</SPAN></td>
<td><SPAN class="math">q_{II}</SPAN></td>
<td></td>
<td COLSPAN="3"><SPAN class="math">\mp 1</SPAN></td>
<td></td>
<td>"</td>
<td>"</td>
<td><SPAN class="math">\pm \fr{2}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\rlap{\stri{#0000F7}}{\sutr{#0000F7}}</SPAN></td>
<td><SPAN class="math">q_{III}</SPAN></td>
<td></td>
<td COLSPAN="3"><SPAN class="math">\pm 1 \;\; \pm \! 1</SPAN></td>
<td></td>
<td>"</td>
<td>"</td>
<td><SPAN class="math">\mp \fr{4}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\rlap{\butr{#999999}}{\btri{#999999}}</SPAN></td>
<td><SPAN class="math">l</SPAN></td>
<td></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\pm 1</SPAN></td>
</tr>
</table>
</td>

<td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>

<td>
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<html>
<center>
<table class="gtable">
<tr border=none>

<td border=none>
<table class="ptable">
<tr>
<th COLSPAN="2"><SPAN class="math">G2+</SPAN></th>
<th></th>
<th><SPAN class="math">x</SPAN></th>
<th><SPAN class="math">y</SPAN></th>
<th><SPAN class="math">z</SPAN></th>
<th></th>
<th><SPAN class="math">g^3</SPAN></th>
<th><SPAN class="math">g^8</SPAN></th>
<th><SPAN class="math">\fr{\sqrt{8}}{\sqrt{3}} B_2</SPAN></th>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{r\bar{g}}</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{r}g}</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{r\bar{b}}</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{r}b}</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{g}b}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">-\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{g\bar{b}}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>

<tr>
<td><SPAN class="math">\btri{#D90000} </SPAN></td>
<td><SPAN class="math">q^r_I</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">\smash{\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#00BF00} </SPAN></td>
<td><SPAN class="math">q^g_I</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#0000F7} </SPAN></td>
<td><SPAN class="math">q^b_I</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\butr{#D90000} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^r_I</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\butr{#00BF00} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^g_I</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">\fr{1}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\butr{#0000F7} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^b_I</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">{\fr{1}{\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">\fr{1}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\rlap{\mutr{#0000F7}}{\mtri{#0000F7}}</SPAN></td>
<td><SPAN class="math">q_{II}</SPAN></td>
<td></td>
<td COLSPAN="3"><SPAN class="math">\mp 1</SPAN></td>
<td></td>
<td>"</td>
<td>"</td>
<td><SPAN class="math">\pm \fr{2}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\rlap{\stri{#0000F7}}{\sutr{#0000F7}}</SPAN></td>
<td><SPAN class="math">q_{III}</SPAN></td>
<td></td>
<td COLSPAN="3"><SPAN class="math">\pm 1 \;\; \pm \! 1</SPAN></td>
<td></td>
<td>"</td>
<td>"</td>
<td><SPAN class="math">\mp \fr{4}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\rlap{\butr{#999999}}{\btri{#999999}}</SPAN></td>
<td><SPAN class="math">l</SPAN></td>
<td></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\pm 1</SPAN></td>
</tr>
</table>
</td>

<td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>

<td>
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<<tiddler HideTags>>
[>img[Garrett at Burning Man, 2004|images/person/Garrett.jpg]]Homepage: http://Li.si
*Email: Gar&#064;Li.si
*Location: Maui, usually
*arXiv papers: http://arxiv.org/find/gr-qc/1/au:+Lisi_A/0/1/0/all/0/1

Selected work:
*This Wiki.
*[[An Exceptionally Simple Theory of Everything]]
*[[Quantum mechanics from a universal action reservoir|http://arxiv.org/abs/physics/0605068]]

Some talks:
*[[talk for ILQGS 07]]
*[[talk for Perimeter Institute 07]]
*[[talk for FQXi 07]]
*[[talk for Loops 07]]





<<tiddler HideTags>>[>img[talks/CSUF09/images/FiberBundle_200.png]]Base [[manifold]]: &nbsp;&nbsp; $M$

A fiber, $F$, is a representation space of a Lie group, $G$.

Entire space of a [[fiber bundle]]: &nbsp;&nbsp; $E \sim M \times F$

For a [[principal bundle]], $G$ is the fiber: &nbsp;&nbsp; $E \sim M \times G$

[>img[images/png/fiber bundle.png]][[Ehresmann principal bundle connection]] over patches of $E$:
$$
\ve{\f{\cal E}}(x,y) = \f{dx^i} A_i^{\p{a}B}(x) \, \ve{T_B}(y) + \f{dy^p} \ve{\pa_p}
$$
Gauge field [[connection]] over $M$,
$$
\f{A}(x) = \f{dx^i} A_i^{\p{a}B}(x) \, T_B
$$
describes how fibers twist: &nbsp;&nbsp; $\f{D} \ps = (\f{d} + \f{A}) \ps$

[[Curvature|curvature]] is how the connection twists over $M$: &nbsp;&nbsp;&nbsp;&nbsp; $\ff{F} = \f{d} \f{A} + \f{A} \f{A}$
<<tiddler HideTags>>$$
\begin{array}{rcl}
S_\ps \!\!&\!\!=\!\!&\!\! 

\int \nf{e} \left\{ \bar{\ps} \ga^\mu \lp e_\mu\rp^i \big(
\pa_i 
+ \ha \om_i^{\p{i}\nu\rh} \ha \ga_{\nu\rh}
+ W_i^{\p{i}\pi} T^W_\pi
+ B_i T^Y
+ g_i^{\p{i}A} T^g_A
\big) \ud{\ps}
+ \bar{\ps} \ph \ud{\ps} \right\} \\[.5em]

\!\!&\!\!=\!\!&\!\!
\int \nf{e} \left\{ \bar{\ps} \ga^\mu \lp e_\mu\rp^i \big(
\pa_i 
+ \ha \om_i^{\p{i}\nu\rh} \ha \ga_{\nu\rh}
+ G_i^{\p{i}\al\be} \ha \ga_{\al\be}
+ \fr{1}{4} (e_i)^\nu \ph^\al \ga_\nu \ga_\al
\big) \ud{\ps}
\right\} \\[.5em]

\!\!&\!\!=\!\!&\!\!
\int \nf{e} \big\{ \bar{\ps} \ve{e} \big( \f{d} + \f{H} \big) \ud{\ps} \big\} \\[.5em]

\!\!&\!\!=\!\!&\!\!
\int \fr{1}{4!} \bar{\ps} \f{e} \f{e} \f{e} \ep \f{D} \ud{\ps}
\end{array}
$$
''Unified bosonic connection'':
$$
\f{H} = {\tiny \frac{1}{2}} \f{\om} + {\tiny \frac{1}{4}} \f{e} \ph + \f{G} \;\; \in spin(?)
$$
With SO(10) GUT:
$$
spin(3,1) + 4 \!\times\! 10 + spin(10) = spin(3,11) \mbox{ or } spin(13,1)
$$
or with Pati-Salam GUT:
$$
spin(3,1) + 4 \!\times\! 4 + spin(4) + spin(6) \subset spin(3,11), spin(13,1), spin(7,7) \mbox{ or } spin(9,5)
$$

One generation of fermions:
$$
64_S^{+\mathbb{R}} \mbox{ of } spin(3,11) \mbox{ or } spin(7,7)
$$
The eight [[trace]]less, Hermitian, ''Gell-Mann matrices'', $\la_A$, are
$$
\begin{array}{cccc}
\la_0 = \la_8 = \fr{1}{\sqrt{3}} \left[\begin{array}{ccc}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & -2
\end{array}\right]
&
\la_1 = \left[\begin{array}{ccc}
0 & 1 & 0\\
1 & 0 & 0\\
0 & 0 & 0
\end{array}\right]
&
\la_2 = \left[\begin{array}{ccc}
0 & -i & 0\\
i & 0 & 0\\
0 & 0 & 0
\end{array}\right]
&
\la_3 = \left[\begin{array}{ccc}
1 & 0 & 0\\
0 & -1 & 0\\
0 & 0 & 0
\end{array}\right]
\\
\la_4 = \left[\begin{array}{ccc}
0 & 0 & 1\\
0 & 0 & 0\\
1 & 0 & 0
\end{array}\right]
&
\la_5 = \left[\begin{array}{ccc}
0 & 0 & -i\\
0 & 0 & 0\\
i & 0 & 0
\end{array}\right]
&
\la_6 = \left[\begin{array}{ccc}
0 & 0 & 0\\
0 & 0 & 1\\
0 & 1 & 0
\end{array}\right]
&
\la_7 = \left[\begin{array}{ccc}
0 & 0 & 0\\
0 & 0 & -i\\
0 & i & 0
\end{array}\right]
\end{array}
$$
<<tiddler HideTags>>
Three generations of fermions in three copies of $64^\mathbb{R}_{S+}$ of $spin(11,3)$, differing only in mass.

Triality?
$\;\;\;\;\;\;\;\;\;$ Maps between three blocks of $64$ in E8.
$\;\;\;\;\;\;\;\;\;$ These blocks have different quantum numbers -- doesn't seem viable.
$\;\;\;\;\;\;\;\;\;$ Look more closely at symmetry breaking.

Larger Lie group or supergroup?
$\;\;\;\;\;\;\;\;\;$ Orthosymplectic, $D(7,3)$ or ?

Larger algebra?
$\;\;\;\;\;\;\;\;\;$ E9. Possible relation to QFT.
$\;\;\;\;\;\;\;\;\;$ Leech lattice. Three E8's as inner shell.
$\;\;\;\;\;\;\;\;\;$ Kac-Moody algebras.

Axions?
$\;\;\;\;\;\;\;\;\;$ Use Peccei-Quinn charge, $w$, in E8 (and E6) and scalars in E8. $\;\;\;\; \th \ff{F} \ff{F} \;\;\;\;\; \big< \bar{\ps} \f{e} \th \f{e} \th \f{e} \th \ep \f{D} \ud{\ps} \big>$
$\;\;\;\;\;\;\;\;\;$ Fermions of different generations as axion-fermion composites. $\;\;\; \th+\ps$
$\;\;\;\;\;\;\;\;\;$ Used successfully in the past for solving the strong CP problem, and dealing with mirror fermions.
$\;\;\;\;\;\;\;\;\;$ E8 appears to come with a nice axion model building kit. 

Something weirder?
<<tiddler HideTags>>$$
\begin{array}{rcl}
S_D \!\!&\!\!=\!\!&\!\! \int \nf{d^4x} |e| \big\{ \bar{\ps} \ga^\mu (e_\mu)^i \big( \pa_i +\fr{1}{4} \om_i^{\p{i}\mu\nu} \ga_{\mu\nu} + G_i^{\p{i}B} T_B \big) \ud{\ps} + \bar{\ps} \ph \ud{\ps} \big\} \\
        \!\!&\!\!=\!\!&\!\! \int \nf{d^4x} |e| \big\{ \bar{\ps} \ga^\mu (e_\mu)^i \big( \pa_i + \fr{1}{4} \om_i^{\p{i}\mu\nu} \ga_{\mu\nu} + \ha G_i^{\p{i}\ps\ch} \ga_{\ps\ch} + \fr{1}{4} (e_i)^\mu \ph^\ps \ga_{\mu\ps} \big) \ud{\ps} \big\} \\
\!\!&\!\!=\!\!&\!\! \int \nf{e} \big\{ \bar{\ps} \ve{e} \big( \f{d} + \f{H} \big) \ud{\ps} \big\}
= \int \fr{1}{4!} \bar{\ps} \f{e} \f{e} \f{e} \ep \f{D} \ud{\ps}
= \int \nf{e} \bar{\ps} D \!\!\!\!/\; \ud{\ps}
\end{array}
$$
Gravity: $\;\;\;\; \f{\om} = \ha \f{dx^i} \, \om_i^{\p{i}\mu\nu} \ga_{\mu\nu}  \;\; \in Cl(3,1)^2 = spin(3,1)
\;\;\;\;\;\;\;\;\; \f{e} = \f{dx^i} (e_i)^\mu \ga_\mu \;\; \in Cl(3,1)^1 = 4 \vp{A_{\big(}}$
GUT: $\;\;\;\; \f{G} \;\; \in \, su(2)_L+u(1)_Y+su(3)$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \subset su(2)_L+su(2)_R+su(4) = spin(4)+spin(6) \;\; \subset spin(10) \vp{A_{\big(}}$
Fermions: $\;\;\;\; \ud{\ps} \;\; \in 2 \!\times\! (2_L \!+\! 2_R) \!\times\! (1+3) = \mathbb{C}^{32} = \mathbb{R}^{64} \;\;\; (\times 3){}^{\p{\big(}}$
Higgs: $\;\;\;\; \ph=\ph^\ps \ga_\ps \;\; \in Cl(4)^1 = 4 = \mathbb{C}^2 \;$ or $\; Cl(N)^1 = N{}^{\p{\big(}}$
ToE: $\;\;\;\; \f{H} = \ha \f{\om} + \fr{1}{4} \f{e} \ph + \f{G} \;\; \in spin(3,1) + 4 \!\times\! 10 + spin(10) {}^{\p{\Big(}}$
Curvature: $\;\;\;\; \ff{F} = \f{d} \f{H} + \f{H} \f{H} 
= \ha (\ff{R} - \fr{1}{8} \f{e}\f{e} \ph^2) + \fr{1}{4} (\f{T} \ph - \f{e} \f{D} \ph) + \ff{F}^G{}^{\p{\big(}}$
Superconnection: $\;\;\;\; \udf{A} = \f{H} + \ud{\ps} \;\; \in spin(3,11) + 64_S^{+\mathbb{R}}   {}^{\p{\big(}}$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\, \subset spin(4,12) + 128_S^{+\mathbb{R}} = E8(-24)$
Supercurvature: $\;\;\;\; \udff{F} = \f{d} \udf{A} + \udf{A} \udf{A} = \ff{F} + \f{D} \ud{\ps} + \ud{\ps} \ud{\ps}{}^{\p{\big(}}$
$$
S = \int \left\{ \ff{\bar{B}} \udff{F} - {\textstyle \fr{\pi G}{4}} \ff{B} \ep \ff{B} + {\small g^2} \ff{B'} \ff{*B'} \right\} \sim \int \left\{ \nf{e} \bar{\ps} D \!\!\!\!/\; \ud{\ps} + {\textstyle \fr{1}{16\pi G}} \nf{e} \big( R - {\textstyle \fr{3}{2}} \ph^2 \big) \ph^2 + {\textstyle \fr{1}{4g^2}} \ff{F'} \ff{*F'} \right\} 
$$
<<tiddler HideTags>>Start with a [[Lie group manifold|Lie group geometry]] (//torsor//), $G$, coordinatized by $y^p$.
Two sets of invariant vector fields (//symmetries, [[Killing vector]] fields//):
[>img[images/png/torsor.png]]$$
\ve{\xi^L_A}(y) \, \f{d} g = T_A \, g(y) \;\;\;\;\;\;\;\; \ve{\xi^R_A}(y) \, \f{d} g = g(y) \, T_A
$$
[[Lie derivative]]: &nbsp;~~&nbsp;&nbsp;~~ $[ \ve{\xi^R_A}, \ve{\xi^R_B} ] = C_{AB}^{\p{AB}C} \ve{\xi^R_C}$
[[Lie bracket|Lie algebra]]: &nbsp;&nbsp;~~&nbsp;~~&nbsp;&nbsp;&nbsp; $\lb T_A, T_B \rb = C_{AB}^{\p{AB}C} T_C$
[[Killing form]] (//[[Minkowski metric]]//): ^^&nbsp;^^ $g_{AB} = C_{AC}^{\p{AC}D} C_{BD}^{\p{BD}C}$
[[Maurer-Cartan form]] (//[[frame]]//): ^^&nbsp;^^&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; $\f{\cal I} = \f{dy^p} ( \xi^R_p )^A T_A$
Entire space of a [[principal bundle]]: &nbsp;&nbsp; $E \sim M \times G^{\p{\big(}}$
[[Ehresmann principal bundle connection]] over patches of $E$:
[>img[images/png/fiber bundle.png]]$$
\ve{\f{\cal E}}(x,y) = \f{dx^i} A_i^{\p{a}B}(x) \, \ve{\xi^L_B}(y) + \f{dy^p} \ve{\pa_p}
$$
Gauge field [[connection]] over $M$:
$$
\f{A}(x) = \si_0^* \ve{\f{\cal E}} \f{\cal I} = \f{dx^i} A_i^{\p{a}B}(x) \, T_B
$$
<<tiddler HideTags>>
$$
\begin{array}{rcl}
L \!\!&\!\!=\!\!&\!\! \bar{\ps} \ve{e} \lp \f{\pa} + {\small \frac{1}{4}} \f{\om}^{a b} \ga_{a b} + {\small \frac{1}{4}} \f{e}^a \ph^m \ga_{a m} + {\small \frac{1}{2}} \f{W}^{m n} \ga_{m n} + {\small \frac{1}{2}} \f{B}^{m n} \ga_{m n} + {\small \frac{1}{2}} \f{g}^{m n} \ga_{m n}  \rp \ps
\end{array}
$$
<html>
<table class="gtable">

<tr>
<td>
<img SRC="talks/CSUF09/images/planes.png">
</td>
<td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>
<td>
<SPAN class="math">
\begin{array}{c}
spin(3) \\
\begin{array}{rcl}
\lb \ga_{zy}, \ga_{xz} \rb \!\!&\!\!=\!\!&\!\! \ga_{xy} \\[2em]
\lb \ga_{xy}, \ga_{zt} \rb \!\!&\!\!=\!\!&\!\! 0
\end{array}

\end{array}
</SPAN>
</td>
</tr>

</table>
</html>
<<tiddler HideTags>>

<html>
<table class="gtable">

<tr>
<td>
<img SRC="talks/CSUF09/images/planes.png">
</td>
<td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>
<td>

<table class="gtable">
<tr><td>
<SPAN class="math">
\begin{array}{c}
spin(3) \\[1.5em]
\large{ \lb \ga_{zy}, \ga_{xz} \rb = \ga_{xy}} \\[1.5em]
\end{array}
</SPAN>
</tr></td>
</table>

</td>
</tr>
</table>
</html>
<html>
<center>
<table class="gtable">

<tr>
<td COLSPAN="3">
<SPAN class="math">su(5) + \bar{5} + 10 + \bar{"}</SPAN>
</td>
</tr>

<tr>
<td>
Proton decay:
<br><br>
<SPAN class="math">p = u^r + u^b + d^g</SPAN>
<br><br>
<SPAN class="math">d^g \to \bar{e} + X^g_{-\fr{4}{3}}</SPAN>
<br><br>
<SPAN class="math">X^g_{-\fr{4}{3}} + u^b \to \bar{u}{}^{\bar{r}} </SPAN>
<br><br>
<SPAN class="math">
\begin{array}{rcl}
p \!\!&\!\!\to\!\!&\!\! \bar{e} + u^r + \bar{u}{}^{\bar{r}} \\
\!\!&\!\!=\!\!&\!\! \bar{e} + \pi^0
\end{array}</SPAN>
</td>
<td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>
<td>
<img SRC="talks/CSUF09/images/SU5Electric.png">
</td>
</tr>

</table>
</center>
</html>
<<tiddler HideTags>>
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/RW12/images/Georgi-Glashow.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math">su(5) \,\,\oplus\,\, \bar{5} \,\oplus\, 10</SPAN>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>>$$
g = g^A T_A =  g^A \fr{i}{2} \la_A
= \fr{i}{2}
\lb
\begin{array}{ccc}
\! g^3 \!+\! {\scriptsize \frac{1}{\sqrt{3}}} g^8 \! & g^1 \!-\! ig^2 \!\! & g^4 \!-\! ig^5 \\
g^1 \!+\! i g^2 & \!\!\! -\! g^3 \!+\! {\scriptsize \frac{1}{\sqrt{3}}} g^8 \!\! & g^6 \!-\! ig^7 \\
g^4 \!+\! i g^5 & \!\! g^6 \!+\! i g^7 & {\scriptsize -\!\frac{2}{\sqrt{3}}} g^8 \!
\end{array}
\rb
$$
Cartan subalgebra: $\qquad C = g^3 T_3 + g^8 T_8 \quad$ (the diagonal)
Roots and root vectors:
$$
\big[ C , V_{g^{g\bar{b}}} \big] = i \lp \Big( -\!\fr{1}{2} \Big) g^3  + \Big( \fr{\sqrt{3}}{2} \Big) g^8 \rp V_{g^{g\bar{b}}}
\qquad
V_{g^{g\bar{b}}} =
\lb \matrix{
0 & 0 & 0 \cr
0 & 0 & 1 \cr
0 & 0 & 0 \cr
} \rb
$$
for the $g^{g\bar{b}}$ gluon. Weights and weight vectors:
$$
C \, V_{q^r} = i \lp \Big( \fr{1}{2} \Big) g^3 + \Big( \fr{1}{2\sqrt{3}} \Big) g^8 \rp V_{q^r}
\qquad
V_{q^r} = [ 1,0,0 ]
$$
for a red quark, $q^r$, and for their duals acted on by $-C^T$, the anti-quarks.
<<tiddler HideTags>>$$
g = g^A T_A =  g^A \fr{i}{2} \la_A
= \fr{i}{2}
\lb
\begin{array}{ccc}
\! g^3 \!+\! {\scriptsize \frac{1}{\sqrt{3}}} g^8 \! & g^1 \!-\! ig^2 \!\! & g^4 \!-\! ig^5 \\
g^1 \!+\! i g^2 & \!\!\! -\! g^3 \!+\! {\scriptsize \frac{1}{\sqrt{3}}} g^8 \!\! & g^6 \!-\! ig^7 \\
g^4 \!+\! i g^5 & \!\! g^6 \!+\! i g^7 & {\scriptsize -\!\frac{2}{\sqrt{3}}} g^8 \!
\end{array}
\rb
$$
Cartan subalgebra: $\qquad C = g^3 T_3 + g^8 T_8 \quad$ (the diagonal)
Root and root vector for the $g^{g\bar{b}}$ gluon in $su(3)$:
$$
\big[ C , T_{g^{g\bar{b}}} \big] = i \lp \Big( -\!\fr{1}{2} \Big) g^3  + \Big( \fr{\sqrt{3}}{2} \Big) g^8 \rp T_{g^{g\bar{b}}}
\qquad
T_{g^{g\bar{b}}} = T_7 -i T_6 =
\lb \matrix{
0 & 0 & 0 \cr
0 & 0 & 1 \cr
0 & 0 & 0 \cr
} \rb
$$
Weight and weight vector for the red quark, $q^r$, in the $3$:
$$
C \, \ps_{q^r} = i \lp \Big( \fr{1}{2} \Big) g^3 + \Big( \fr{1}{2\sqrt{3}} \Big) g^8 \rp \ps_{q^r}
\qquad
\ps_{q^r} =
\lb \matrix{
1 \cr
0 \cr
0 \cr
} \rb
$$
The dual anti-quarks in the $\bar{3}$ are acted on by $-C^T$ and have the opposite weights.
<<tiddler HideTags>>Embed the standard model gauge algebra and fermion representation space in the Lie algebra and representation space of a larger group.
[>img[talks/CSUF09/images/coupling.png]]
''Standard Model'' $\p{F^{\big(}}$
\begin{eqnarray}
G_{SM} &=& su(2)_L \,\oplus\, u(1)_Y \,\oplus\, su(3) \\[.25em]
\ps_{SM} &=& (2_L\!\,\oplus\,\!2_R) \!\otimes\! (1\!\,\oplus\,\!3) \\[.75em]
\end{eqnarray}
''Georgi-Glashow SU(5)''
\begin{eqnarray}
G_{SM} \subset G_{SU(5)} &=& su(5) \\[.25em]
\ps_{SM} \supset \ps_{SU(5)} &=& \bar{5} \,\oplus\, 10 \\[.75em]
\end{eqnarray}
''Pati-Salam''
\begin{eqnarray}
G_{SM} \subset G_{PS} &=& su(2)_L \,\oplus\, su(2)_R \,\oplus\, su(4) = spin(4) \,\oplus\, spin(6) \\[.25em]
\ps_{SM} = \ps_{PS} &=& (2_L\!\,\oplus\,\!2_R) \!\otimes\! 4 = 4 \!\otimes\! 4 \\[.75em]
\end{eqnarray}
''Spin(10)''
\begin{eqnarray}
G_{SU(5)} \subset G_{Spin(10)} \;\;\;\;\;\;\; G_{PS} \subset G_{Spin(10)} \;\;\;\;\;\;\; G_{Spin(10)} &=& spin(10) \\[.25em]
G_{SM} = G_{SU(5)} \cap G_{PS} \subset G_{Spin(10)} \;\;\;\;\;\;\;\;\;\;\;\;\;\; \ps_{SM} &=& 16^\mathbb{C}_{S+}
\end{eqnarray}
<<tiddler HideTags>>Gauge field Lie algebra (not including gravity) embeds in a simpler Lie algebra.

$$\begin{array}{rcl}
\f{A} \,\in\, G_{SM} \!\!&\!\!=\!\!&\!\! su(2)_L \oplus u(1)_Y \oplus su(3) \\[.5em]
\!\!&\!\!\subset\!\!&\!\! su(2)_L \oplus su(2)_R \oplus su(4) \\[.5em]
\!\!&\!\!=\!\!&\!\! spin(4) \oplus spin(6) \\[.5em]
\!\!&\!\!\subset\!\!&\!\! spin(10)
\end{array}
$$

$$\begin{array}{rcl}
\ps_{SM} \!\!&\!\!\in\!\!&\!\! (2_L\!\oplus\!2_R) \!\otimes\! (1\!\oplus\!3) \\[.5em]
\!\!&\!\!=\!\!&\!\! 16^\mathbb{C}_{S+}
\end{array}
$$
Real ''Grassmann numbers'', $\ud{a},\ud{b} \in \mathbb{G}$, are like real numbers but they anti-commute with each other, $\ud{a} \ud{b} = - \ud{b} \ud{a}$, and commute with reals.  The square of a Grassmann number is necessarily zero, $\ud{a} \ud{a} = 0$.  The product of two real Grassman numbers is a real number (acording to [[Ramond|http://www.amazon.com/Field-Theory-Modern-Frontiers-Physics/dp/0201304503/ref=pd_bbs_sr_1/104-9709999-3726336?ie=UTF8&s=books&qid=1177293245&sr=8-1]]),
$$
\lp \ud{a} \ud{b} \rp^* = - \ud{b}^* \ud{a}^* = - \ud{b} \ud{a} = \ud{a} \ud{b} \in \mathbb{R}
$$
Since Grassmann numbers square to zero, the Taylor expansion of any function of Grassmann variables terminates at the first order,
$$
f(\ud{c}) = a + b \, \ud{c}  
$$
Derivatives work as for real numbers (but make sure to change the sign when commuting them past other Grassmann numbers). Using the example above,
$$
\fr{\pa}{\pa \ud{c}} f(\ud{c}) = b
$$
Integrals are effectively the same as derivatives,
$$
\int{\ud{dc} \, f(\ud{c})} = b
$$
Using this rule, for two sets of Grassmann variables, $\ud{a^i},\ud{b^j}$, and a real matrix, $A$, integration gives the [[determinant]],
$$
\int{\ud{da} \, \ud{db} \, \exp(\ud{a^i} A_{ij} \ud{b^j}}) = \det A
$$

For a compex Grassmann number, $\ud{z} = \ud{x} + i \ud{y}$, its square and norm are:
\begin{eqnarray}
\ud{z} \ud{z} &=& i \ud{x} \ud{y} + i \ud{y} \ud{x} = 0 \\
\ud{z}^* \ud{z} &=& i \ud{x} \ud{y} - i \ud{y} \ud{x} = 2 i \ud{x} \ud{y} = - \ud{z} \ud{z}^* \in \mathbb{I} 
\end{eqnarray}

An ''anti-Grassmann number'', $\od{a}$, contracts with a Grassmann number to give a real, $\od{a} \ud{b} \in \mathbb{R}$, just like in [[vector-form algebra]]. In fact, a Grassmann number may be thought of as a [[1-form]] in the space of functions. With this interpretation, the product of two Grassmann numbers is not a real, but a ''Grassmann grade two number''.
<<tiddler HideTags>>$$\begin{array}{rcl}
0 \!\!&\!\!=\!\!&\!\! \ve{e} \lp \f{d} + {\textstyle \fr{1}{4}} \f{\om}{}^{ab} \Ga_{ab}  + \ha \f{A}{}^{xy} \Ga_{xy} \rp \ps + \ph^x \Ga_x \, \ps \\[.5em]
 \!\!&\!\!=\!\!&\!\! \Ga^a \ve{e}{}_a \lp \f{d} + {\textstyle \fr{1}{4}} \f{\om}{}^{bc} \Ga_{bc}  + \fr{1}{4} \f{e}{}^b \ph^x \Ga_{bx}  + \ha \f{A}{}^{xy} \Ga_{xy} \rp \ps \\[.5em]
 \!\!&\!\!=\!\!&\!\! \ve{e} \lp \f{d} + {\textstyle \fr{1}{2}} \f{\om} + \fr{1}{4} \f{e} \ph + \f{A} \rp \ps \\[.5em]
 \!\!&\!\!=\!\!&\!\! \ve{e} \lp \f{d} + \f{H} \rp \ps
\end{array}
$$
| $\; \f{H} = \ha \f{\om} + \fr{1}{4} \f{e} \ph + \f{A}  \; $ |$\in \; spin(1,3) \,\oplus\, 4 \otimes 10 \,\oplus\, spin(10) = spin(11,3)$ |unified bosonic connection |
| $\; \ps \; $ |$\in \; 64^\mathbb{R}_{S+} \;\;\;\;\; (\otimes 3)$ |all spinor fermion multiplets |

''Curvature''
$$
\ff{F} = \f{d} \f{H} + {\textstyle \fr{1}{2}} [ \f{H}, \f{H} ] = {\textstyle \ha} (\ff{R} - {\textstyle \fr{1}{8}} \f{e}\f{e} \ph^2) + {\textstyle \fr{1}{4}} (\f{T} \ph - \f{e} \f{D} \ph) + \ff{F}{}^A
$$
| $\; \ff{R} = \f{d} \f{\om} + \ha \f{\om} \f{\om}  \; $ |$\in \; spin(1,3) $ |Riemann curvature |
| $\; \ff{T} = \f{d} \f{e} + \ha \f{\om} \f{e} + \ha \f{e} \f{\om}   \; $ |$\in \; 4_V $ |Torsion |
| $\; \f{D} \ph = \f{d} \ph + \f{A} \ph - \ph \f{A}  \; $ |$\in \; 10_V$ |Covariant Higgs derivative |
| $\; \ff{F}{}^A = \f{d} \f{A} + \f{A} \f{A} \; $ |$\in \; spin(10)$ |Gauge curvature |
<<tiddler HideTags>>$$
\begin{array}{rcl}
S_\ps \!\!&\!\!=\!\!&\!\! 

\int \nf{e} \left\{ \bar{\ps} \ga^a \lp e_a \rp^\mu \big(
\pa_\mu 
+ \ha \om_\mu^{\p{\mu}bc} \ha \ga_{bc}
+ W_\mu^{\p{\mu}I} T^W_I
+ B^Y_\mu T^Y
+ g_\mu^{\p{\mu}A} T^g_A
\big) \ud{\ps}
+ \bar{\ps} \ph \ud{\ps} \right\} \\[.5em]

S^U_\ps \!\!&\!\!=\!\!&\!\!
\int \nf{e} \left\{ \bar{\ps} \Ga^a \lp e_a \rp^\mu \big(
\pa_\mu 
+ \ha \om_\mu^{\p{\mu}bc} \ha \Ga_{bc}
+ A_\mu^{\p{\mu}xy} \ha \Ga_{xy}
+ \fr{1}{4} (e_\mu)^b \ph^x \Ga_b \Ga_x
\big) \ud{\ps}
\right\} \\[.5em]

\!\!&\!\!=\!\!&\!\!
\int \nf{e} \big\{ \bar{\ps} \ve{e} \big( \f{d} + \f{H} \big) \ud{\ps} \big\} \\[.5em]

\!\!&\!\!=\!\!&\!\!
\int \fr{1}{4!} \bar{\ps} \f{e} \f{e} \f{e} \ep \f{D} \ud{\ps}
\end{array}
$$
''Unified bosonic connection'':
$$
\f{H} = {\tiny \frac{1}{2}} \f{\om} + {\tiny \frac{1}{4}} \f{e} \ph + \f{A} \;\; \in spin(?)
$$
With SO(10) GUT:
$$
spin(1,3) \,\oplus\, 4 \!\otimes\! 10 \,\oplus\, spin(10) = spin(11,3) \mbox{ or } spin(13,1)
$$
or with Pati-Salam GUT:
$$
spin(1,3) \,\oplus\, 4 \!\otimes\! 4 \,\oplus\, spin(4) \,\oplus\, spin(6) \subset spin(11,3), spin(13,1), spin(7,7) \mbox{ or } spin(9,5)
$$

One generation of fermions:
$$
64^\mathbb{R}_{S^+} \mbox{ of } spin(11,3) \mbox{ or } spin(7,7)
$$
<html>
<center>
<table class="gtable">
<tr border=none>

<td><div class="math">
\begin{array}{l}
\om = \ha {\om}{}^{\mu \nu} \ga_{\mu\nu} = 
\lb \begin{array}{cc}
{\om}{}_L &  \\
 & {\om}{}_R
\end{array} \rb \vp{|_{\Big(}}
\\
\qquad \qquad \qquad \qquad
{\om}{}_{L/R} =
\lb \begin{array}{cc}
i {\om}{}_{L/R}^3 & {\om}{}_{L/R}^\wedge \\
{\om}{}_{L/R}^\vee & -i {\om}{}_{L/R}^3
\end{array} \rb \vp{|_{\Big(_{\big(}}}
\\
{e} = {e}{}^\mu \ga_\mu
=
\lb \begin{array}{cc}
 & {e}{}_R \\
{e}{}_L &  
\end{array} \rb \vp{|_{\Big(}}
\\
\qquad \qquad \qquad \qquad
{e}{}_{L/R} =
\lb \begin{array}{cc}
{e}{}_T^{\wedge/\vee} & \mp {e}{}_S^\wedge \\
\mp  {e}{}_S^\vee & {e}{}_T^{\vee/\wedge}
\end{array} \rb \vp{|_{\Big(_{\big(}}}
\\
{f} =
\lb \begin{array}{c}
{f}{}_L \\ {f}{}_R
\end{array} \rb
\qquad \quad \;
{f}{}_{L/R} =
\lb \begin{array}{c}
{f}{}_{L/R}^\wedge \\ {f}{}_{L/R}^\vee
\end{array} \rb
\end{array}
</div></td>

<td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>

<td border=none>
<table class="ptable">
<tr>
<th ALIGN=CENTER COLSPAN="2"><SPAN class="math">SO(3,1)</SPAN></th>
<th></th>
<th ALIGN=CENTER><SPAN class="math">\ha \om_L^3</SPAN></th>
<th ALIGN=CENTER><SPAN class="math">\ha \om_R^3</SPAN></th>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\mcir{#59FF59} </SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\om_L^\wedge</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">1</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\mcir{#59FF59} </SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\om_L^\vee</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">-1</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\mcir{#59FF59} </SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\om_R^\wedge</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">1</SPAN></td>
</tr>
<tr class="butt">
<td ALIGN=CENTER><SPAN class="math">\mcir{#59FF59} </SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\om_R^\vee</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-1</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\msqu{#FF5959}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">e_S^\wedge</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\ha</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\ha</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\msqu{#FF5959}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">e_S^\vee</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">-\ha</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-\ha</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\msqu{#FF5959}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">e_T^\wedge</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">-\ha</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\ha</SPAN></td>
</tr>
<tr class="butt">
<td ALIGN=CENTER><SPAN class="math">\msqu{#FF5959}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">e_T^\vee</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\ha</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-\ha</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\btri{#F2F200}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">f_L^\wedge</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\ha</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\btri{#F2F200}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">f_L^\vee</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">-\ha</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\btri{#999999}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">f_R^\wedge</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\ha</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\btri{#999999}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">f_R^\vee</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-\ha</SPAN></td>
</tr>
</table>
</td>

</tr>
</table>
</center>
</html>
<<tiddler HideTags>>
<<tiddler HideTags>>\begin{eqnarray}
S_s &=& \int \big< \ff{B_s} \ff{F_s} + \Ph_s(\ff{B_s}) \big> \,
= \int \big< \ff{B_s} {\scriptsize \frac{1}{2}} \big( \ff{R} + {\scriptsize \frac{1}{8}}M^2 \f{e}\f{e} \big) - {\scriptsize \frac{1}{4}} \ff{B_s} \ff{B_s} \ga \big> \\
&& \de \ff{B_s} \rightarrow \ff{B_s} = \big( \ff{R} + {\scriptsize \frac{1}{8}} M^2 \f{e}\f{e} \big) \ga^- 
\;\;\;\;\;\; \text{pseudoscalar:} \;\; \ga = {\ga_0 \ga_1 \ga_2 \ga_3}{\phantom{\Bigg(}} \\

S_s &=& {\scriptsize \frac{1}{4}} \int \big< \big( \ff{R} + {\scriptsize \frac{1}{8}} M^2 \f{e}\f{e} \big) \big( \ff{R} + {\scriptsize \frac{1}{8}} M^2 \f{e}\f{e} \big) \ga^- \big>
= \int \big< \ff{F_s} \ff{F_s} \ga^- \big> \\
&& \big< \ff{R} \ff{R} \ga^- \big> = \f{d} \big< \big( \f{\om} \f{d} \f{\om} + {\scriptsize \frac{1}{3}} \f{\om} \f{\om} \f{\om} \big) \ga^- \big> 
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,
 \leftarrow \text{Chern-Simons} \\
&& {\scriptsize \frac{1}{4!}} \big< \f{e}\f{e} \f{e} \f{e} \ga^- \big> = \nf{e} 
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
 \leftarrow \text{volume element} \\
&& \big< \f{e}\f{e} \ff{R} \, \ga^- \big> = \nf{e} R_{\p{\Big(}}
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
 \leftarrow \text{curvature scalar} \\

S_s &=& {\scriptsize \frac{\La}{12}} \int \nf{e} \lp R + 2 \La \rp
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\text{cosmological constant:}_{\p{\big(}} \;\; \La = {\scriptsize \frac{3}{4}} M^2
\end{eqnarray}
<<tiddler HideTags>>\begin{eqnarray}
S_G &=& 
\int \big< \ff{B}{}_G \ff{F}{}_G + {\scriptsize \frac{\pi G}{4}} \ff{B}{}_G \ff{B}{}_G \ga \big>
\qquad
\ff{F}{}_G = {\scriptsize \frac{1}{2}} \big( \ff{R} - {\scriptsize \frac{1}{8}} \f{e} \f{e} \ph^2 \big)
\in \ff{so}(3,1)
\\
&& \de \ff{B}{}_G \rightarrow \ff{B}{}_G = \fr{1}{\pi G} \big( \ff{R} - {\scriptsize \frac{1}{8}} \f{e}\f{e} \ph^2 \big) \ga
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\ga = {\ga_1 \ga_2 \ga_3 \ga_4}{\phantom{\Bigg(}} \\

S_G &=& 
{\scriptsize \frac{1}{\pi G}} \int \big< \ff{F}{}_G \ff{F}{}_G \ga \big>
=
{\scriptsize \frac{1}{4 \pi G}} \int \big< \big( \ff{R} - {\scriptsize \frac{1}{8}} \f{e}\f{e} \ph^2 \big) \big( \ff{R} - {\scriptsize \frac{1}{8}} \f{e}\f{e} \ph^2 \big) \ga \big> \\[.6em]
&& \big< \ff{R} \ff{R} \ga \big> = \f{d} \big< \big( \f{\om} \f{d} \f{\om} + {\scriptsize \frac{1}{3}} \f{\om} \f{\om} \f{\om} \big) \ga \big> 
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,
 \leftarrow \text{Chern-Simons} \\
&& {\scriptsize \frac{1}{4!}} \big< \f{e}\f{e} \f{e} \f{e} \ga \big> = - \nf{e} 
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,
 \leftarrow \text{volume element} \\
&& \big< \f{e}\f{e} \ff{R} \, \ga \big> = - \nf{e} R_{\p{\Big(}}
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,
 \leftarrow \text{curvature scalar} \\

S_G &=& {\scriptsize \frac{1}{16\pi G}} \int \nf{e} \, \ph^2 \lp R - {\scriptsize \frac{3}{2}} \ph^2 \rp
\;\;\;\;\;\;
\text{cosmological constant:}_{\p{\big(}} \;\; \La = {\scriptsize \frac{3}{4}} \ph^2
\end{eqnarray}
<<tiddler HideTags>>Using [[chiral]] (//Weyl//) $\mathbb{C}(4 \times 4)$ representation of [[Cl(1,3)]] [[Dirac matrices]]:
$$
\begin{array}{rclrcl}
\ga_0 \!\!&\!\!=\!\!&\!\! \si_1 \otimes 1
=
\lb \begin{array}{cc}
 & 1 \\
1 &
\end{array} \rb_{\p{\big(}} 
& \;\;\;\;
\ga_\pi \!\!&\!\!=\!\!&\!\! - i \si_2 \otimes \si_\pi
=
\lb \begin{array}{cc}
 & -\si_\pi \\
\si_\pi &
\end{array} \rb
\\

\ga_{0\va} \!\!&\!\!=\!\!&\!\! \ga_0 \ga_\va
=
\lb \begin{array}{cc}
\si_\va &  \\
 & -\si_\va
\end{array} \rb_{\p{(}}
& \;\;\;\;
\ga_{\va\pi} \!\!&\!\!=\!\!&\!\! \ga_\va \ga_\pi
=
\lb \begin{array}{cc}
 -i \ep_{\va\pi\ta} \si_\ta & \\
&  -i \ep_{\va\pi\ta} \si_\ta
\end{array} \rb

\end{array}
$$
[[Spacetime frame|spacetime frame]] and [[spin connection|spacetime spin connection]]:
$$
\begin{eqnarray}
\f{\om} + \f{e} &=&
\f{dx^a} {\scriptsize \frac{1}{2}} \om_a^{{\p{a}}\mu\nu} \ga_{\mu\nu} + \f{dx^a} ( e_a )^\mu \ga_\mu {}_{\p{(}} \\
&=&
\lb \begin{array}{cc}
( \f{\om^{0 \va}} \si_\va - {\small \frac{i}{2}} \f{\om^{\va \pi}} \ep_{\va \pi \ta} \si_\ta ) &   ( \f{e^0} - \f{e^\pi} \si_\pi ) \\
( \f{e^0} + \f{e^\pi} \si_\pi ) & (- \f{\om^{0 \va}} \si_\va - {\small \frac{i}{2}} \f{\om^{\va \pi}} \ep_{\va \pi \ta} \si_\ta )
\end{array} \rb_{\p{(}}
\\
&=&
\lb \begin{array}{cc}
\f{\om_L} & \f{e_L} \\
\f{e_R} & \f{\om_R}
\end{array} \rb_{\p{(}}
\;\; \in \;\; \f{Cl}^{1+2}(1,3)
\end{eqnarray}
$$
Note algebraic equivalence: &nbsp;&nbsp; $Cl^{1+2}(1,3) = Cl^2(1,4) = so(1,4)_{\p{(}}$
<<tiddler HideTags>>





@@display:block;text-align:center;font-size:24pt;Gravity is geometry.@@







@@display:block;text-align:center;"I am convinced, however, that the distinction between geometrical
and other kinds of fields is not logically founded." -- A.E. $\;\;\;\;\;\;\;\;\;\;\;\;\vp{)^{\big(}}$@@
<<tiddler HideTags>>

<html><center>
<img src="talks/IfA11/images/Gravity.png" height="400">
</center></html>

@@display:block;text-align:center;Gravitational frame bundle over 4D base.@@

<<tiddler HideTags>>The rest frame, $\f{e} \in \f{Cl^1}(1,3)$, is the fiber of spacetime: &nbsp;&nbsp; $\f{e} = \f{dx^i} (e_i)^\mu \ga_\mu$

This determines the metric: &nbsp;&nbsp; $g_{ij} = (e_i)^\mu \eta_{\mu \nu} (e_j)^\nu$

The gravitational spin connection, $\f{\om} \in \f{Cl^2}(1,3) = \f{so}(1,3)$, determines how the frame twists over the spacetime base manifold,
$$
\ff{T} = \f{D} \f{e} = \f{d} \f{e} + {\small \frac{1}{2}} \big[ \f{\om}, \f{e} \big]
$$

But the unified bosonic connection includes the frame:
$$
\f{H} = \ha \f{\om} + \f{G} + {\small \frac{1}{4}} \f{e} \ph
$$

This gives good dynamics for gravity and meshes perfectly with the standard model gauge group, and the Pati-Salam GUT, to act on fermion spinor multiplets:
$$
\big( so(1,3) + su(2)_L + u(1)_R + 4 \!\times\! (2\!+\!2) + u(1)_B + su(3) \big) + 8 \!\times\! 8
$$
<html><center><table class="gtable">
<tr border=none>

<td align="left">A <b>triality</b> rotation, <span class="math">T</span>, of <span class="math">D4</span>:
<div class="math">
{\small
\lb \begin{array}{c}
\ha {\om'}^3_L \\ \ha {\om'}^3_R \\ {W'}^3 \\ {B'}_1^3
\end{array} \rb
=
\lb \begin{array}{cccc}
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \\
1 & 0& 0 & 0
\end{array} \rb
\lb \begin{array}{c}
\ha \om^3_L \\ \ha \om^3_R \\ W^3 \\ B_1^3
\end{array} \rb
=
\lb \begin{array}{c}
\fr{1}{2} \om^3_R \\ B_1^3 \\ W^3 \\ \ha \om^3_L  
\end{array} \rb }
</div>
<div class="math">
T \, T \, T \, \om_R^\wedge = T \, T \,  \om_L^\wedge = T \, B_1^+ = \om_R^\wedge
</div>
Roots invariant under this <span class="math">T</span>:
<div class="math">
\{
W^+, \,
W^- , \,
e_S^\wedge\ph_+, \,
e_S^\wedge \ph_0, \,
e_S^\vee \ph_-, \,
e_S^\vee \ph_1
\}
</div>
Rotations to triality-equivalent vector and negative chiral spinor representation spaces:
<div class="math">
T \, 8_{S+} = 8_V \quad \; T \, 8_V = 8_{S-}  \quad \; T \, 8_{S-} = 8_{S+}
</div>
Three generations, related by triality:
<div class="math">
T \, e_L^\wedge = \mu_L^\wedge
\quad \;
T \, \mu_L^\wedge = \ta_L^\wedge
\quad \;
T \, \ta_L^\wedge = e_L^\wedge
</div>
</td>

<td>&nbsp;&nbsp;&nbsp;</td>

<td border=none>
<table class="ptable">
<tr>
<th COLSPAN="2"><SPAN class="math">8_V</SPAN></th>
<th></th>
<th><SPAN class="math">\ha \om_L^3</SPAN></th>
<th><SPAN class="math">\ha \om_R^3</SPAN></th>
<th><SPAN class="math">W^3</SPAN></th>
<th><SPAN class="math">B_1^3</SPAN></th>
</tr>
<tr>
<th COLSPAN="2">&nbsp;&nbsp;&nbsp;&nbsp;tri&nbsp;&nbsp;&nbsp;</th>
<th></th>
<th><SPAN class="math">\ha \om_R^3</SPAN></th>
<th><SPAN class="math">B_1^3</SPAN></th>
<th><SPAN class="math">W^3</SPAN></th>
<th><SPAN class="math">\ha \om_L^3</SPAN></th>
</tr>
<tr>
<td><SPAN class="math">\mtri{#B2B200}</SPAN></td>
<td><SPAN class="math">\nu_{\mu L}^{\wedge/\vee}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mtri{#F2F200}</SPAN></td>
<td><SPAN class="math">\mu_L^{\wedge/\vee}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mtri{#999999}</SPAN></td>
<td><SPAN class="math">\nu_{\mu R}^{\wedge/\vee}</SPAN></td>
<td></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mtri{#D9D9D9}</SPAN></td>
<td><SPAN class="math">\mu_R^{\wedge/\vee}</SPAN></td>
<td></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
</table>
<br>
<table class="ptable">
<tr>
<th COLSPAN="2"><SPAN class="math">8_{S-}</SPAN></th>
<th></th>
<th><SPAN class="math">\ha \om_L^3</SPAN></th>
<th><SPAN class="math">\ha \om_R^3</SPAN></th>
<th><SPAN class="math">W^3</SPAN></th>
<th><SPAN class="math">B_1^3</SPAN></th>
</tr>
<tr>
<th COLSPAN="2">&nbsp;&nbsp;&nbsp;&nbsp;tri&nbsp;&nbsp;&nbsp;</th>
<th></th>
<th><SPAN class="math">B_1^3</SPAN></th>
<th><SPAN class="math">\ha \om_L^3</SPAN></th>
<th><SPAN class="math">W^3</SPAN></th>
<th><SPAN class="math">\ha \om_R^3</SPAN></th>
</tr>
<tr>
<td><SPAN class="math">\stri{#B2B200}</SPAN></td>
<td><SPAN class="math">\nu_{\ta L}^{\wedge/\vee}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\stri{#F2F200}</SPAN></td>
<td><SPAN class="math">\ta_L^{\wedge/\vee}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\stri{#999999}</SPAN></td>
<td><SPAN class="math">\nu_{\ta R}^{\wedge/\vee}</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\stri{#D9D9D9}</SPAN></td>
<td><SPAN class="math">\ta_R^{\wedge/\vee}</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
</tr>
</table>

</td>

</tr>
</table>
</center>
</html>
<<tiddler HideTags>>
<html>
<center>
<table class="gtable">
<tr border=none>
<td><div class="math">
\begin{array}{l}
H_1 = (\ha \om + \fr{1}{4} e \ph + W + B_1) \vp{|_{(}} \\[.5em]
\quad \;\; \in so(3,1) + 4 \times 4 + \big( su(2)+su(2) \big) \vp{|_{(}} \\[.5em]
\quad \;\; = Cl^2(7,1) = so(7,1) = d4\\[3em]
8_{S+} \quad \to \qquad \quad  H_1 \, (\nu_e + e)\\[.5em]
\qquad \qquad \qquad \qquad \quad =\\[.5em]
{\small
\lb \begin{array}{cccc}
\! \fr{1}{2} \om_L \!+\! \fr{i}{2} W^3 \!\!\! & W^+ & - \! \fr{1}{4} e_R \ph_1 & \fr{1}{4} e_R \ph_+ \\
W^- & \!\!\! \fr{1}{2} \om_L \!-\! \fr{i}{2} W^3 \!\!\! & \p{-} \fr{1}{4} e_R \ph_- & \fr{1}{4} e_R \ph_0 \\
-\fr{1}{4} e_L \ph_0 & \fr{1}{4} e_L \ph_+ & \!\!\! \fr{1}{2} \om_R \!+\! \fr{i}{2} B_1^3 \!\!\! & B_1^+ \\
\p{-}\fr{1}{4} e_L \ph_- & \fr{1}{4} e_L \ph_1 & B_1^- & \!\!\! \fr{1}{2} \om_R \!-\! \fr{i}{2} B_1^3 \!
\end{array} \rb
\lb \begin{array}{c}
\nu_{eL} \\ e_L \\ \nu_{eR} \\ e_R
\end{array} \rb }
\end{array}
</div></td>

<td>&nbsp;&nbsp;</td>

<td>
<table class="ptable">
<tr>
<th COLSPAN="2"><SPAN class="math">D4</SPAN></th>
<th></th>
<th><SPAN class="math">\ha \om_L^3</SPAN></th>
<th><SPAN class="math">\ha \om_R^3</SPAN></th>
<th><SPAN class="math">W^3</SPAN></th>
<th><SPAN class="math">B_1^3</SPAN></th>
</tr>
<tr>
<td><SPAN class="math">\mcir{#59FF59} </SPAN></td>
<td><SPAN class="math">\om_L^{\wedge/\vee}</SPAN></td>
<td></td>
<td><SPAN class="math">\pm 1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#59FF59} </SPAN></td>
<td><SPAN class="math">\om_R^{\wedge/\vee}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\pm 1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#FFFF00} </SPAN></td>
<td><SPAN class="math">\smash{W^\pm}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\pm 1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#FFFFFF} </SPAN></td>
<td><SPAN class="math">\smash{B_1^\pm}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\pm 1</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\msqu{#B2B200} </SPAN></td>
<td><SPAN class="math">e_T^{\wedge/\vee} \ph_+</SPAN></td>
<td></td>
<td><SPAN class="math">\mp \ha</SPAN></td>
<td><SPAN class="math">\pm \ha</SPAN></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">\ha</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\msqu{#B2B200} </SPAN></td>
<td><SPAN class="math">e_S^{\wedge/\vee} \ph_+</SPAN></td>
<td></td>
<td><SPAN class="math">\pm \ha</SPAN></td>
<td><SPAN class="math">\pm \ha</SPAN></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">\ha</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mdia{#F2F200} </SPAN></td>
<td><SPAN class="math">e_T^{\wedge/\vee} \ph_-</SPAN></td>
<td></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mdia{#F2F200} </SPAN></td>
<td><SPAN class="math">e_S^{\wedge/\vee} \ph_-</SPAN></td>
<td></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\msqu{#F77C00}</SPAN></td>
<td><SPAN class="math">e_T^{\wedge/\vee} \ph_0</SPAN></td>
<td></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\msqu{#F77C00}</SPAN></td>
<td><SPAN class="math">e_S^{\wedge/\vee} \ph_0</SPAN></td>
<td></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mdia{#BF6000}</SPAN></td>
<td><SPAN class="math">e_T^{\wedge/\vee} \ph_1</SPAN></td>
<td></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\mdia{#BF6000}</SPAN></td>
<td><SPAN class="math">e_S^{\wedge/\vee} \ph_1</SPAN></td>
<td></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#B2B200}</SPAN></td>
<td><SPAN class="math">\nu_{eL}^{\wedge/\vee}</SPAN></td>
<td></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#F2F200}</SPAN></td>
<td><SPAN class="math">e_L^{\wedge/\vee}</SPAN></td>
<td></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#999999}</SPAN></td>
<td><SPAN class="math">\nu_{eR}^{\wedge/\vee}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#D9D9D9}</SPAN></td>
<td><SPAN class="math">e_R^{\wedge/\vee}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
</tr>
</table>
</td>

</tr>
</table>
</center>
</html>
<<tiddler HideTags>>
<<tiddler HideTags>><html><center>
<img SRC="talks/IfA11/images/hst.jpg" height=450px>
</center></html>
<<tiddler HideTags>><html>
<table class="gtable">

<tr>
<td>General Relativity</td>
<td></td>
<td>Quantum Field Theory</td>
</tr>

<tr>
<td></td>
<td></td>
<td></td>
</tr>

<tr>
<td>
<SPAN class="math">
\begin{array}{rcl}
R_{\mu}^{\p{\mu} a} - \ha (e_\mu)^a R \!\!&\!\!=\!\!&\!\! (e_\mu)^a \La - 8 \pi G \, T_{\mu}^{\p{\mu} a} \\[.3em]
R_{\mu}^{\p{\mu} a} \!\!&\!\!=\!\!&\!\! (e_b)^\nu ( \pa_{\lb \nu \rd} \om_{\ld \mu \rb}{}^{ba} + \om_{\lb \nu \rd}{}^{bc} \om_{\ld \mu \rb c}{}^a ) \\[.7em]
\ha T_{\mu \nu}{}^a \!\!&\!\!=\!\!&\!\! \pa_{\lb \mu \rd} (e_{\ld \nu \rb})^a + \om_{\lb \mu \rd}{}^{ab} (e_{\ld \nu \rb})_b \\[.7em]
\end{array}
</SPAN>
</td>
<td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>
<td>
<SPAN class="math">
\begin{array}{rcl}
0 \!\!&\!\!=\!\!&\!\! \ga^a (e_a)^\mu \big( \pa_\mu +\fr{1}{4} \om_\mu^{\p{\mu}bc} \ga_{bc} + A_\mu^{\p{\mu}B} T_B \big) \ud{\ps} + \ph \ud{\ps} \\[.3em]
0 \!\!&\!\!=\!\!&\!\! D^\mu D_\mu \ph + \mu^2 \ph - \la \ph^3 \\[.7em]
0 \!\!&\!\!=\!\!&\!\! D^\mu F_{\mu \nu}^{\p{\mu \nu} B} + \bar{\ps} \ga^a (e_a)_\nu T^B \ps + (\pa_\nu \ph) T^B \ph \\[.7em]
\end{array}
</SPAN>
</td>
</tr>

<tr>
<td>
<img SRC="talks/IfA11/images/hst2.jpg" height=320>
</td>
<td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>
<td>
<table class="gtable">
<tr><td>
<img SRC="talks/IfA11/images/lhc2.jpg" height=320>
</tr></td>
</table>
</td>
</tr>

</table>
</html>
/%
|Name|HideTags|
|Source|http://www.TiddlyTools.com/#HideTiddlerTags|
|Version|0.0.0|
|Author|Eric Shulman - ELS Design Studios (edited by Garrett)|
|License|http://www.TiddlyTools.com/#LegalStatements <<br>>and [[Creative Commons Attribution-ShareAlike 2.5 License|http://creativecommons.org/licenses/by-sa/2.5/]]|
|~CoreVersion|2.1|
|Type|script|
|Requires|InlineJavascriptPlugin|
|Description|hide a tiddler's tagged/tagging/references display elements|

Usage: <<tiddler HideTags>>

%/<script>
	var t=story.findContainingTiddler(place);
	if (t && t.id!="tiddlerHideTags")
		for (var i=0; i<t.childNodes.length; i++)
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				{t.childNodes[i].style.height="554.5px";}}
</script>
Every [[differential form]] [[field|cotangent bundle]] may be decomposed as
$$
\nf{F} = \nf{\Om} + \f{d} \nf{\Phi} + \ve{\de} \nf{\Psi} 
$$
in which $\nf{\Om}$ is [[harmonic]], $\f{d}$ is the [[exterior derivative]], and $\ve{\de}$ is the [[codifferential]]. 

Every [[closed]] differential form field, $\f{d} \nf{F} = 0$, may be decomposed as
$$
\nf{F} = \nf{\Om} + \f{d} \nf{\Phi} 
$$
Therefore, the [[cohomology]] is the same as the space of harmonic forms -- Hodge's theorem. That's kind of strange, since cohomology doesn't require a [[metric]], while the [[Hodge dual]] does.

Ref:
*http://en.wikipedia.org/wiki/De_Rham_cohomology
*[[Vector Calculus and the Topology of Domains in 3-Space|papers/vectorcalc.pdf]]
There are duality transformations in and between the tangent and cotangent spaces similar to [[Clifford dual]]ity.  For any [[differential form]] of grade $p$,
$$
\nf{a} = \fr{1}{p!} a_{\al \dots \be} \f{e^\al} \dots \f{e^\be}
$$
its ''vector dual'' (-p)-form is
$$
\bar{a} = \fr{1}{p!} a_{\al \dots \be} \et^{\al \ga} \dots \et^{\be \de} \ve{e_\ga} \dots \ve{e_\de}
$$
By multiplying any p-form by the (-n)-form, $\bar{e} = \ve{e_0} \ve{e_1} \dots \ve{e_{n-1}}$, one gets its ''Hodge vector dual'' (p-n)-form,
$$
\bar{* a} = \bar{e} \nf{a} = \ve{e_0} \ve{e_1} \dots \ve{e_{n-1}} \fr{1}{p!} a_{\al \dots \be} \f{e^\al} \dots \f{e^\be} 
= \fr{1}{p! \lp n-p \rp!} a_{\al \dots \be} \ep^{\al \dots \be \ga \dots \de} \ve{e_\ga} \dots \ve{e_\de}
$$
using [[vector-form algebra]] and the [[permutation symbol]].  And finally, by taking the form dual to this, one gets the ''Hodge dual'' (n-p)-form,
$$
\begin{eqnarray}
* \nf{a} = \nf{*a} &=& \fr{1}{p! \lp n-p \rp!} a_{\al \dots \be} \ep^{\al \dots \be \ga \dots \de} \f{e_\ga} \dots \f{e_\de} \\
&=& \fr{\ll \et \rl}{\ll e \rl p! \lp n-p \rp!} a_{i \dots j} \va^{i \dots jk \dots l} g_{km} \dots g_{ln} \f{dx^m} \dots \f{dx^n}
\end{eqnarray}
$$
The Hodge dual, which is only defined in the presence of a [[frame]] or [[metric]], is quite useful and allows the construction of the n-form product of any two p-forms, $\nf{a}$ and $\nf{b}$,
\begin{eqnarray}
\nf{*a}\nf{b} &=& \fr{1}{p! \lp n-p \rp!} a_{\al \dots \be} \ep^{\al \dots \be \ga \dots \de} \f{e_\ga} \dots \f{e_\de} \fr{1}{p!} b_{\ep \dots \up} \f{e^\ep} \dots \f{e^\up}\\
&=& \fr{\ll \et \rl}{p! \lp n-p \rp!} \fr{1}{p!} a_{\al \dots \be} b_{\ep \dots \up} \ep^{\al \dots \be \ga \dots \de} \ep_{\ga \dots \de}{}^{\ep \dots \up} \nf{e}\\
&=& \nf{e} \fr{1}{p!} a_{\al \dots \be} b^{\al \dots \be} = \nf{e} \fr{1}{p!} a_{i \dots j} b^{i \dots j} = \nf{e} \lp \bar{a} \nf{b} \rp = \nf{a} \nf{*b}
\end{eqnarray}
relying on [[permutation identities]].  Just as the Clifford dual squares to $\pm 1$ depending on [[signature|Minkowski metric]], the Hodge dual of a p-form similarly squares to
\[ \nf{**a} = \ll \et \rl \lp -1 \rp^{p \lp n-p \rp} \nf{a} \]

There is an example important enough to address specifically. If $\ff{F} = \ha \f{e^\mu} \f{e^\nu} F_{\mu  \nu}$ is a 2-form over a four dimensional space (or spacetime), then its Hodge dual is:
$$
\ff{*F} = \fr{1}{4} F_{\mu \nu} \ep^{\mu \nu \rh \si} \f{e_\rh} \f{e_\si} = \ff{\vv{\ep}} \ff{F}
$$
in which the ''Hodge dual projector'' is a 2-vector valued 2-form defined as
$$
\ff{\vv{\ep}} = - \f{e^\rh} \f{e^\si} \ep_{\rh \si}^{\p{\rh \si} \mu \nu} \ve{e_\mu} \ve{e_\nu} = - \f{e_\rh} \f{e_\si} \ep^{\rh \si \mu \nu} \ve{e_\mu} \ve{e_\nu}
= \left< \f{e} \f{e} \ga \ve{e} \ve{e} \right>
$$
which contracts with $\ff{F}$ via the [[vector-form algebra]]. Other Hodge dual projectors may be built corresponding to other cases.

There is a somewhat awkward but coordinate free expression for the Hodge dual, taking the angle brackets to group the enclosed Clifford elements and the parenthesis to group the form elements in
\begin{eqnarray}
\nf{*a} &=& \fr{1}{p! \lp n-p \rp!} < \lp \ve{e} \rp^p \ga^- ( \lp \f{e} \rp^{n-p} > \nf{a})\\
&=& \fr{1}{p! \lp n-p \rp!} \ve{e_\al} \dots \ve{e_\be} \lp \f{e^\ga} \dots \f{e^\de} \f{e^\ep} \dots \f{e^\up} \rp \li \ga^\al \dots \ga^\be \ga^- \ga_\ga \dots \ga_\de \ri \fr{1}{p!} a_{\ep \dots \up}\\
&=& \fr{\ll \et \rl}{p! \lp n-p \rp!} \ve{e_\al} \dots \ve{e_\be} \nf{e} \ep^{\ga \dots \de \ep \dots \up} \ep_{\ga \dots \de}{}^{\al \dots \be} a_{\ep \dots \up}\\
&=& \fr{1}{p! \lp n-p \rp!} \f{e_\al} \dots \f{e_\be} \ep^{\ga \dots \de \al \dots \be} a_{\ga \dots \de}
\end{eqnarray}
For a [[tangent vector]], $\ve{v}$, and a grade $p$ [[differential form]], $\nf{f}$, the [[vector-form algebra]] contraction can be equated with an expression involving the [[Hodge dual]],
$$\ve{v} \nf{f} = (-1)^{p(n-p)} * \lp \f{v} \nf{*f} \rp$$
in which $\f{v} = v^\al \et_{\al \be} \f{e^\be}$ is the form dual of $\ve{v}$.

(//add more as needed//)
David Ritz Finkelstein
http://arxiv.org/abs/gr-qc/0608086
*proposes a "flexing" of [[Lie algebra]] structure constants to go from one level of physical theory (some struct const = 0) to another (some not 0, or all not 0 ([[simple]])).
*hey, isn't this the same as Lie algebra deformation?
A horizontal dividing line.
----
{{{----}}}
<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
<embed src="talks/TED08/anim/charge.mov" width="620" height="620" controller="false" autoplay="false" loop="false"></embed>
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<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
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</center></html>@@
<<<
A whole block
of text to be quoted.
<<<
or
>>>Multiple levels of indented quotes.
>>Just like [[Bullet Points]].
>yep
>>or like [[Numbered Lists]]
That's what they said.
{{{
<<<
A whole block
of text to be quoted.
<<<
or
>>>Multiple levels of indented quotes.
>>Just like [[Bullet Points]].
>yep
>>or like [[Numbered Lists]]
That's what they said.
}}}
TiddlyWiki lets you write ordinary HTML by enclosing it in {{{<html>}}} and {{{</html>}}}:
<html>
<a href="javascript:;" onclick="onClickTiddlerLink(event);" 
tiddlyLink="Welcome"
style="background-color: yellow;">
Link to Welcome constructed in HTML</a>
</html>
{{{
<html>
<a href="javascript:;" onclick="onClickTiddlerLink(event);" 
tiddlyLink="Welcome"
style="background-color: yellow;">
Link to Welcome constructed in HTML</a>
</html>
}}}
HTML can enable some exotic new features (like [[embedding GMail and Outlook|http://groups.google.com/group/TiddlyWiki/browse_thread/thread/d363303aff5868d0/056269d8409d121f?lnk=st&q=embedding+gmail&rnum=1#056269d8409d121f]] in a TiddlyWiki). But, care needs to be taken with including things like JavaScript code.
/***
|Name|InlineJavascriptPlugin|
|Source|http://www.TiddlyTools.com/#InlineJavascriptPlugin|
|Documentation|http://www.TiddlyTools.com/#InlineJavascriptPluginInfo|
|Version|1.9.6|
|Author|Eric Shulman|
|License|http://www.TiddlyTools.com/#LegalStatements|
|~CoreVersion|2.1|
|Type|plugin|
|Description|Insert Javascript executable code directly into your tiddler content.|
''Call directly into TW core utility routines, define new functions, calculate values, add dynamically-generated TiddlyWiki-formatted output'' into tiddler content, or perform any other programmatic actions each time the tiddler is rendered.
!!!!!Documentation
>see [[InlineJavascriptPluginInfo]]
!!!!!Revisions
<<<
2010.12.15 1.9.6 allow (but ignore) type="..." syntax
|please see [[InlineJavascriptPluginInfo]] for additional revision details|
2005.11.08 1.0.0 initial release
<<<
!!!!!Code
***/
//{{{
version.extensions.InlineJavascriptPlugin= {major: 1, minor: 9, revision: 6, date: new Date(2010,12,15)};

config.formatters.push( {
	name: "inlineJavascript",
	match: "\\<script",
	lookahead: "\\<script(?: type=\\\"[^\\\"]*\\\")?(?: src=\\\"([^\\\"]*)\\\")?(?: label=\\\"([^\\\"]*)\\\")?(?: title=\\\"([^\\\"]*)\\\")?(?: key=\\\"([^\\\"]*)\\\")?( show)?\\>((?:.|\\n)*?)\\</script\\>",
	handler: function(w) {
		var lookaheadRegExp = new RegExp(this.lookahead,"mg");
		lookaheadRegExp.lastIndex = w.matchStart;
		var lookaheadMatch = lookaheadRegExp.exec(w.source)
		if(lookaheadMatch && lookaheadMatch.index == w.matchStart) {
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			var tip=lookaheadMatch[3];
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			var show=lookaheadMatch[5];
			var code=lookaheadMatch[6];
			if (src) { // external script library
				var script = document.createElement("script"); script.src = src;
				document.body.appendChild(script); document.body.removeChild(script);
			}
			if (code) { // inline code
				if (show) // display source in tiddler
					wikify("{{{\n"+lookaheadMatch[0]+"\n}}}\n",w.output);
				if (label) { // create 'onclick' command link
					var link=createTiddlyElement(w.output,"a",null,"tiddlyLinkExisting",wikifyPlainText(label));
					var fixup=code.replace(/document.write\s*\(/gi,'place.bufferedHTML+=(');
					link.code="function _out(place,tiddler){"+fixup+"\n};_out(this,this.tiddler);"
					link.tiddler=w.tiddler;
					link.onclick=function(){
						this.bufferedHTML="";
						try{ var r=eval(this.code);
							if(this.bufferedHTML.length || (typeof(r)==="string")&&r.length)
								var s=this.parentNode.insertBefore(document.createElement("span"),this.nextSibling);
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								s.innerHTML=this.bufferedHTML;
							if((typeof(r)==="string")&&r.length) {
								wikify(r,s,null,this.tiddler);
								return false;
							} else return r!==undefined?r:false;
						} catch(e){alert(e.description||e.toString());return false;}
					};
					link.setAttribute("title",tip||"");
					var URIcode='javascript:void(eval(decodeURIComponent(%22(function(){try{';
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					URIcode+='}catch(e){alert(e.description||e.toString())}})()%22)))';
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					link.style.cursor="pointer";
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					if (out && out.length) wikify(out,w.output,w.highlightRegExp,w.tiddler);
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			}
			w.nextMatch = lookaheadMatch.index + lookaheadMatch[0].length;
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} )
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// // Backward-compatibility for TW2.1.x and earlier
//{{{
if (typeof(wikifyPlainText)=="undefined") window.wikifyPlainText=function(text,limit,tiddler) {
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	var wikifier = new Wikifier(text,formatter,null,tiddler);
	return wikifier.wikifyPlain();
}
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// // GLOBAL FUNCTION: $(...) -- 'shorthand' convenience syntax for document.getElementById()
//{{{
if (typeof($)=='undefined') { function $(id) { return document.getElementById(id.replace(/^#/,'')); } }
//}}}
[>img[images/person/John Baez.jpg]]Homepage: http://math.ucr.edu/home/baez/
*Location: UCRiverside

A wonderfully prolific mathematical physicist, and all around good guy.
Access keys are shortcuts to common functions accessed by typing a letter with either the 'alt' (PC) or 'control' (Mac) key:
|!PC|!Mac|!Function|
|Alt-F|Ctrl-F|Search|
|Alt-J|Ctrl-J|NewJournal|
|Alt-N|Ctrl-N|NewNote|
|Alt-S|Ctrl-S|SaveChanges|
These access keys are provided by the associated internal [[Macros]] for the functions above. The macro needs to be used in an open note (or the [[MainMenu]] or SideBar) in order for the access keys to work.

While editing a note:
* ~Control-Enter or ~Control-Return accepts your changes and switches out of editing mode (use ~Shift-Control-Enter or ~Shift-Control-Return to stop the date and time being updated for MinorChanges)
* Escape abandons your changes and reverts the note to its previous state

In the search box:
* Escape clears the search term
A [[Lie algebra]] is a [[vector space]] as well as an algebra, spanned by the basis vectors, $T_A$, and one may use the structure constants to build a natural [[metric]] giving a scalar result from two Lie algebra elements,
$$
\lp B, C \rp = {\rm Tr}({\rm Ad}_B {\rm Ad}_C) = T^D \lb B, \lb C, T_D \rb \rb = B^A C^B g_{AB} 
$$
using the [[trace]] and Lie algebra adjoint action, ${\rm Ad}_B C = \lb B, C \rb$. The resulting metric coefficients are those of the ''Killing form'',
$$
g_{AB} = \lp T_A, T_B \rp = C_{AC}{}^D C_{BD}{}^C
$$
It is always possible to transform to a new set of generators, $T'_A = L_A{}^B T_B$, producing a new set of structure constants and hence a new metric, $\et_{AB} = L_A{}^C L_B{}^D g_{CD}$. As long as the metric is non-degenerate, which happens iff $Lie(G)$ is semi-[[simple]], it is possible to transform to a set of generators such that this metric is unit diagonal (with $+1$ and $-1$ entries like in the [[Minkowski metric]]) via the methods of [[spectral decomposition|eigen]]. This has already been done for most common generator representations used in physics, up to a constant factor, so that often $g_{AB} = \et_{AB} = \de_{AB}$. 

Using the Jacobi identity, the Killing form is symmetric, $g_{AB} = g_{BA}$, and is ''adjoint invariant'',
\begin{eqnarray}
\lp \lb A, B \rb, C \rp &=& \lp A, \lb B, C \rb \rp \\
\lp g A g^-, C \rp &=& \lp A, g^- C g \rp
\end{eqnarray}
which implies the structure constants are antisymmetric in the last two indices,
$$
C_{ABC} = -C_{ACB}
$$
when the Killing forms $g_{AB}$ (and $g^{AB}$) are used to lower (and raise) Lie algebra indices, $C_{ABC}=C_{AB}{}^D g_{DC}$. Since we always have $C_{AB}{}^C = - C_{BA}{}^C$, the structure constants are completely [[antisymmetric|index bracket]],
$$
C_{ABC} = C_{\lb ABC \rb} = C_{BCA} = C_{CAB} = -C_{BAC} = -C_{ACB} = -C_{CBA}
$$
which is useful enough to call the ''Killing form identity''.

The ''inverse Killing form'', $g^{AB}$, is used to define the ''generator duals'', $T^A = g^{AB} T_B$, satisfying $\lp T^A, T_B \rp = \de^A_B$.

For some Lie algebras with [[Clifford algebra]] or matrix generators, the scalar part or trace gives the orthonormality relations
$$
\lp T_A, T_B \rp = g_{AB} \sim \li T_A T_B \ri = \de_{AB}
$$
A ''Killing spinor'' is a [[spinor]] field satisfying
$$
\f{\na} \ps = \la \f{e} \ps
$$
for some constant ''Killing number'', $\la$, in which $\f{\na}$ is the [[spinor covariant derivative]] and $\f{e}$ is the [[coframe|frame]].

The [[tangent vector]] field corresponding to a Killing spinor is a [[Killing vector]],
$$
\ve{\xi} = \left< \bar{\ps} \ga^\al \ps \right> \ve{e_\al}
$$
in which $\bar{\ps}$ is the [[Clifford conjugate]] of the spinor and $\ve{e_\al}$ are [[frame]] vectors. 
A ''Killing vector'' field, $\ve{\xi}(x)$, is the generator of a [[flow]], $\ph_t = e^{t {\cal L}_{\ve{\xi}}}$, that leaves the geometry of a [[manifold]] invariant &mdash; constituting a symmetry of the geometry.  It is a [[tangent vector field|tangent bundle]] satisfying ''Killing's equation'',
$$
L_{\ve{\xi}} \f{e} = B \times \f{e}
$$
The [[Lie derivative]] of the [[frame]] along a Killing vector field gives a [[rotation|Clifford rotation]] of the frame by some corresponding Clifford bivector field, $B (x) \in Cl^2$,
$$
L_{\ve{\xi}} \f{e^\al} = B_\be{}^\al \f{e^\be}
$$
with ''Killing rotation coefficients'', $B_\be{}^\al = - B^\al{}_\be$. This version of Killing's equation, or equivalently,
$$
L_{\ve{\xi}} \ve{e_\al} = - B_\al{}^\be \ve{e_\be}
$$
matches the usual definition that the Lie derivative of the [[metric]] along a Killing vector field vanishes.

Any set of Killing vector fields is related through the [[Lie bracket|Lie derivative]],
$$
\lb \ve{\xi_A}, \ve{\xi_B} \rb_L = C_{AB}{}^C \ve{\xi_C}
$$
with $C_{AB}{}^C$ the set of ''structure constants'' for the symmetries. The manifold [[diffeomorphism]]s (''isometries'') built from the flows generated by a set of Killing vector fields constitute a [[Lie group]].

A Killing vector field has many nice [[properties|Killing vector identities]].
The defining equation of a [[Killing vector]] field, $\ve{\xi}$, is
\begin{eqnarray}
B_\be{}^\al \f{e^\be} &=& {\cal L}_{\ve{\xi}} \f{e^\al} = \ve{\xi} \lp \f{d} \f{e^\al} \rp + \f{d} \xi^\al \\
&=& \ve{\xi} \lp \f{w}{}_\be{}^\al \f{e^\be} + \ff{T^\al} \rp + \f{d} \xi^\al \\
&=& \f{e^\be} \lp \xi^\de w_{\de \be}{}^\al - \xi^\de w_{\be \de}{}^\al + 2 \xi^\de T_{\de \be}{}^\al + \pa_\be \xi^\al \rp
\end{eqnarray}
by virtue of the defining equations for the [[Lie derivative]], the [[cotangent bundle connection]], and the [[torsion]]. This gives a useful expression for the derivative of the Killing vector field coefficients,
$$
\pa_\be \xi_\al = B_{\be \al} - \xi^\de w_{\de \be \al} + \xi^\de w_{\be \de \al} - 2 \xi^\de T_{\de \be \al}
$$
The [[1-form dual|frame]] to the Killing vector field is $\f{\xi} = \xi_\al \f{e^\al} = \xi^\be \et_{\be \al} \f{e^\al}$. The cotangent bundle covariant derivative of this field is
$$
\f{\na} \f{\xi} = \f{e^\be} \na_\be \lp \xi_\al \f{e^\al} \rp
= \f{e^\be} \f{e^\ga} \lp \pa_\be \xi_\ga + \xi_\al w_{\be \ga}{}^\al \rp
= \f{e^\be} \f{e^\ga} \lp B_{\be \ga} - \xi^\de w_{\de \be \ga} - 2 \xi^\de T_{\de \be \ga} \rp
$$
Similarly,
$$
\f{\na} \ve{\xi} = \f{e^\be} \na_\be \lp \xi^\al \ve{e_\al} \rp
= \f{e^\be} \ve{e_\ga} \lp \pa_\be \xi^\ga + \xi^\al w_\be{}^\ga{}_\al \rp
= \f{e^\be} \ve{e_\ga} \lp B_\be{}^\ga - \xi^\de w_{\de \be}{}^\ga - 2 \xi^\de T_{\de \be}{}^\ga \rp
$$
If $\ve{v}$ is the velocity along a [[geodesic]],
$$
0 = \ve{v} \f{\na} \ve{v} = v^\al \lp \pa_\al v^\de + v^\be w_\al{}^\de{}_\be \rp \ve{e_\de}
$$
the component of this velocity along any Killing vector field, $p = \lp \ve{v}, \ve{\xi} \rp = \ve{v} \f{\xi}$, is constant along the geodesic,
$$
\ve{v} \f{d} p = v^\al \pa_\al \lp v^\be \xi_\be \rp 
= v^\al \lp \pa_\al v^\be \rp \xi_\be + v^\al v^\be \lp \pa_\al \xi_\be \rp
= v^\al \lp \pa_\al v^\de \rp \xi_\de + v^\al v^\be \lp w_{\al \de \be} \xi^\de - 2 \xi^\de T_{\de \al \be} \rp
= 0
$$
as long as the torsion vanishes, or at least $T_{\de \lp \al \be \rp} = 0$.

If the Killing vector field is of constant length, $\ve{\xi} \f{\xi} = \xi^\al \xi_\al = c$, then
$$
0 = \pa_\be \lp \xi^\al \xi_\al \rp = 2 \xi^\al \lp \pa_\be \xi_\al \rp
= 2 \xi^\al \lp B_{\be \al} - \xi^\de w_{\de \be \al} - 2 \xi^\de T_{\de \be \al} \rp
$$
and the integral curves of the Killing vector field are geodesics,
$$
\ve{\xi} \f{\na} \ve{\xi} = \xi^\be \ve{e_\ga} \lp B_\be{}^\ga - \xi^\de w_{\de \be}{}^\ga \rp = 0
$$
as long as the torsion vanishes or $T_{\de \lp \al \be \rp} = 0$.
The ''Kronecker product'' (//''tensor product''//), $\otimes$, of an $m$ by $n$ matrix, $A$, and a $p$ by $q$ matrix, $B$, is an $mp$ by $nq$ matrix, $C = A \otimes B$,
$$
C_{\lp \lp a - 1 \rp p + x \rp}{}^{\lp \lp b - 1 \rp q + y \rp} = A_a{}^b B_x{}^y
$$
It is a ''block'' matrix of $B$'s multiplied by the entries of $A$. For example,
$$
\lb \begin{array}{cc}
1 & 2\\
3 & 1
\end{array} \rb
\otimes
\lb \begin{array}{cc}
0 & 3\\
2 & 1
\end{array} \rb
=
\lb \begin{array}{cc}
1
\lb \begin{array}{cc}
0 & 3\\
2 & 1
\end{array} \rb
 & 2
\lb \begin{array}{cc}
0 & 3\\
2 & 1
\end{array} \rb
\\
3
\lb \begin{array}{cc}
0 & 3\\
2 & 1
\end{array} \rb
 & 1
\lb \begin{array}{cc}
0 & 3\\
2 & 1
\end{array} \rb
\end{array} \rb
=
\lb \begin{array}{cccc}
0 & 3 & 0 & 6\\
2 & 1 & 4 & 2\\
0 & 9 & 0 & 3\\
6 & 3 & 2 & 1
\end{array} \rb
$$

This product spawns several identities, including:
$$
\lp A \otimes B \rp \lp C \otimes D \rp = A C \otimes B D
$$

Ref:
http://en.wikipedia.org/wiki/Kronecker_product
<<tiddler HideTags>><html><center>
<img SRC="talks/IfA11/images/lhc2.jpg" height=500px>
</center></html>
<<tiddler HideTags>><html><center><iframe src="talks/IfA11/anim/LHCanim.html" width="560" height="540" frameborder="0"></iframe>
</center></html>
//Use the first method in each example below, unless you have some reason not to.//
Mathematical symbols, such as \(e^{x^2}\), may be inserted inline.
{{{
Mathematical symbols, such as $e^{x^2}$, may be inserted inline.
Mathematical symbols, such as \(e^{x^2}\), may be inserted inline.
}}}
Or as displayed math,$$e^{x^2}$$ on its own line.
{{{
Or as displayed math, \[e^{x^2}\] on its own line.
Or as displayed math, $$e^{x^2}$$ on its own line.
Or as displayed math, \begin{equation}e^{x^2}\end{equation} on its own line.
}}}
Or as an equation array,
\begin{eqnarray}A &=& e^{x^2}\\&=&C\end{eqnarray}
{{{
Or as an equation array,\begin{eqnarray}A &=& e^{x^2}\\&=& C\end{eqnarray}
}}}

Some of the available TeX symbols can be found at [[jsMath|http://www.math.union.edu/~dpvc/jsMath/symbols/welcome.html]], the best method I could find  for displaying TeX online.  The small button in the lower right corner of this window opens its control planel.  I'm not sure how many LaTeX and AMSTeX commands are supported -- play around.

TeX substitution macros such as $\f{A}$, ({{{$\f{A}$}}}), may be inserted into the [[jsMathPlugin]] just before the jsMath.process call.  See that plugin for abbreviated commands I've included.
<<tiddler HideTags>><html>
<table class="gtable">

<tr>
<td>
<img SRC="talks/Cate2010/Laser.jpg" width=300>
</td>
<td>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>
<td>

<table class="gtable">
<tr><td> &nbsp;</td></tr>
<tr><td>&nbsp; </td></tr>
<tr><td>&nbsp;</td></tr>
<tr><td>&nbsp;</td></tr>
<tr><td>&nbsp;</td></tr>
<tr><td>&nbsp;</td></tr>
<tr><td>
<img SRC="talks/Cate2010/EM field.png" width=300>
</tr></td>
</table>

</td>
</tr>
</table>
</html>
[>img[images/person/Laurent Freidel.jpg]]Homepage: http://www.perimeterinstitute.ca/index.php?option=com_content&task=view&id=30&Itemid=72&pi=Laurent_Freidel
*Location: $\Pi$
*arXiv papers: http://arxiv.org/find/gr-qc/1/au:+Freidel_L/0/1/0/all/0/1

Often coauthors with: Artem Starodubtsev.

Selected work:
*[[Quantum gravity in terms of topological observables|papers/0501191.pdf]]
*[[Particles as Wilson lines of gravitational field]]
[>img[images/person/Leonardo Castellani.jpg]]Homepage: http://www.mfn.unipmn.it/~castella/
*Location: Turin
*Papers: http://www-library.desy.de/cgi-bin/spiface/find/hep/www?rawcmd=find++fa+castellani%2Cl+and+not+a+martinelli+or+a+aschieri+and+castellani&FORMAT=WWW&SEQUENCE=

Selected work:
*[[Group Geometric Methods in Supergravity and Superstring Theories|papers/Group Geometric Methods in Supergravity and Superstring Theories.pdf]]
**nice job on [[reductive Lie group geometry]]
**BRST by adding a (Grassmann?) piece to the Lie group geometry
*[[A Geometric Interpretation of BRST symmetry|papers/A Geometric Interpretation of BRST symmetry.pdf]]
**just the BRST part from paper above
**add generator, $Q$, and $\f{d \th} g^A T_A$.
**This reminds me of the geometry of Lagrange multipliers
*[[Gravity on Finite Groups|papers/9909028.pdf]]
**differential geometry over disconnected spaces
*lots of [[Kaluza-Klein]] stuff
A ''Lie algebra'' consists of elements of a $N$ dimensional [[vector space]], $B=B^A T_A \in {\rm Lie}(G)$, composed of real (or complex, for a complex Lie algebra) coefficients, $B^A$, multiplying $N$ ''Lie algebra generators'', $T_A$. The elements are closed under a ''Lie algebra bracket'', equal to the Lie algebra ''adjoint action'' of one element on another,
$$
{\rm Ad}_A B = \lb A, B \rb = C
$$
and equivalent to the [[commutator]] relation. The Lie algebra brackets of the generators,
$$
\lb T_A , T_B \rb = T_A  T_B - T_B  T_A = C_{AB}{}^C T_C
$$
give the real or complex ''structure constants'' (//''structure coefficients''//), $C_{AB}{}^C = - C_{BA}{}^C$, for the Lie algebra. The generators may be [[Clifford basis elements]] ($T_A=\ga_A$), square matrices, or abstract operators. The bracket must satisfy the ''Jacobi identity'',
\begin{eqnarray}
0 &=& \lb A, \lb B , C \rb \rb + \lb B, \lb C , A \rb \rb + \lb C, \lb A , B \rb \rb \\
0 &=& C_{AD}{}^E C_{BC}{}^D + C_{BD}{}^E C_{CA}{}^D + C_{CD}{}^E C_{AB}{}^D 
\end{eqnarray}
identical to the [[Clifford Jacobi identity|Clifford product identities]].

Although the choice of generators is somewhat arbitrary, every Lie algebra has a specific [[Lie algebra structure]].
It has been seen that an $n$ dimensional [[Clifford algebra]] can be interpreted as a $2^n$ dimensional [[Lie algebra]], with the [[Clifford basis elements]] identified as generators, $\ga_{\al \dots \be} \sim T_A$, and the Lie bracket given by the [[Clifford basis product identities]]. It is also the case that any $n$ dimensional Lie algebra may be identified as a subalgebra of the bivector algebra of an $n$ dimensional Clifford algebra. This representation may be made as:
$$
T_A = - \fr{1}{4} C_A{}^{BC} \ga_{BC}
$$
employing the Lie algebra structure constants, the Clifford bivector basis elements, and using the unit diagonal [[Killing form]], $g^{BD}=\et^{BD}$, as the Clifford algebra metric and to raise the structure constant indices. This representation faithfully gives
\begin{eqnarray}
\lb T_A, T_D \rb &=& \fr{1}{8} C_A{}^{BC} C_D{}^{EF} \ga_{BC} \times \ga_{EF} \\
&=& \ha C_A{}^{BC} C_{DC}{}^E \ga_{BE} \\
&=& - \fr{1}{4} C_{AD}{}^C C_C{}^{BE} \ga_{BE} \\
&=& C_{AD}{}^C T_C
\end{eqnarray}
via the [[Jacobi identity|Lie algebra]] and Clifford basis identities. These Lie algebra generators are orthogonal,
\begin{eqnarray}
\li T_A T_D \ri &=& \fr{1}{16} C_A{}^{BC} C_D{}^{EF} \li \ga_{BC} \ga_{EF} \ri \\
&=& \fr{1}{16} C_A{}^{BC} C_D{}^{EF} \lp \et_{BF} \et_{CE} - \et_{BE} \et_{CF} \rp \\
&=& \fr{1}{8} C_A{}^{BC} C_{DCB} \\
&=& \fr{1}{8} \et_{AD}
\end{eqnarray}
in the [[scalar part|Clifford grade]] (trace) operator. Note that the $2^{[n/2]}$ dimensional [[Clifford matrix representation]] gives a corresponding matrix representation for the Lie algebra.
The structure of every [[simple]] [[Lie algebra]] may be described by arranging the generators to span different related sub-spaces. The ''rank'', $R$, of a $N$ dimensional Lie algebra is the maximal number if inter-commuting generators, $T^C_a$,
$$
\big[ T^C_a, T^C_b \big] = 0 \;\;\;\; \forall \;\; 1 \le a,b \le R
$$
These generators span a $R$ dimensional [[vector space]], the ''Cartan subalgebra'', $C = c^a T^C_a$. [[Exponentiating|exponentiation]] the Cartan subalgebra gives the ''maximal torus'' -- a $R$ dimensional [[submanifold]] of the [[Lie group manifold|Lie group geometry]].

Every element of the Cartan subalgebra may be represented by a $(N \times N)$ matrix, $M$, related to the adjoint action, ${\rm Ad}_C$ , which acts on other Lie algebra elements, $B = b^A T_A$, as vectors,
$$
[ C , B ] = {\rm Ad}_C \, B = c^a \big[ T^C_a , T_A \big] b^A = T_D \big( c^a C_{aA}^{\p{aA}D} \big) b^A = T_D M^D_{\p{D}A} b^A 
$$
The components of $M$ may be written in terms of the structure constants as $M^D_{\p{D}A} = c^a C_{aA}^{\p{aA}D}$. As a vector space, the Lie algebra structure may be understood in terms of the [[eigen]]values and eigenvectors of $M$.

Since $M \, C_1 = 0$ for any $C_1$ in the Cartan subalgebra, $R$ of the eigenvalues of $M$ will be $0$, with that null eigenspace equal to $C$. The remaining $(N-R)$ eigenvectors of $M$ come in pairs, $V^\pm_i$, with each corresponding to a unique complex eigenvalue,
$$
M(c) \, V^\pm_i = \pm \al_i(c) V^\pm_i
$$
(no sum over $i$) Since $M(c)$ depends linearly on the $c^a$, each of the eigenvalues, $\al_i(c)$, also depends linearly on the $c^a$. Note that the eigenvectors, $V^\pm_i$, do not depend on $c^a$ -- they depend only on the whole Cartan subalgebra. The eigenvalues may be related to elements, $\al_i \in C^*$, of a dual vector space to the Cartan subalgebra, $\al_i : C \to \mathbb{C}$. Written out in terms of dual basis elements, $T^a$, they are $\al_i = \al_{ia} T^a$, so
$$
\al_i(c) = \al_i \, C = \al_{ia} T^a \, c^b T_b = \al_{ia} c^a \in \mathbb{C}
$$
These dual space elements, $\al_i$, are called the ''roots''. Each root has a negative partner root, and may not be twice any other root, $\al_i \ne 2 \al_j$.

Since a Lie algebra has a natural metric, the [[Killing form]], every element, $\al \in C^*$, of the dual space has a corresponding element in the Cartan subalgebra, $h_\al = h_\al^a T_a = ( g^{ab} \al_b ) T_a \in C$, such that
$$
\al \, C = \al_a T^a \, c^b T_b = \al_a c^a = h_\al^a c^b g_{ab} = h_\al^a c^b \big( T_a , T_b) = \big( h_\al , C \big)
$$
In this way, the Killing form gives a ''root space metric'', $< \al, \be > = (h_\al, h_\be)$. Using this metric, the roots for any Lie algebra may be seen to have a nicely symmetric arrangement which describes the algebra.

The complete set of non-zero Lie brackets for the algebra are:
\begin{eqnarray}
\big[ C_1 , C_2 \big] &=& 0 \\
\big[ C , V^\pm_i \big] &=& \pm \al_i(c) V^\pm_i \\
\big[ V^\pm_i , V^\mp_i \big] &=& \big( V^\pm_i , V^\mp_i \big) h_{\al_i} \\
\big[ V^\pm_i , V^\pm_j \big] &=& N_{ij} \, V^\pm_k
\end{eqnarray}
in which $N_{ij}$ are normalization constants and it must be the case that $\al_i + \al_j = \al_k$ in the last bracket above. This algebra has $R$ $su(2)$ subalgebras,
\begin{eqnarray}
\big[ h_{\al_i} , V^+_i \big] &=& < \al_i , \al_i > V^+_i \\
\big[ h_{\al_i} , V^-_i \big] &=& - < \al_i , \al_i > V^-_i \\
\big[ V^+_i , V^-_i \big] &=& \big( V^+_i , V^-_i \big) h_{\al_i}
\end{eqnarray}
//(More on root diagrams, Dynkin, etc. -- use examples.)//

Ref:
*Robert Cahn
**[[Semi-Simple Lie Algebras and Their Representation|papers/Semi-Simple Lie Algebras and Their Representation.pdf]]
link from [[Maurer-Cartan form]]

ref:
http://en.wikipedia.org/wiki/Lie_algebroid
[[Differential Operators and Actions of Lie Algebroids|papers/0209337.pdf]]
The ''Lie derivative'' is the rate of change of any field perceived by an observer as she moves along a path with some [[velocity|tangent vector]], $\ve{v}$. Basically, the field where she is going is pulled back and compared with the field where she's at. This description is extended to give the Lie derivative, ${\cal L}_{\ve{v}}$, with respect to a [[flow]], $\ph_t$, giving the rate of change of any field perceived by observers at every manifold point as they move according to the velocity field, $\ve{v}(x)$. The parameterized flow is given to first order in $t$ by
$$
\ph_t^i(x) \simeq x^i + t v^i(x)
$$
The Lie derivative of any field, $X$, is a [[natural]] operator defined as
$$
{\cal L}_{\ve{v}} X = \fr{d}{d t} \ph_t^* X = \fr{d}{d t} X(t) = \lim_{t \to 0} \fr{\ph_t^*X - X}{t} = \lim_{t \to 0} \fr{ \lp \ph_{-t} \rp_* X - X}{t}
$$
in which
$$
\ph_t^*X = \ph_t^*\lb X \rl_{\ph_t} \simeq \ph_t^* \lp \lb X \rl_x + t v^i \pa_i \lb X \rl_x \rp
$$
is the [[pullback]] of the field from where the flow is going back to the initial points, expanded to first order in $t$.

If the field is a [[function]] over the manifold, the Lie derivative of this field is the same as the [[directional derivative|tangent vector]] of this function with respect to the velocity field at every point,
$$
{\cal L}_{\ve{v}} f = \fr{d}{d t} f(x) = \lim_{t \to 0} \fr{\ph_t^*f - f}{t} = v^i \pa_i f = \ve{v} \f{d} f 
$$
If the field is a [[1-form|cotangent bundle]] field, $\f{f} = \f{dx^i} f_i(x)$, the pullback along the flow is
$$
\ph_t^*\f{f} = \f{dx^i} \lb \fr{\pa \ph_t^j}{\pa x^i} \rb f_j(\ph_t) 
\simeq \f{dx^i} \lb \de^j_i + t \pa_i v^j \rb \lb f_j(x) + t v^k \pa_k f_j(x) \rb
\simeq \f{dx^i} \lp f_i + t v^k \pa_k f_i + t f_j \pa_i v^j \rp
$$
and the Lie derivative is thus
\begin{eqnarray}
{\cal L}_{\ve{v}} \f{f} &=& \lim_{t \to 0} \fr{\ph_t^*\f{f} - \f{f}}{t}
= \f{dx^i} \lp v^k \pa_k f_i + f_j \pa_i v^j \rp \\
&=& \lp \ve{v} \f{\pa} \rp \f{f} + \lp \f{\pa} \ve{v} \rp \f{f} \\
&=& \ve{v} \lp \f{d} \f{f} \rp + \f{d} \lp \ve{v} \f{f} \rp
\end{eqnarray}
using the [[exterior derivative]], [[partial derivative]], and [[vector-form algebra]]. For any differential form or [[Clifform]] field with form grade greater than zero this generalizes to give ''Cartan's formula'' for the Lie derivative,
\begin{eqnarray}
{\cal L}_{\ve{v}} \, \nf{F}
&=& \lp \ve{v} \f{\pa} \rp \nf{F} + \lp \f{\pa} \ve{v} \rp \nf{F} \\
&=& \ve{v} \lp \f{d} \nf{F} \rp + \f{d} \lp \ve{v} \nf{F} \rp
\end{eqnarray}
and another nice formula (easier for computations) obtained via the power of vector-form algebra and the partial derivative operator.
If the field is a [[vector|tangent bundle]] field, $\ve{u}(x)$, the pushforward along the negative flow of the velocity at where the flow goes is
$$
\lp \ph_{-t} \rp_* \ve{u} = u^j(\ph_t) \lb \fr{\pa \ph_{-t}^i}{\pa x^j} \rb \ve{\pa_i}
\simeq \lb u^j + t v^k \pa_k u^j \rb \lb \de^i_j - t \pa_j v^i \rb \ve{\pa_i}
\simeq \lp u^j + t v^k \pa_k u^i - t u^j \pa_j v^i \rp \ve{\pa_i}
$$
and the the Lie derivative of a vector field is thus
$$
{\cal L}_{\ve{v}} \ve{u} = \lim_{t \to 0} \fr{ \lp \ph_{-t} \rp_* \ve{u} - \ve{u}}{t}
= \lp v^k \pa_k u^i - u^j \pa_j v^i \rp \ve{\pa_i}
= \ve{v} \f{\pa} \ve{u} - \ve{u} \f{\pa} \ve{v}
$$

The Lie derivative of one velocity field with respect to another defines the ''Lie bracket'',
$$
\lb \ve{v}, \ve{u} \rb_L = {\cal L}_{\ve{v}} \ve{u} = - \lb \ve{u}, \ve{v} \rb_L
$$

The Lie derivative has a number of nice [[properties|Lie derivative identities]].
The [[Lie derivative]] is a natural, fundamental derivative operator on any geometric field on a manifold.

By virtue of Cartan's formula, it commutes with the [[exterior derivative]] operator when acting on [[function]]s, [[differential form]]s or [[Clifform]]s,
$$
{\cal L}_{\ve{v}} \f{d} \nf{F} = \f{d} \lp \ve{v} \lp \f{d} \nf{F} \rp \rp = \f{d} {\cal L}_{\ve{v}} \nf{F} 
$$
but not always when acting on vector fields.

It is linear in both the velocity field,
$$
{\cal L}_{\ve{v} + \ve{u}} X = {\cal L}_{\ve{v}} X + {\cal L}_{\ve{u}} X
$$
and argument,
$$
{\cal L}_{\ve{v}} \lp X + Y \rp = {\cal L}_{\ve{v}} X + {\cal L}_{\ve{v}} Y
$$

The Lie derivative is a grade $0$ [[derivation]], acting on various products of fields via the Liebniz rule,
\begin{eqnarray}
{\cal L}_{\ve{v}} \lp a \nf{F} \rp &=& \lp {\cal L}_{\ve{v}} a \rp \nf{F} + a \lp {\cal L}_{\ve{v}} \nf{F} \rp \\
{\cal L}_{\ve{v}} \lp \f{a} \nf{F} \rp &=& \lp {\cal L}_{\ve{v}} \f{a} \rp \nf{F} + \f{a} \lp {\cal L}_{\ve{v}} \nf{F} \rp \\
{\cal L}_{\ve{v}} \lp \ve{a} \nf{F} \rp &=& \lp {\cal L}_{\ve{v}} \ve{a} \rp \nf{F} + \ve{a} \lp {\cal L}_{\ve{v}} \nf{F} \rp \\
\end{eqnarray}

Scaling the velocity field by a function results in
\begin{eqnarray}
{\cal L}_{f \ve{v}} \nf{F} &=& f {\cal L}_{\ve{v}} \nf{F} + \lp \f{d} f \rp \lp \ve{v} \nf{F} \rp \\
{\cal L}_{f \ve{v}} \ve{u} &=& f {\cal L}_{\ve{v}} \ve{u} - \lp \ve{u} \f{d} f \rp \ve{v}
\end{eqnarray}

The [[commutator]] of Lie derivatives with respect to two velocity fields acting on anything is equal to the Lie derivative with respect to the Lie bracket of the two velocity fields,
$$
\lb {\cal L}_{\ve{v}}, {\cal L}_{\ve{u}} \rb = {\cal L}_{\ve{v}} {\cal L}_{\ve{u}} - {\cal L}_{\ve{u}} {\cal L}_{\ve{v}} = {\cal L}_{\lb \ve{v}, \ve{u} \rb_L}
$$
This gives the [[Jacobi identity|Lie algebra]] for the Lie bracket when acting on a vector field,
$$
\lb \lb \ve{v}, \ve{u} \rb_L, \ve{w} \rb_L = - \lb \lb \ve{u}, \ve{w} \rb_L, \ve{v} \rb_L + \lb \lb \ve{v}, \ve{w} \rb_L, \ve{u} \rb_L
$$

The Lie derivative of [[fiber bundle]] basis elements (other than tangent or cotangent bundle basis), such as [[Clifford basis elements]], is zero,
$$
{\cal L}_{\ve{v}} \ga_\al = 0
$$

The definition of the Lie derivative in terms of the flow allows the flow to be written as the [[exponentiation]] of the Lie derivative,
$$
X(t) = \ph_t^* X = e^{t {\cal L}_{\ve{v}}} X
$$

For a surface, $\Si_t$, carried along with the flow, the time derivative of an integral over that surface is
$$
\fr{d}{d t} \int_{\Si_t} \nf{F} = \int_{\Si_t} {\cal L}_{\ve{v}} \nf{F}
$$
which combines consistently with [[Stoke's theorem|integration]],
$$
\fr{d}{d t} \int_{\pa \Si_t} \nf{F}
= \fr{d}{d t} \int_{\Si_t} \f{d} \nf{F}
= \int_{\Si_t} {\cal L}_{\ve{v}} \f{d} \nf{F} 
= \int_{\pa \Si_t} {\cal L}_{\ve{v}} \nf{F}
$$
An $n$ dimensional ''Lie group'' is a [[group]] of infinitely many elements, $g(x) \in G$, parametrized by $n$ real (or complex, for a complex Lie group) parameters, $x \in \Re^n$. A Lie group is also a [[manifold]], with points, $x$, corresponding to the parameters (in various patches).

Near the identity, $1=g(0)$, group elements may be described in terms of coordinates multiplying the [[Lie algebra]] generators associated with the group,
$$
g(x) \simeq 1 + x^A T_A
$$
The Lie algebra completely describes the local geometry of the group. For a ''connected'' manifold, and Lie group, all manifold points (and corresponding group elements) may be connected to the identity by smooth paths. [[Exponentiation|exponentiation]] of Lie algebra generators gives the ''universal cover'' of the corresponding connected Lie group,
$$
g(x)=e^{x^A T_A} \in G
$$
Points of a connected Lie group manifold may be described by the $x^A$ coordinates, with global group structure (manifold topology) determined by the ranges and matchings of $x^A$. A connected manifold, and Lie group, is ''simply connected'' if all paths are contractible. For example, a sphere is simply connected while a torus is not. The universal cover of a connected Lie group is simply connected.

The [[Lie group geometry]] is the Lie group manifold with geometry and symmetries corresponding to the action of the Lie algebra generators as vector fields. (Lie group geometry may alternatively be described as a [[Lie group bundle]], with the base manifold taken to be the Lie group manifold, the Lie group as typical fiber, and a special "identity" section, $g_I(x)$, defined.)
A [[Lie group]] is an $n$ dimensional group, $G$, with elements, $g$, that can be identified with points, $x$, on an $n$ dimensional manifold, $M$. Typically the group elements, $g(x) \in G$, are understood to be written as a function of parameters; however, it is possible to consider this identification as a bijective section, the ''identity section'', $g_I(x)$, of a [[fiber bundle]]. This bundle clearly has $M$ as base and $G$ as typical fiber. But what is the structure group and action? It is desirable to preserve the group structure of the Lie group fiber: if any three fiber/group elements satisfy $g_1 g_2 = g_3$ the corresponding elements, after transformation by an element of the structure group, should satisfy $g'_1 g'_2 = g'_3$. The structure group and action is thus identified as the [[automorphism]] group and action for $G$, and the ''Lie group bundle'' is defined as the [[automorphism bundle]] for an $n$ dimensional $G$ over an $n$ dimensional base, along with an identity section.

Since the Lie group bundle comes with a special identity section, $g_I(x)$, it has a particular automorphism bundle connection, the ''Lie group connection'', $\f{A}$, similar to the [[Maurer-Cartan connection]], such that the identity section is horizontal,
$$
0 = \f{\na} g_I = \f{d} g_I + \ha \f{A} g_I - \ha g_I \f{A}
$$
This Lie group connection can be calculated in a few steps. Presuming the matrix of connection coefficients has an [[inverse|matrix inverse]], $A_B{}^i$, this set of vectors is identified as the adjoint action vectors of the [[Lie group geometry]],
$$
\ve{A_B} = \ve{\xi^A_B} = \ha \ve{\xi^L_B} - \ha \ve{\xi^R_B}
$$
in which the left and right action vector field matrices may be calculated via the defining equations for their inverses, $\f{\xi_L^B} T_B = \lp \f{d} g_I \rp g_I^-$ and $\f{\xi_R^B} T_B = g_I^- \lp \f{d} g_I \rp$. The curvature of the Lie group connection vanishes. //(check that)// The identity section transforms under a gauge transformation to $g'_I = h g_I h^-$.

//Will outer automorphisms make things more complicated?//
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="images/png/torifibers.png" height="400">
</td></tr>
<tr><td>
Maximal tori of <SPAN class="math">SU(3) \otimes SU(2)_L \otimes U(1)_Y \otimes Spin(1,3)</SPAN> over spacetime.
</td></tr>
</table>
</center></html>
[<img[images/png/torsor.png]]A [[Lie group]] has elements $g(x) \in G$ specified by the points, $x$ (corresponding to coordinates, $x^i$), of an $n$ dimensional [[manifold]]. This manifold naturally acquires a geometry corresponding to the structure of the Lie group.

The [[group action|group]] -- left, right, or adjoint -- of any group element, $h$, on all other elements of the group provides a set of [[autodiffeomorphism|diffeomorphism]]s, $ \left\{ \ph^L_h,\ph^R_h,\ph^A_h \right\} $, on the group manifold. However, only the adjoint action is a group [[automorphism]] as well as a manifold autodiffeomorphism. A group element, $h$, near the identity,
$$h \simeq 1 + t v^B T_B$$
is specified by a [[Lie algebra]] vector, $v=v^B T_B \in Lie \, G$, and a small parameter, $t$. These parameterized actions on the group correspond to [[flow]]s,
$$
e^{t \ve{\xi} \f{d}}
$$
on the group manifold, with each flow corresponding to a vector field generator, $\ve{\xi}$, over the manifold. For example, for the right action,
\begin{eqnarray}
R_h g &=& g h = g(\ph^R_h(x)) \\
&\simeq& g + t v^B g T_B = g + t v^B \lp \xi^R_B \rp^i \pa_i g(x)
\end{eqnarray}
So, for each group action, $\left\{ L, R, A \right\}$, and each Lie algebra generator, $T_B$, there is a corresponding vector field,
\begin{eqnarray}
T_B g &=& {\cal L}_{\ve{\xi^L}} g = \ve{\xi^L_B} \f{d} g = \lp \xi^L_B \rp^i \pa_i g(x) \\
g T_B &=& {\cal L}_{\ve{\xi^R}} g = \ve{\xi^R_B} \f{d} g = \lp \xi^R_B \rp^i \pa_i g(x) \\
\ha T_B g - \ha g T_B  &=& {\cal L}_{\ve{\xi^A}} g = \ve{\xi^A_B} \f{d} g = \lp \xi^A_B \rp^i \pa_i g(x)
\end{eqnarray}
which acts via the [[Lie derivative]] and corresponds to a flow on the group manifold. The action of the Lie algebra generators and the flow by the corresponding right action vector field generators may be conceptually identified as the same, $T_B \sim \ve{\xi^R_B}$. When working with a specific group representation and coordinatization, the vector field component matrices, $\lp \xi_B \rp^i$, may be found explicitly by solving the above defining equations. Using the defining equations, the [[Lie bracket|Lie derivative]]s between vector field generators are:
\begin{eqnarray}
\lb \ve{\xi^R_B}, \ve{\xi^R_C} \rb_L &=& C_{BC}{}^D \ve{\xi^R_D} \\
\lb \ve{\xi^L_B}, \ve{\xi^L_C} \rb_L &=& - C_{BC}{}^D \ve{\xi^L_D} \\
\lb \ve{\xi^L_B}, \ve{\xi^R_C} \rb_L &=& 0
\end{eqnarray}
with the same structure constants as from the Lie algebra brackets. Also tiddler that
$$\ve{\xi^A_B} = \ha \ve{\xi^L_B} - \ha \ve{\xi^R_B}$$
allows the Lie brackets of adjoint action vector fields with the others to be easily determined. To make things more confusing, the ''left action vector fields'', $\ve{\xi^L_B}$, are ''[[right invariant]] vector fields'', while the ''right action vector fields'', $\ve{\xi^R_B}$, are ''[[left invariant]] vector fields'',
\begin{eqnarray}
R_{h*} \ve{\xi^L_B}(x) &=& \ve{\xi^L_B}(\Phi^R_h(x)) \\
L_{h*} \ve{\xi^R_B}(x) &=& \ve{\xi^R_B}(\Phi^L_h(x))
\end{eqnarray}
These vector fields have [[1-form]] duals, the ''right invariant 1-forms'', $\f{\xi_L^B}=\f{dx^i} \lp \xi^L_i \rp^B$, satisfying $\ve{\xi^L_A} \f{\xi_L^B} = \de_A^B$ and
$$
\f{\xi_L^B} T_B = \lp \f{d} g \rp g^-
$$
and the ''left invariant 1-forms'', $\f{\xi_R^B} = \f{dx^i} \lp \xi^R_i \rp^B$, satisfying $\ve{\xi^R_A} \f{\xi_R^B} = \de_A^B$ and
$$
\f{\xi_R^B} T_B = g^- \f{d} g = \f{\cal I}
$$
in which $\f{\cal I}$ is the [[Maurer-Cartan form]] over the Lie group manifold. (Note that the $L$ and $R$ are just labels that can move around, and not [[indices]] to be summed over.)

A [[geodesic]] in a Lie group is found by [[exponentiation]] of the flow from a point, resulting in a parameterized [[path]] with [[tangent vector]] $\vec{\xi}$ satisfying the geodesic equation, $0 = (\vec{\xi} \f{\na}) \vec{\xi}$, which for a Lie group is
$$
\pa_t \vec{\xi} = ad^*_{\vec{\xi}} \vec{\xi}
$$
in which the ''coadjoint operator'' is defined as
$$
<ad^*_{\vec{\xi}} \vec{u}, \vec{w}> = <\vec{u}, ad_{\vec{\xi}} \vec{w}> = <\vec{u}, [\vec{\xi},\vec{w}]>
$$
producing the local coordinate expression
$$
(ad^*_{\vec{\xi}} \vec{u})^A = u^E \xi^B C_{BC}{}^D g_{ED} g^{CA}
$$

The left invariant vector fields, along with their natural [[Lie algebra metric|Lie algebra]], provide a natural [[Lie group tangent bundle geometry]], including a frame, connection, and curvature. Note that on the ''Lie group geometry'', a manifold with a collection of special vector fields on it, all points are identical since the Lie brackets are the same between the vector fields at every point. In particular, there is no special ''identity point'' on the Lie group geometry -- a Lie group geometry is also called a //''torsor''//, //''G-torsor''//, or //''principal homogeneous space''//.
<<tiddler HideTags>>[>img[talks/CSUF09/images/group.png]]The $N$ generators, $\ve{T_A}$, are orthogonal vector fields on the $N$ dimensional Lie group manifold, $G$.
$$
\lb \ve{T_A} , \ve{T_B} \rb =
{\cal L}_{\ve{T_A}} \ve{T_B} = C_{AB}^{\p{AB}C} \ve{T_C}
$$
Following a generator around $G$ from any point, the resulting path is a circle.


[>img[talks/CSUF09/images/spiral.png]]Eigen-generators (root vectors) twist integrally around this circle.
$$
{\cal L}_{\ve{T_3}} \ve{T_+} = i \al_{3+} \ve{T_+}
$$

[>img[talks/CSUF09/images/torus.png]]Following all the commuting generators in the Cartan subalgebra produces a $R$ dimensional ''maximal torus'' in the Lie group. The $R$ twist numbers of weight vectors around each orthogonal circle in the maximal torus are the coordinates of that weight.

Weight diagrams specify how generators are twisting around each other.
A [[metric]] for the [[tangent bundle]] over the [[Lie group]] manifold may be defined such that the [[left invariant]] vector fields of a [[Lie group geometry]] have the same scalar product as the [[Lie algebra]] generators using the [[Killing form]]:
$$
\lp \ve{\xi^R_B}, \ve{\xi^R_C} \rp = \lp T_B, T_C \rp = g_{BC} = C_{BD}{}^E C_{CE}{}^D
$$
The Lie algebra metric, $g_{BC}$, may be made diagonal (like the [[Minkowski metric]] and Kronecker delta) by transforming the generators by a constant matrix (as long as $G$ is semi-[[simple]]), found via the methods of [[spectral decomposition|eigen]]. In this way, the left invariant vector fields are identified as the set of [[orthonormal basis vector fields|frame]] on the Lie group manifold, $\ve{e_B} = \ve{\xi^R_B}$, and the left invariant 1-form coefficients,
\begin{eqnarray}
\lp e_i \rp^B &=& \lp \xi^R_i \rp^B = {\cal I}_i{}^B \\
\f{e^B} &=& \f{\xi_R^B} = \f{{\cal I}^B} 
\end{eqnarray}
are the coefficients of the frame 1-form and the [[Maurer-Cartan form]], $\f{\cal I} = g^- \f{d} g$. The resulting metric on the manifold is
$$
g_{ij} = \lp e_i \rp^B \lp e_j \rp^C g_{BC} 
$$
The right invariant vector fields, $\ve{\xi_B} = \ve{\xi^L_B}$, are [[Killing vector]]s for this Lie group geometry, since
$$
{\cal L}_{\ve{\xi_B}} \ve{e_C} = \lb \ve{\xi^L_B}, \ve{\xi^R_C} \rb_L = 0
$$
And, since
$$
{\cal L}_{\ve{e_B}} \ve{e_C} = \lb \ve{\xi^R_B}, \ve{\xi^R_C} \rb_L = C_{BC}{}^A \ve{e_A}
$$
the left invariant vector fields (the orthonormal basis vectors) are also Killing, by the [[Killing form]] identity, $\lp B_B \rp_C{}^A = C_{BC}{}^A = -C_B{}^A{}_C$.

The [[torsion]]less [[tangent bundle connection]], $\f{w}{}^A{}_B$, for the Lie group manifold may be found by solving [[Cartan's equation]],
$$
0 = \f{d} \f{e^C} + \f{w}^C{}_B \f{e^B} 
$$
We can cheat a little by seeing that, since $\f{e^B}=\f{{\cal I}^B}$, this is the same as the Maurer-Cartan equation,
$$
0 = \f{d} \f{{\cal I}^C} + \ha \f{{\cal I}^A} \f{{\cal I}^B} C_{AB}{}^C
$$
giving the ''Lie group tangent bundle connection'',
$$
\f{w}^C{}_B = \ha \f{e^A} C_{AB}{}^C = - \ha \f{e^A} C_A{}^C{}_B
$$
with the indices of the structure constants raised and lowered by the diagonal matrices, $g^{AB}$ and $g_{AB}$, and using the Killing form identity. The connection coefficients directly relate to the structure constants, $w_{AC}{}^B = - \ha C_{AC}{}^B$. Alternatively, a connection with torsion could be defined if desired.

Using one of the [[Killing vector identities]], the covariant derivative of any of the right invariant vector fields or some combination, $\ve{\xi} = \xi^B \ve{\xi^L_B}$, along itself vanishes
$$
\ve{\xi} \f{\na} \ve{\xi} = \ve{\xi} \f{e^A} \ve{e_C} \lp \xi^B \lp B_B \rp_A{}^C - \xi^B w_{B A}{}^C \rp
= \fr{3}{2} \xi^A \xi^B \ve{e_C} C_{BA}{}^C 
= 0
$$
This implies the integral curves of the [[flow]]s along any right invariant field are [[geodesic]]s. By another Killing vector identity, all the right invariant vector fields are constant length.  

The [[Riemann curvature]] for the Lie group tangent bundle connection is
\begin{eqnarray}
\ff{R}^A{}_B &=& \f{d} \f{w}^A{}_B + \f{w}^A{}_C \f{w}^C{}_B \\
&=& - \ha \lp \f{d} \f{e^C} \rp C^A{}_B{}_C + \f{w}^A{}_C \f{w}^C{}_B \\
&=& \ha \lp \f{w}^C{}_D \f{e^D} \rp C^A{}_B{}_C + \f{w}^A{}_C \f{w}^C{}_B \\
&=& \fr{1}{4} \f{e^F} \f{e^D} \lp - C^C{}_{DF} C^A{}_{BC} + C^A{}_{CF} C^C{}_{BD} \rp \\
&=& - \fr{1}{4} \f{e^F} \f{e^D} C_{BCF} C_D{}^{AC}
\end{eqnarray}
using the Jacobi identity. The [[Ricci curvature]] is
$$
\f{R}{}_B = \ve{e_A} \ff{R}^A{}_B = - \fr{1}{4} \ve{e_A} \f{e^F} \f{e^D} C_{BCF} C_D{}^{AC}
= - \fr{1}{4} \f{e^D} C_{BCA} C_D{}^{AC} = - \fr{1}{4} \f{e^D} g_{BD} = - \fr{1}{4} \f{e}{}_B 
$$
showing that a Lie group geometry is an [[Einstein space|Einstein's equation]]. The [[curvature scalar]] is $R = \ve{e^B} \f{R}{}_B = - \fr{1}{4} n$.

The [[volume form]] over the Lie group manifold is the ''Haar measure'',
$$
\f{e^1} \dots \f{e^n} = \nf{d^n x} \left| e \right|
$$
The complete list of real, [[simple]], compact, connected [[Lie group]]s was completed around 1890. They fall into four infinite families (the classical Lie groups) and five exceptional groups. Sorted by rank, $r$, they are:
| !rank | !group | !a.k.a. | !dim | !name |
| $r$ | $A_r$ | $SU(r+1)$ | $r(r+2)$ | [[special unitary group]] |
| $r$ | $B_r$ | $SO(2r+1)$ | $r(2r+1)$ | odd [[special orthogonal group]] |
| $r$ | $C_r$ | $Sp(2r)$ | $r(2r+1)$ | symplectic group |
| $r>2$ | $D_r$ | $SO(2r)$ | $r(2r-1)$ | even [[special orthogonal group]] |
| $2$ | $G_2$ |  | $14$ | [[G2]] |
| $4$ | $F_4$ |  | $52$ | [[F4]] |
| $6$ | $E_6$ |  | $78$ | [[E6]] |
| $7$ | $E_7$ |  | $133$ | E7 |
| $8$ | $E_8$ |  | $248$ | [[E8]] |
<<tiddler HideTags>>The complete list of real, [[simple]], compact, connected [[Lie group]]s was completed around 1890. They fall into four infinite families (the classical Lie groups) and five exceptional groups. Sorted by rank, $r$, they are:

| !rank | !group | !a.k.a. | !dim | !name |
| $r$ | $A_r$ | $SU(r+1)$ | $r(r+2)$ | [[special unitary group]] |
| $r$ | $B_r$ | $Spin(2r+1)$ | $r(2r+1)$ | odd spin group |
| $r$ | $C_r$ | $Sp(2r)$ | $r(2r+1)$ | symplectic group |
| $r>2$ | $D_r$ | $Spin(2r)$ | $r(2r-1)$ | even spin group |
| $2$ | $G_2$ |  | $14$ | [[G2]] |
| $4$ | $F_4$ |  | $52$ | [[F4]] |
| $6$ | $E_6$ |  | $78$ | [[E6]] |
| $7$ | $E_7$ |  | $133$ | E7 |
| $8$ | $E_8$ |  | $248$ | [[E8]] |
A ''Lieform'' is a ''[[Lie algebra]] valued [[differential form]]'', having a single form grade, $p$. In terms of [[coordinate basis forms]] and Lie algebra generators, an arbitrary Lieform may be written as
$$
\nf{A} = \f{dx^i} \dots \f{dx^k} \fr{1}{p!} A_{i \dots k}{}^B T_B \in \nf{\rm Lie}(G)
$$
The basis forms and Lie algebra generators act in different algebras. By convention, the form basis elements will be collected on the left and the Lie algebra generators on the right. The most common type of Lie form is a ''Lie algebra valued 1-form'', $\f{A} = \f{dx^i} A_i{}^B T_B \in \f{\rm Lie}(G)$.

The most common operation between Lieforms is the graded [[commutator]], equivalent to the graded [[Lie algebra bracket|Lie algebra]],
$$
\lb \nf{A}, \nf{B} \rb = \nf{A^C} \nf{B^D} \lb T_C, T_D \rb = \nf{A^C} \nf{B^D} C_{CD}{}^E T_E = \nf{A} \nf{B} - \lp -1 \rp^{pq} \nf{B} \nf{A}
$$
which produces a grade $(pq)$ Lieform from the bracket of grade $p$ and $q$ Lieforms.
Link to notes, such as [[Horizontal Rule]].
{{{
Link to notes, such as [[Horizontal Rule]].
}}}
Link to [[external sites|http://www.osmosoft.com]] or [[ordinary notes|Horizontal Rule]] with ordinary words,
without the messiness of the full URL appearing.
{{{
Link to [[external sites|http://www.osmosoft.com]] or [[ordinary notes|Horizontal Rule]] with ordinary words,
without the messiness of the full URL appearing.
}}}
Or just type out http://www.osmosoft.com and it will be automatically linkified.
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[[Loops '07|http://www.matmor.unam.mx/eventos/loops07/]] was held in Morelia, Mexico. My [[talk for Loops 07]] has links to my slides and the accompanying audio.

Here are some talks I went to, and some personal impressions (these are just brief tiddlers to myself, please don't take them too seriously)
*Monday
**[[Lucien Hardy|http://www.perimeterinstitute.ca/index.phpindex.php?option=com_content&task=view&id=30&Itemid=7&view_directory=1&pi=1078]]${}^*$, The causaloid formalism: a tentative framework for quantum gravity
***Compression of measurement data
***Obtain probabilistic distribution over temporal orderings of measurements maybe?
**[[Rafael Sorkin|http://www.phy.syr.edu/~sorkin/]], Quantum reality and anhomomorphic logic
***Wants to use a "quantum" version of probability by discarding preclusion or inference rules
****discarding logical "and" (multiplication) and/or discarding logical "or" (addition)
***Talked to him about complex probability distributions over paths.
**[[John F. Donoghue|http://www.fqxi.org/aw-donoghue.html]]${}^*$, Effective field theory and quantum general relativity
***He argued that the Effective Field Theory of gravity could be used to perturb around a Newtonian potential to get the classical (GR) corrections and the quantum GR correction proportional to $h$ -- in two different ways.
***Quantum GR will have to reproduce this.
**[[Rodolfo Gambini|http://www.fqxi.org/aw-pullin.html]]${}^*$, Relational physics with real rods and clocks, (A)
***Works with Jorge Pullin
***Err, it was hard to understand what he was saying, and his talk was kind of scattered.
**[[Johannes Tambornino|http://www.perimeterinstitute.ca/index.php?option=com_content&task=view&id=30&Itemid=72&pi=4735]], Taming observables in GR: A perturbative approach, (B)
***Tries to quantize symmetry reduced theory the right way.
**[[Frederic P. Schuller|http://www.nucleares.unam.mx/~f.p.schuller/]], Area metric gravity, (B)
***Reformulate everything in terms of $G_{[ab][cd]}$ "area metric."
**[[Merced Montesinos|http://www.fis.cinvestav.mx/~geogravi/gr_mphysics/Nuevos/faculty.html]], Cartan's equations define a topological field theory, (B)
***Main idea: include Riemann curvature and Torsion as independent degrees of freedom. Them put extra "topological" terms, like Euler characteristic and Pontryagin characteristic, in the action to ensure $\ff{R}$ and $\ff{T}$ satisfy Cartan equations.
***Possibly related paper: http://arxiv.org/abs/gr-qc/0603076
**Marat Reyes, Generalized path dependent representations for gauge theories, (B)
****Rats, I missed this one.
**Yuya Sasai, Braided quantum field theories and their symmetries, (A)
***Err, very hard to tell what the heck he was doing since his English wasn't very good.
**[[Garrett Lisi|http://sifter.org/~aglisi/]]${}^*$, Deferential Geometry, (A)
*** :) My talk went very well.
***"This is very interesting." -- L.S. "It's not bullshit." -- S.H.
*Tuesday
**Thomas Thiemann, Elements of Loop Quantum Gravity
***Has a new book coming out in September -- should be good.
***Talk was packed full of good stuff.
***Start wit Palatini formulation, combine constraints into master $M$, convert variables to $su(2)$ holonomies and fluxes, build solutions using coherent states, minimize expectation value of master constraint, $M$, rather than making it strictly $0$.
**Abhay Ashtekar, LQG: Lessons from models
***Symmetry reduced models. Spherically symmetric black holes, 1+1.
**[[Carlo Rovelli]]${}^*$, Vertex amplitude and propagator in loop quantum gravity
***Work in $so(4)$. Barrett/Crane model has problem with off diagonal (vertex) terms -- B.C. doesn't give intertwiners in perturbation calc. so it's wrong. Need to use GFT to get better model.
**Jan Ambjorn, 4d quantum gravity as a sum over histories
***His talk went long and he got cut off.
***I'm kind of unimpressed with CDT -- it's an odd approximation that forces plausible numerical results.
**Dan Christensen, Computations involving spin networks, spin foams, quantum gravity and lattice gauge theory (B)
***Kind of odd: gets positive real values for QM amplitude of spinfoam areas.
**[[Wade Cherrington|http://arxiv.org/abs/0705.2629]], Numerical Spin Foam Computation of Pure Yang-Mills Theory (B)
***YM on a lattice, in an odd way. Holonomies. Expand amplitudes into group characters.
**Seth Major
***Kodama state, approximating flat spacetime, from exponentiating Chern-Simons action.
**John Swain, Spin Networks and Simplicial Quantum Gravity (B)
***Cool guy. 3+1 D Regge -> simplicial, areas and angles.
*Wednesday
**Moshe Rozali, Background Independence in String Theory
***No action for $g$ in string theory. Have to choose a metric and perturb around that.
**Klaus Fredenhagen, General covariance in quantum field theory and the background problem in perturbative quantum gravity
***Sited [[new Stefan Hollands paper|http://arxiv.org/abs/0705.3340]] on BRST and Yang-Mills in curved spacetime.
***Path integral is not covariant when you define it in detail.
***Admissible embeddings are ones that preserve causal curves.
**Alejandro Perez, Regulator dependence in quantum gravity and non perturvative renormalizability: possible new perspectives
***missed it to talk to S.H.
**Martin Reuter, Asymptotically safe quantum gravity and cosmology
***Missed it. :( I came down with flu. But watched it online.
***Using truncated action, with running coupling constants $G(k)=g(k)/k^2$ and $\La(k)=k^2 \la(k)$, there is a Gaussian fixed point at $k \to 0$ and a non-Gaussian fixed point, $\big( g(\infty), \la(\infty) \big) \to {\rm const}$.
***Using more action terms, with many running parameters, the EH action appears to emerge as the unique fixed point at high energy. Just comes from fields and symmetries.
***Large cosmo constant at high energy can drive inflation, without inflaton.
***His recent [[paper|http://arxiv.org/abs/0708.1317]] on this just came out.
**No parallel sessions
**Quarantined myself because of flu.
*Thursday
**Feeling better -- think I'll venture out of hotel room again.
**Daniele Oriti, Group field theory: spacetime from quantum discreteness to an emergent continuum
***G's live at vertices. Graphically beautiful slides -- equations and figures.
***My guess on what GFT is: start with many copies of a Lie gp, G. State is a collection of reps for each gp. When these states "match" these gps are linked, giving a spin network approximating a base manifold for a principal G-bundle.
****No, I talked with Thomas Thiemann and he said this isn't how it works. There is only a G for each dimension of the spacetime (e.g. four, or three for just space), $(D-1)=4?$
**Artem Starodubtsev, Some physical results from spinfoam models
***According to a discussion at Physics Forums, this talk will actually be about BF gravity.
***He didn't show up to the conference. Had some travel holdup in the US.
**Martin Bojowald, Loop quantum cosmology and effective theory
***Skipped it to talk with L and S.
**Jorge Pullin${}^*$, Uniform discretizations and spherically symmetric loop quantum gravity
**Sundance Bilson-Thompson, Braids, loops, and the emergence of the standard model (B)
***Leader of the Braidy Bunch.
***Poor guy had a plague of audio problems.
***gluons are two stacked braids, with a plus and minus charge.
**Jonathan Hackett, ribbon networks, (B)
***reduced link invariants
**Yidun Wan, ribbons, (B)
***nice circle diagrams for tetra, and links could describe topology, but why are they braided?
**Jonathan Engle, cosmology, (A)
**Ileana Naish-Guzman, On the regularizability of the Ponzano-Regge model, (A)
***Works with John Barrett. Soft Brittish accent. Twisted cohomology groups over a cell complex.
**James Ryan, Aspects of Group Field Theory (A)
***Excellent intro to GFT. with group su(1,1) or su(2)
**Winston J. Fairbairn, Quantization of string-like sources coupled to BF theory: transition amplitudes and topological invariance, (A)
***(d-3) branes.
*Friday
**Fotini Markopoulou${}^*$, Quantum gravity and emergent locality
***Quantum Graphity
**Lee Smolin${}^*$, Chiral excitations of quantum geometries as a possible route to unification
***"New theories should include surprises."
***Ribbons as framed graphs, consistent with a cosmological constant.
***We don't really know the relationship between spin network Hamiltonian constraint and spin foam with evolution moves.
***Working with Sundance's braids 
**Sabine Hossenfelder, Phenomenological Quantum Gravity
***"Top down inspired bottom up approaches"
***Pessimistic Freeman Dyson quote on QFT/GR independence.
***Colider constraints on KK models. (Black hole production)
***Zero black holes in standard setup (I'm not sure that's true).
***Minimal length scale as UV cutoff. (Doesn't this relate to discrete deSitter modes?)
**William Donnelly, Entanglement Entropy in Loop Quantum Gravity (B)
***Black hole entropy. Spin networks as the boundary/horizon in two part spinfoam/spacetime.
**Olaf Dreyer${}^*$, Internal Relativity: A progress report (A)
***Start with something like Ising model on lattice with Lorentz group as internal structure.
**Florian Girelli, 2-Groups and Topological Action (changed talk title!) (A)
***parallel transp of strings. defined 2-group, 2-Lie algebra, 2-principal bundle. Need 2-Peter-Weil theorem.
***one of not many DSR talks...
**Roberto Pereira, The loop-quantum-gravity vertex-amplitude (A)
***Works with Rovelli.
**Emanuele Alesci, Graviton propagator: the non diagonal terms (A)
***Works with Rovelli. (I get the impression this guy does the grind work of the calculations.) on {10J} propogator.
***Had to make up a term so that intertwiners would be involved when calculating the off diagonal part of the propogator.
**[[Isabeau Premont-Schwarz|http://arxiv.org/abs/hep-th/0508168]], Quantum Evolution in an Expanding Hilbert Space (Talk title at last minute) (A)
***Is this just using a non-square $U$ for evolution?
*Saturday
**John Stachel, Projective and Conformal Structures in General Relativity
***Older guy. Einstein biographer.
***Cecile deWitt has new book out, "Functional Integration"
***Affine space, affine connection and curvature. Need to break geometric variables into smaller pieces before quantizing.
***(His slides were messed up, missing most of the math symbols, which kinda wrecked it)
**Michael Reisenberger, Canonical gravity with free null initial data
***Free (unconstrained) gravitational initial data variables are known for initial hypersurfaces consisting of two intersecting null hypersurfaces. Recently the Poisson bracket on functions of such data has been obtained. This opens the prospect of a constraint free canonical formulation of general relativity.
**David Rideout, Can the supercomputer provide new insights into quantum gravity?
***Cactus. Causal sets and spin networks.

${}^*$ FQXi member -- might see the same talk in Iceland
(A,B) detiddlers which parallel session


During last two hours, questions were asked of the Plenary speakers, based on a book that had been passed around the audience. Carlo Rovelli moderated.
#You're at Loops '17, presenting your talk. Presuming your research program has been fully successful, describe your talk. What likelihood do you estimate for this happening? (This question was saved until the end, so speakers could prepare.)
#Topology change in QG?
##Abhay: not possible in canonical framework, but possible in LQG spin network.
#Does QG say anything about QM? Do we need a deeper framework, or a new interpretation, or different GR?
##Lucien: Need new math.
##John S: Need process QMI. (use paths)
##Thomas Thiemann: No. (just use conservative approach)
##Bianca: Relational framework
##John D: Path integrals OK, need to change GR.
#QG has fluctuating causal structure -- would this have any measurable effect?
##Sabine: photon spreading
##Michael: Matter sees only one spacetime. (yep)
#What is finite in spin foam models?
##Alejandro Perez: There are ambiguities in the theory.
#What would be a graph theory of nature? Spinfoam fundamental, or embedded in a manifold?
##Thomas: Topology change, so not embedded. He thinks spinfoam.
##Sundance: Braids.
#The first question was answered last:
##Lucien: Causaloids.
##John D: Background dependent, GR + SM emergent from spin substrate, with perturbative breaking of general covariance. 60%
##Thomas T: Complete description of QGR, as well as experiments done by himself. (ha) 5%
##Ashtekar: Resolution of all ambiguities in LQG, and establishment of all relationships between branches.
##A Perez: Thiemann was wrong. (ha!) "We have to make the road by walking."
##M Reuter: Same underlying theory, with non-Gaussian fixed point. 20%
##Daniele:
###100% - statistical GFT, with low temperature equilibrium phase
###80% - derive effective dynamics from GFT
###50% - one model singled out as successful
##Sabine: her talk would have the same title (ha) Measurements to support one model or another.
###New, unexpected data. 50%
###She'd find a permanent position. (ha)
##John S: Quantized Conformal Structures. But odds were low he'd be with it in 10 years. :(
##David Rideout: Causel sets give QM and GR. Probability: epsilon. (ha)
##Carlo Rovelli: closing words. "Let each of a hundred flowers bloom" -- quote from Michael's talk.

Random tiddlers:
*Many people used [[Beamer|http://latex-beamer.sourceforge.net/]] for their slides.

Loops '08 will be in Nottingham, England, in July (unless I misheard?)
Loops '09 probably in Beijing (or maybe P.I.)

I met a LOT of people
**PI grad students
***Chanda
***Jean Christian Boileau
***Jonathan Hackett (social guy at the end of table)
***Joel  Brownstein (inflation guy)
***Bruno Hartmann (skinny sharp german guy with glasses, works with Thiemann)
***Isabeau Premont-Schwarz (funny german guy with glasses)  
***Sean (ultimate frisbee guy)
***Cecilia
***Alejandro Satz (http://realityconditions.blogspot.com/)
***Joel  Brownstein (inflation)
***William Donnely (http://williamdonnelly.blogspot.com/)
**other PI people
***Lucien Hardy
***Fotini
***Olaf?
***Lee Smolin
***Sabine
***Sundance
***Hans Westman (big baldish german with glasses)
***Rafael Sorkin
**misc
***John Stachel
***Daniele Oriti
****Alejandro Perez (kind of wild looking)
****Florian Gireli
***John Swain
***Michael Reisenberger
***Jonathon Engle
***Wayne Bomstad (grad student, works with John Klauder in Florida, said he admires for being well rounded)
The generalized ''special orthochronous Lorentz group'', $\mbox{SO}{}^+(1,n-1)$, is a [[Lie group]] composed of [[Lorentz rotation]]s in $n$ dimensions, including one of time.  The generalized ''Lorentz group'', $\mbox{O}(1,n-1)$, has four disconnected components &mdash; two are special ($\mbox{S}$) and two are orthochronous (${}^+$).  The special orthochronous Lorentz group is the only group component containing the identity.

The Wikipedia article is quite thorough:
http://en.wikipedia.org/wiki/Lorentz_group
A rotation is a smoothly operating linear transformation acting on vectors that leaves the scalar product between vectors invariant.  ("Vectors" in this case may stand for [[tangent vector]]s, [[1-form]]s, [[Clifford vectors|Clifford element]], or any other appropriately [[indexed|indices]] object.)  For example, two vectors with components $u^\al$ and $v^\al$ may have the scalar product, $u^\al \et_{\al \be} v^\be$.  A linear transformation of the vector components by the ''Lorentz matrix'' maps the vector components to
\begin{eqnarray}
{u'}^\al &=& L^\al {}_\be u^\be\\
{v'}^\al &=& L^\al {}_\be v^\be
\end{eqnarray}
which must preserves the scalar product,
\[ u^\al \et_{\al \be} v^\be = {u'}^\al \et_{\al \be} {v'}^\be = L^\al{}_\be u^\be \et_{\al \ga} L^\ga{}_\de v^\de \]
So the Lorentz matrix must be ''orthogonal'',
\[ L^\al{}_\be \et_{\al \ga} L^\ga{}_\de = \et_{\be \de} \]
or, with the [[Minkowski metric]] raising and lowering indices, $L_{\ga \be} L^{\ga \de} = \de_\be^\de$ or $L^T L = I$.  A transformation satisfying this restriction is a ''Lorentz transformation''.  But such a transformation could also include reflections, and would then not be "smoothly operating" (not connected to the identity).  To exclude this possibility, $L$ is restricted to have positive determinant, $|L|=1$, in which case it is called "special" or "proper", and $L$ is also restricted to preserver the direction of time (no reflection of the $0$ components), in which case it is called "orthochronous".  A special orthochronous Lorentz transformation is called a ''Lorentz rotation''.  It is "smoothly operating" or "connected to the identity" in that it may be built up by many small rotations,
\[ L = \lim_{N \to \infty} \lp I + \fr{1}{N} l \rp^N \]
in which $l$ is an antisymmetric matrix, $l_{\al \be} = l_{\lb \al \be \rb}$.  A Lorentz transformation built this way is special and orthochronous.

The group of Lorentz rotations forms the [[special orthochronous Lorentz group|Lorentz group]].

Although the matrix representation is more standard, rotations are better described and carried out as [[Clifford rotation]]s.
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&nbsp;[[About]]&nbsp;&nbsp;&nbsp;[[Tags]]&nbsp;&nbsp;&nbsp;[[Symbols]]&nbsp;&nbsp;&nbsp;[[To Do]]
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| $\; \ga_\mu \; $ |[[Clifford basis vectors]] for [[Cl(1,3)]] |
| $\; \ga_{\mu\nu} = \ga_\mu \ga_\nu \; $ |[[Clifford basis bivectors|Clifford basis elements]] |
| $\; T_A \; $ |[[Lie algebra]] basis elements (//generators//) |
| $\; ( e_\mu )^a \; $ |[[orthonormal basis vector|frame]] components (//frame, vierbein//) |
| $\; \om_a^{\p{a}\nu\rh} \; $ |[[spin connection]] components |
| $\; B_a^{\p{a}A}, W_a^{\p{a}A}, G_a^{\p{a}A} \; $ |Yang-Mills [[gauge field|principal bundle]] components (//connections//) |
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$$
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//}}}
<<tiddler HideTags>>First $\mathbb{C}(8\times8)$ quadrant of a $\mathbb{C}(16\times16)$ [[chiral]] rep of $Cl(1,7)$ bivectors:
\begin{eqnarray}
\f{H^+} &=& \big( \ha \f{w} + \fr{1}{4} \f{e} \ph + \f{W} + \f{B} \big)^+ \\
&=& \fr{1}{4} \f{w^{\mu\nu}} \ga^+_{\mu\nu} + \fr{1}{4} \f{e^\mu} \ph^\ph \ga^+_{\mu\ph} \\
&& - \f{W^\pi} \fr{1}{4} \big( \ep_{\pi (\ph-4)(\ps-4)} \ga^+_{\ph \ps} + \ga^+_{(\pi+4)8} \big) 
+ \f{B} \ha \big( \ga^+_{78} - \ga^+_{56} \big)_{\phantom{\Big(}} \\

&=&
\lb \begin{array}{cccc} 
\ha \f{\om_L} \!+\! i \f{W^3} & i \f{W^1} \!+\! \f{W^2} & - \fr{1}{4} \f{e_R} \ph_0^* & \fr{1}{4} \f{e_R} \ph_+ \\
i \f{W^1} \!-\! \f{W^2} & \ha \f{\om_L} \!-\! i \f{W^3} & \fr{1}{4} \f{e_R} \ph_+^* & \fr{1}{4} \f{e_R} \ph_0 \\
-\fr{1}{4} \f{e_L} \ph_0 & \fr{1}{4} \f{e_L} \ph_+ & \ha \f{\om_R} \!+\! i \f{B} & \\
\fr{1}{4} \f{e_L} \ph_+^* & \fr{1}{4} \f{e_L} \ph_0^* & & \ha \f{\om_R} \!-\! i \f{B}
\end{array} \rb^{\phantom{\big(}}
\end{eqnarray}

with $\ph_0 = (\ph^7 + i \ph^8)$ abd $\ph_+ = (-\ph^5 + i \ph^6)$.

<<tiddler HideTags>>$$
S_\ps = \int \nf{e} \left\{ \bar{\ps} \ga^\mu \lp e_\mu\rp^i \big(
\pa_i 
+ {\tiny \frac{1}{2}} \om_i^{\p{i}\nu\rh} {\tiny \frac{1}{2}} \ga_{\nu\rh}
+ W_i^{\p{i}\pi} T^W_\pi
+ B_i T^Y
+ g_i^{\p{i}A} T^g_A
\big) \ud{\ps}
+ \bar{\ps} \ph \ud{\ps} \right\} \vp{A_{\Big(}}
$$ $$
\begin{array}{lcrcc}
\ga_1 \!\!&\!\! = \!\!&\!\! \si_2 \!\!&\!\! \otimes \!\!&\!\! \si_1 \\
\ga_2 \!\!&\!\! = \!\!&\!\! \si_2 \!\!&\!\! \otimes \!\!&\!\! \si_2 \\
\ga_3 \!\!&\!\! = \!\!&\!\! \si_2 \!\!&\!\! \otimes \!\!&\!\! \si_3 \\
\ga_4 \!\!&\!\! = \!\!&\!\! i \si_1 \!\!&\!\! \otimes \!\!&\!\! 1
\end{array}
\;\;\;\;\;\;
\begin{array}{l}
Cl(3,1) \subset GL(4,\mathbb{C}) \\[.25em]
\ga_{\mu \nu} = \ga_\mu \ga_\nu \in spin(3,1) \\[.25em]
\ep = \ga_1 \ga_2 \ga_3 \ga_4 = i \si_3 \otimes 1 \\[.25em]
P_{R/L} = \ha (1 \pm i \ep)
\end{array}
\;\;\;\;\;\;
\begin{array}{rcl}
\La_A \!\!&\!\! = \!\!&\!\!
\lb
\matrix{
0 & 0 \\
0 & \la_A
}
\rb \\
 \!\!&\!\! \in \!\!&\!\! GL(4,\mathbb{C})\vp{A^{\big(}}
\end{array} \vp{A_{\Big(}}
$$

$$
\begin{array}{lcl}
T^\om_{\mu \nu} \!\!&\!\! = \!\!&\!\! 1 \otimes 1 \otimes \ga_{\mu \nu} \;\;\;\;\;\;\; \in GL(32,\mathbb{C}) \\[.5em]
T^W_\pi \!\!&\!\! = \!\!&\!\! 1 \otimes \fr{i}{2} \si_\pi \otimes P_L \\[.5em]
T^g_A \!\!&\!\! = \!\!&\!\! \fr{i}{2} \La_A \otimes 1 \otimes 1 \\[.5em]
T^Y \!\!&\!\! = \!\!&\!\! 1 \otimes i \si_3 \otimes P_R \\[.25em]
 \!\!&\!\!  \!\!&\!\! - i \, \mbox{diag}(1,-\fr{1}{3},-\fr{1}{3},-\fr{1}{3}) \otimes 1 \otimes 1
\end{array}
\;\;\;\;\;\;\;
\ud{\ps} =
\lb
\begin{array}{c}
\nu \\ e \\ u^r \\ d^r \\ u^g \\ d^g \\ u^b \\ d^b
\end{array}
\rb
\, \in 32^\mathbb{C} \vp{A_{\Big(}}
$$

$$
\mbox{complex structure:} \;\; i \to
\lb
\matrix{
0 & -1 \\
1 & 0 }
\rb
= -i \si_2 \;\; \Rightarrow \; T \in GL(64,\mathbb{R}), \; \ud{\ps} \in 64^\mathbb{R}
$$
<<tiddler HideTags>>$$
\begin{array}{c}
\mbox{Real } Cl^1(11,3) \mbox{ basis elements in } \mathbb{R}(128)  \\[.25em]
\begin{array}{rcrccccccccccccc}
\Ga_1 \!\!&\!\! = \!\!&\!\! i \!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\Ga_2 \!\!&\!\! = \!\!&\!\! -i \!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
\Ga_3 \!\!&\!\! = \!\!&\!\! i \!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\Ga_4 \!\!&\!\! = \!\!&\!\!  \!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\Ga_5 \!\!&\!\! = \!\!&\!\!  \!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
\Ga_6 \!\!&\!\! = \!\!&\!\!  \!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\Ga_7 \!\!&\!\! = \!\!&\!\!  \!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
\Ga_8 \!\!&\!\! = \!\!&\!\! - \!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\Ga_9 \!\!&\!\! = \!\!&\!\!  \!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \cr
\Ga_{10} \!\!&\!\! = \!\!&\!\! - \!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \cr
\Ga_{11} \!\!&\!\! = \!\!&\!\! - \!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \cr
\Ga_{12} \!\!&\!\! = \!\!&\!\!  \!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \cr
\Ga_{13} \!\!&\!\! = \!\!&\!\!  \!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \cr
\Ga_{14} \!\!&\!\! = \!\!&\!\! - \!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \cr
\end{array}
\\[.75em]
\Ga = \Ga_1 \Ga_2 \dots \Ga_{14} = \si_3  \otimes  1 \\[.75em]
P_\pm = \ha (1 \pm \Ga) \\[.75em]
\Ga^+_{ij} = P_+ \Ga_i \Ga_j \, \mbox{ in } \, \mathbb{R}(64)
\end{array}
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\begin{array}{l}
\!\!\!\!\!\! \mbox{18 Standard Model generators in } spin(11,3) \\[.25em]
\begin{array}{lcl}

T^\om_{ab} \!\!&\!\! = \!\!&\!\! \Ga^+_{a \, b} \\[.4em]

T^W_1 \!\!&\!\! = \!\!&\!\! \fr{1}{4} \Ga^+_{5 \, 8} - \fr{1}{4} \Ga^+_{6 \, 7} \\
T^W_2 \!\!&\!\! = \!\!&\!\! -\fr{1}{4} \Ga^+_{5 \, 7} - \fr{1}{4} \Ga^+_{6 \, 8} \\
T^W_3 \!\!&\!\! = \!\!&\!\! - \fr{1}{4} \Ga^+_{5 \, 6} + \fr{1}{4} \Ga^+_{7 \, 8} \\[.4em]

T^g_1 \!\!&\!\! = \!\!&\!\! \fr{1}{4} \Ga^+_{9 \, 12} - \fr{1}{4} \Ga^+_{10 \, 11} \\
T^g_2 \!\!&\!\! = \!\!&\!\! -\fr{1}{4} \Ga^+_{9 \, 11} - \fr{1}{4} \Ga^+_{10 \, 12} \\
T^g_3 \!\!&\!\! = \!\!&\!\! -\fr{1}{4} \Ga^+_{9 \, 10} + \fr{1}{4} \Ga^+_{11 \, 12} \\
T^g_4 \!\!&\!\! = \!\!&\!\! -\fr{1}{4} \Ga^+_{9 \, 14} + \fr{1}{4} \Ga^+_{10 \, 13} \\
T^g_5 \!\!&\!\! = \!\!&\!\! \fr{1}{4} \Ga^+_{9 \, 13} + \fr{1}{4} \Ga^+_{10 \, 14} \\
T^g_6 \!\!&\!\! = \!\!&\!\! \fr{1}{4} \Ga^+_{11 \, 14} - \fr{1}{4} \Ga^+_{12 \, 13} \\
T^g_7 \!\!&\!\! = \!\!&\!\! -\fr{1}{4} \Ga^+_{11 \, 13} - \fr{1}{4} \Ga^+_{12 \, 14} \\
T^g_8 \!\!&\!\! = \!\!&\!\! \fr{1}{4\sqrt{3}} ( - \Ga^+_{9 \, 10} - \Ga^+_{11 \, 12} + 2 \, \Ga^+_{13 \, 14} ) \\[.4em]

T^Y \!\!&\!\! = \!\!&\!\!  \fr{1}{4} ( \Ga^+_{5 \, 6} + \Ga^+_{7 \, 8} ) \\
& &\!\! + \fr{1}{6} ( \Ga^+_{9 \, 10} + \Ga^+_{11 \, 12} +  \Ga^+_{13 \, 14} )
\end{array} \\[.5em]
\ps = \ps^{\, \io} Q_\io \in 64^\mathbb{R}_{S+}
\end{array}
$$
<<tiddler HideTags>>$$
\begin{array}{c}
\mbox{Real } Cl(3,11) \mbox{ basis elements in } GL(128,\mathbb{R})  \\[.25em]
\begin{array}{lcrccccccccccccc}
\Ga_1 \!\!&\!\!\! = \!\!\!&\!\!  \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_2 \!\!&\!\!\! = \!\!\!&\!\!  \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \\
\Ga_3 \!\!&\!\!\! = \!\!\!&\!\!  \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_3 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_4 \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_5 \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_1 \\
\Ga_6 \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_3 \\
\Ga_7 \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_3 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_8 \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_9 \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_{10} \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_3 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_{11} \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_{12} \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_3 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_{13} \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_3 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_{14} \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\end{array}
\\[.75em]
\begin{array}{rcl}
\Ga \!\!&\!\!\! = \!\!\!&\!\! \Ga_1 \Ga_2 \dots \Ga_{14} \\
 \!\!&\!\!\! = \!\!\!&\!\! \si_3  \otimes  1  \otimes  1  \otimes  1  \otimes  1  \otimes  1  \otimes  1
\end{array} \\[.75em]
P_\pm = \ha (1 \pm \Ga) \\[.75em]
\Ga^+_{\al \be} = P_+ \Ga_\al \Ga_\be \mbox{ in } GL(64,\mathbb{R})
\end{array}
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\begin{array}{l}
\!\!\!\!\!\! \mbox{18 Standard Model generators in } spin(3,11) \\[.25em]
\begin{array}{lcl}

T^\om_{\mu \nu} \!\!&\!\! = \!\!&\!\! \sqrt{2} \, \Ga^+_{\mu \nu} \\[.25em]

T^W_1 \!\!&\!\! = \!\!&\!\! \Ga^+_{5 , 8} - \Ga^+_{6 , 7} \\
T^W_2 \!\!&\!\! = \!\!&\!\! \Ga^+_{5 , 7} + \Ga^+_{6 , 8} \\
T^W_3 \!\!&\!\! = \!\!&\!\! - \Ga^+_{5 , 6} + \Ga^+_{7 , 8} \\[.25em]

T^g_1 \!\!&\!\! = \!\!&\!\! \Ga^+_{9 , 12} - \Ga^+_{10 , 11} \\
T^g_2 \!\!&\!\! = \!\!&\!\! \Ga^+_{9 , 11} + \Ga^+_{10 , 12} \\
T^g_3 \!\!&\!\! = \!\!&\!\! -\Ga^+_{9 , 10} + \Ga^+_{11 , 12} \\
T^g_4 \!\!&\!\! = \!\!&\!\! \Ga^+_{9 , 14} - \Ga^+_{10 , 13} \\
T^g_5 \!\!&\!\! = \!\!&\!\! \Ga^+_{9 , 13} + \Ga^+_{10 , 14} \\
T^g_6 \!\!&\!\! = \!\!&\!\! \Ga^+_{11 , 14} - \Ga^+_{12 , 13} \\
T^g_7 \!\!&\!\! = \!\!&\!\! \Ga^+_{11 , 13} + \Ga^+_{12 , 14} \\
T^g_8 \!\!&\!\! = \!\!&\!\! \fr{1}{\sqrt{3}} ( - \Ga^+_{9 , 10} - \Ga^+_{11 , 12} + 2 \, \Ga^+_{13 , 14} ) \\[.25em]

T^Y \!\!&\!\! = \!\!&\!\!  \fr{\sqrt{3}}{\sqrt{5}} ( \Ga^+_{5 , 6} + \Ga^+_{7 , 8} ) \\
& &\!\! + \fr{2}{\sqrt{15}} ( \Ga^+_{9 , 10} + \Ga^+_{11 , 12} + \Ga^+_{13 , 14} )
\end{array} \\[.5em]
\ps \in 64^{+\mathbb{R}}_S
\end{array}
$$
<<tiddler HideTags>>$$
\begin{array}{c}
\mbox{Real } Cl(4,12) \mbox{ basis elements in } GL(256,\mathbb{R})  \\[.25em]
\begin{array}{lcrccccccccccccccc}
\Ga_1 \!\!&\!\!\! = \!\!\!&\!\!  \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_3 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_1 \\
\Ga_2 \!\!&\!\!\! = \!\!\!&\!\!  \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_3 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \\
\Ga_3 \!\!&\!\!\! = \!\!\!&\!\!  \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_3 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_3 \\
\Ga_4 \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_3 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \\
\Ga_5 \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_6 \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_7 \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_3 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_8 \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_3 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_3 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \\
\Ga_9 \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_{10} \!\!&\!\!\! = \!\!\!&\!\!  \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_{11} \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_{12} \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_3 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_{13} \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_{14} \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_3 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_{15} \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_3 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_{16} \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\end{array}
\end{array}
\;\;\;\;\;\;
\begin{array}{l}
\mbox{120 generators in } spin(4,12) \\[.25em]
\;\;\;\;\;\; \mbox{91 in } spin(3,11) \\[.25em]
\;\;\;\;\;\; \;\;\;\;\;\; \mbox{6 for } \om \mbox{ in } spin(3,1) \\[.25em]
\;\;\;\;\;\; \;\;\;\;\;\; \mbox{45 in } spin(10) \\[.25em]
\;\;\;\;\;\; \;\;\;\;\;\; \;\;\;\;\;\; \mbox{12 for } W, B, g \\[.25em]
\;\;\;\;\;\; \;\;\;\;\;\; \;\;\;\;\;\; \mbox{3 for } W', Z' \\[.25em]
\;\;\;\;\;\; \;\;\;\;\;\; \;\;\;\;\;\; \mbox{30 for colored } X \mbox{ bosons} \\[.25em]
\;\;\;\;\;\; \;\;\;\;\;\; \mbox{40 for } e\ph \mbox{ frame (4)} \times \mbox{Higgs (10) } \\[.25em]
\;\;\;\;\;\; \mbox{1 for Peccei-Quinn } w \mbox{ in } spin(1,1) \\[.25em]
\;\;\;\;\;\; \mbox{8 for } e\th \mbox{ ''axion'' frame (4)} \times \mbox{Higgs (2)}\\[.25em]
\;\;\;\;\;\; \mbox{20 for more } X \mbox{ bosons} \\[.25em]

\mbox{128 generators in } 128_S^{+\mathbb{R}} \mbox{ of } spin(4,12) \\[.25em]
\;\;\;\;\;\; \mbox{64 for SM fermions in } 64_S^{+\mathbb{R}} \mbox{ of } spin(3,11) \\[.25em]
\;\;\;\;\;\; \mbox{64 for ''mirror'' fermions, with opposite } w \\[.25em]
\end{array}
$$
<<tiddler HideTags>>$$
S_\ps = \int \nf{e} \left\{ \bar{\ps} \ga^a \lp e_a \rp^\mu \big(
\pa_\mu 
+ {\textstyle \frac{1}{2}} \om_\mu^{\p{\mu}bc} {\textstyle \frac{1}{2}} T^\om_{bc}
+ W_\mu^{\p{\mu}I} T^W_I
+ B_\mu^Y T^Y
+ g_\mu^{\p{\mu}A} T^g_A
\big) \ud{\ps}
+ \bar{\ps} \ph \ud{\ps} \right\} \vp{A_{\Big(}}
$$ $$
\begin{array}{rcrccc}
\ga_1 \!\!&\!\!=\!\!&\!\!  i \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \cr
\ga_2 \!\!&\!\!=\!\!&\!\!  i \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\ga_3 \!\!&\!\!=\!\!&\!\!  i \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \cr
\ga_4 \!\!&\!\!=\!\!&\!\!    \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
\end{array}
\;\;\;\;\;\;\;\;\;
\begin{array}{l}
Cl(1,3) \subset \mathbb{C}(4) \\[.25em]
\ga_{ab} = \ha ( \ga_a \ga_b - \ga_b \ga_a) \in spin(1,3)  \\[.25em]
\ep = \ga_1 \ga_2 \ga_3 \ga_4 = -i \, \si_3 \otimes 1 \\[.25em]
P_{L/R} = \ha (1 \pm i \, \ep)
\end{array}
\;\;\;\;\;\;\;\;\;
\begin{array}{rcl}
\La_A \!\!&\!\! = \!\!&\!\!
\lb
\matrix{
0 & 0 \\
0 & \la_A
}
\rb \\
 \!\!&\!\! \in \!\!&\!\! \mathbb{C}(4)\vp{A^{\big(}} \\
\end{array}
$$

$$
\begin{array}{rlcl}
spin(1,3) \;\;\;\; \!\!&\!\! T^\om_{ab} \!\!&\!\! = \!\!&\!\! 1 \otimes 1 \otimes \ga_{ab} \;\;\;\;\;\;\; \in \mathbb{C}(32) \\[.5em]
su(2)_L \;\;\;\; \!\!&\!\! T^W_I \!\!&\!\! = \!\!&\!\! 1 \otimes \fr{i}{2} \si_I \otimes P_L \\[.5em]
su(3) \;\;\;\; \!\!&\!\! T^g_A \!\!&\!\! = \!\!&\!\! \fr{i}{2} \La_A \otimes 1 \otimes 1 \\[.5em]
u(1)_Y \;\;\;\; \!\!&\!\! T^Y \!\!&\!\! = \!\!&\!\! 1 \otimes \fr{i}{2} \si_3 \otimes P_R  -  \fr{i}{2} \,{\small \mbox{diag}(1,-\fr{1}{3},-\fr{1}{3},-\fr{1}{3})} \otimes 1 \otimes 1 \\[.25em]
\!\!&\!\! \!\!&\!\! = \!\!&\!\! \fr{i}{2} \,{\small \mbox{diag}(-1,0,-1,-2,\fr{1}{3},\fr{4}{3},\fr{1}{3},-\fr{2}{3},\fr{1}{3},\fr{4}{3},\fr{1}{3},-\fr{2}{3}.\fr{1}{3},\fr{4}{3},\fr{1}{3},-\fr{2}{3}) } \otimes 1 \\
\end{array}
\,
\ud{\ps} =
\lb
\begin{array}{c}
\nu \\ e \\ u^r \\ d^r \\ u^g \\ d^g \\ u^b \\ d^b
\end{array}
\rb
\begin{array}{c} 
\! \left\{
\,
\lb
\begin{array}{c}
u_{L}^{r\wedge} \\
u_{L}^{r\vee} \\
u_{R}^{r\wedge} \\
u_{R}^{r\vee} \\
\end{array}
\rb
\right.
\\[4em]
\end{array}
\, \in 32^\mathbb{C} \vp{A_{\Big(}}
$$

$$
\mbox{complex structure:} \;\; i \mapsto
\lb
\matrix{
0 & -1 \\
1 & 0 }
\rb
= -i \si_2 , \;\; u^{r\wedge}_L \mapsto
\lb
\matrix{
u^{r\wedge}_{Lr} \\
u^{r\wedge}_{Li} }
\rb
\;\;
\Longrightarrow \; T \in \mathbb{R}(64), \; \ud{\ps} \in 64^\mathbb{R}
$$
<<tiddler HideTags>>$$
S_\ps = \int \nf{e} \left\{ \bar{\ps} \ga^a \lp e_a \rp^\mu \big(
\pa_\mu 
+ {\textstyle \frac{1}{2}} \om_\mu^{\p{\mu}bc} {\textstyle \frac{1}{2}} T^\om_{bc}
+ W_\mu^{\p{\mu}I} T^W_I
+ B_\mu^Y T^Y
+ g_\mu^{\p{\mu}A} T^g_A
\big) \ud{\ps}
+ \bar{\ps} \ph \ud{\ps} \right\} \vp{A_{\Big(}}
$$ $$
{\small
\begin{array}{rcrccc}
\ga_{23} \!\!&\!\!=\!\!&\!\! -i \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \cr
\ga_{13} \!\!&\!\!=\!\!&\!\!  i \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\ga_{12} \!\!&\!\!=\!\!&\!\! -i \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \cr
\ga_{14} \!\!&\!\!=\!\!&\!\!    \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \cr
\ga_{24} \!\!&\!\!=\!\!&\!\!    \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\ga_{34} \!\!&\!\!=\!\!&\!\!    \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \cr
\end{array}
}
\;\;\;
\begin{array}{l}
{\small
\si_1 \!=\! \lb \begin{matrix} 0 & 1 \cr 1 & 0 \cr \end{matrix} \rb
\;
\si_2 \!=\! \lb \begin{matrix} 0 & -i \cr i & 0 \cr \end{matrix} \rb
\;
\si_3 \!=\! \lb \begin{matrix} 1 & 0 \cr 0 & -1 \cr \end{matrix} \rb
}
\\[1em]
P_{L/R} = \ha (1 \pm \si_3) \otimes 1 \;\;\;\;\;\; P_L= \lb \begin{matrix} 1 & 0 \cr 0 & 0 \cr \end{matrix} \rb
\end{array}
\;\;\;
g^A \La_A =
{\small
\lb
\matrix{
0 & 0 & 0 & 0 \\
0 & g^3 \!+\! \fr{1}{\sqrt{3}} g^8 \! & g^1\!-\!ig^2 & g^4\!-\!ig^5 \\
0 & g^1\!+\!ig^2 & \! -g^3 \!+\! \fr{1}{\sqrt{3}} g^8 & g^6\!-\!ig^7 \\
0 & g^4\!+\!ig^5 & g^6\!+\!ig^7 & \fr{-2}{\sqrt{3}} g^8
}
\rb
}
$$

$$
\begin{array}{rlcl}
spin(1,3) \;\;\;\; \!\!&\!\! T^\om_{ab} \!\!&\!\! = \!\!&\!\! 1 \otimes 1 \otimes \ga_{ab} \;\;\;\;\;\;\; \in \mathbb{C}(32) \\[.5em]
su(2)_L \;\;\;\; \!\!&\!\! T^W_I \!\!&\!\! = \!\!&\!\! 1 \otimes \fr{i}{2} \si_I \otimes P_L \\[.5em]
su(3) \;\;\;\; \!\!&\!\! T^g_A \!\!&\!\! = \!\!&\!\! \fr{i}{2} \La_A \otimes 1 \otimes 1 \\[.5em]
u(1)_Y \;\;\;\; \!\!&\!\! T^Y \!\!&\!\! = \!\!&\!\! 1 \otimes \fr{i}{2} \si_3 \otimes P_R  -  \fr{i}{2} \,{\small \mbox{diag}(1,-\fr{1}{3},-\fr{1}{3},-\fr{1}{3})} \otimes 1 \otimes 1 \\[.25em]
\!\!&\!\! \!\!&\!\! = \!\!&\!\! \fr{i}{2} \,{\small \mbox{diag}(-1,0,-1,-2,\fr{1}{3},\fr{4}{3},\fr{1}{3},-\fr{2}{3},\fr{1}{3},\fr{4}{3},\fr{1}{3},-\fr{2}{3}.\fr{1}{3},\fr{4}{3},\fr{1}{3},-\fr{2}{3}) } \otimes 1 \\
\end{array}
\,
\ud{\ps} =
\lb
\begin{array}{c}
\nu \\ e \\ u^r \\ d^r \\ u^g \\ d^g \\ u^b \\ d^b
\end{array}
\rb
\begin{array}{c} 
\! \left\{
\,
\lb
\begin{array}{c}
u_{L}^{r\wedge} \\
u_{L}^{r\vee} \\
u_{R}^{r\wedge} \\
u_{R}^{r\vee} \\
\end{array}
\rb
\right.
\\[4em]
\end{array}
\, \in 32^\mathbb{C} \vp{A_{\Big(}}
$$

$$
\mbox{complex structure:} \;\; i \mapsto
\lb
\matrix{
0 & -1 \\
1 & 0 }
\rb
= -i \si_2 , \;\; u^{r\wedge}_L \mapsto
\lb
\matrix{
u^{r\wedge}_{Lr} \\
u^{r\wedge}_{Li} }
\rb
\;\;
\Longrightarrow \; T \in \mathbb{R}(64), \; \ud{\ps} \in 64^\mathbb{R}
$$
<<tiddler HideTags>>$$
S_\ps = \int \nf{e} \left\{ \bar{\ps} \ga^a \lp e_a \rp^\mu \big(
\pa_\mu 
+ {\textstyle \frac{1}{2}} \om_\mu^{\p{\mu}bc} {\textstyle \frac{1}{2}} T^\om_{bc}
+ W_\mu^{\p{\mu}I} T^W_I
+ B_\mu^Y T^Y
+ g_\mu^{\p{\mu}A} T^g_A
\big) \ud{\ps}
+ \bar{\ps} \ph \ud{\ps} \right\} \vp{A_{\Big(}}
$$ $$
{\small
\begin{array}{rcrccc}
\ga_{23} \!\!&\!\!=\!\!&\!\! -i \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \cr
\ga_{13} \!\!&\!\!=\!\!&\!\!  i \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\ga_{12} \!\!&\!\!=\!\!&\!\! -i \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \cr
\ga_{14} \!\!&\!\!=\!\!&\!\!    \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \cr
\ga_{24} \!\!&\!\!=\!\!&\!\!    \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\ga_{34} \!\!&\!\!=\!\!&\!\!    \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \cr
\end{array}
}
\;\;\;
\begin{array}{l}
{\small
\si_1 \!=\! \lb \begin{matrix} 0 & 1 \cr 1 & 0 \cr \end{matrix} \rb
\;
\si_2 \!=\! \lb \begin{matrix} 0 & -i \cr i & 0 \cr \end{matrix} \rb
\;
\si_3 \!=\! \lb \begin{matrix} 1 & 0 \cr 0 & -1 \cr \end{matrix} \rb
}
\\[1em]
P_{L/R} = \ha (1 \pm \si_3) \otimes 1 \;\;\;\;\;\; P_L= \lb \begin{matrix} 1 & 0 \cr 0 & 0 \cr \end{matrix} \rb
\end{array}
\;\;\;
g^A \La_A =
{\small
\lb
\matrix{
0 & 0 & 0 & 0 \\
0 & g^3 \!+\! \fr{1}{\sqrt{3}} g^8 \! & g^1\!-\!ig^2 & g^4\!-\!ig^5 \\
0 & g^1\!+\!ig^2 & \! -g^3 \!+\! \fr{1}{\sqrt{3}} g^8 & g^6\!-\!ig^7 \\
0 & g^4\!+\!ig^5 & g^6\!+\!ig^7 & \fr{-2}{\sqrt{3}} g^8
}
\rb
}
$$

$$
\begin{array}{rlcl}
spin(1,3) \;\;\;\; \!\!&\!\! T^\om_{ab} \!\!&\!\! = \!\!&\!\! 1 \otimes 1 \otimes \ga_{ab} \;\;\;\;\;\;\; \in \mathbb{C}(32) \\[.5em]
su(2)_L \;\;\;\; \!\!&\!\! T^W_I \!\!&\!\! = \!\!&\!\! 1 \otimes \fr{i}{2} \si_I \otimes P_L \\[.5em]
su(3) \;\;\;\; \!\!&\!\! T^g_A \!\!&\!\! = \!\!&\!\! \fr{i}{2} \La_A \otimes 1 \otimes 1 \\[.5em]
u(1)_Y \;\;\;\; \!\!&\!\! T^Y \!\!&\!\! = \!\!&\!\! 1 \otimes \fr{i}{2} \si_3 \otimes P_R  -  \fr{i}{2} \,{\small \mbox{diag}(1,-\fr{1}{3},-\fr{1}{3},-\fr{1}{3})} \otimes 1 \otimes 1 \\[.25em]
\!\!&\!\! \!\!&\!\! = \!\!&\!\! \fr{i}{2} \,{\small \mbox{diag}(-1,0,-1,-2,\fr{1}{3},\fr{4}{3},\fr{1}{3},-\fr{2}{3},\fr{1}{3},\fr{4}{3},\fr{1}{3},-\fr{2}{3}.\fr{1}{3},\fr{4}{3},\fr{1}{3},-\fr{2}{3}) } \otimes 1 \\
\end{array}
\,
\ud{\ps} =
\lb
\begin{array}{c}
\nu \\ e \\ u^r \\ d^r \\ u^g \\ d^g \\ u^b \\ d^b
\end{array}
\rb
\begin{array}{c} 
\! \left\{
\,
\lb
\begin{array}{c}
u_{L}^{r\wedge} \\
u_{L}^{r\vee} \\
u_{R}^{r\wedge} \\
u_{R}^{r\vee} \\
\end{array}
\rb
\right.
\\[4em]
\end{array}
\, \in 32^\mathbb{C} \vp{A_{\Big(}}
$$

$$
\mbox{complex structure:} \;\; i \mapsto
\lb
\matrix{
0 & -1 \\
1 & 0 }
\rb
= -i \si_2 , \;\; u^{r\wedge}_L \mapsto
\lb
\matrix{
u^{r\wedge}_{Lr} \\
u^{r\wedge}_{Li} }
\rb
\;\;
\Longrightarrow \; T \in \mathbb{R}(64), \; \ud{\ps} \in 64^\mathbb{R}
$$
<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
<embed src="talks/TED08/images/G_tahiti_620.jpg" width="827" height="620"></embed>
</center></html>@@
The Maurer-Cartan connection is like the [[Lie group bundle]] connection, but for a [[principal bundle]]. The relevant ''principal Lie group bundle'' is a fiber bundle with $n$ dimensional base manifold, $M$, and $n$ dimensional Lie group, $G$, as typical fiber and structure group acting on the fiber from the right -- it is a principal bundle. The structure group action (and maybe the structure group) differs from that of a Lie group bundle. Like for a Lie group bundle, a bijective identity section, $g_I(x)$, maps base manifold points to group/fiber elements. The ''Maurer-Cartan connection'', $\f{M}$, is the connection such that the covariant derivative of the identity section is horizontal,
$$
0 = \f{\na} g_I = \f{d} g_I - g_I \f{M}
$$
which gives
$$
\f{M} = g_I^- \f{d} g_I
$$

The [[exterior derivative]] of an inverse element comes from
$$
0 = \f{d} \lp g g^- \rp = g \f{d} g^- + \lp \f{d} g \rp g^-
$$
and is $\f{d} g^- = - g^- \lp \f{d} g \rp g^-$. So the Maurer-Cartan connection satisfies the ''Maurer-Cartan equation'',
$$
\f{d} \f{M} = \f{d} \lp g_I^- \lp \f{d} g_I \rp  \rp = \lp \f{d} g_I^- \rp \lp \f{d} g_I \rp = - \f{M} \f{M} = -\f{M} \times \f{M} 
$$
and the related principal bundle curvature vanishes,
$$
\ff{F} = \f{d} \f{M} + \f{M} \f{M} = 0
$$

Interestingly, the Maurer-Cartan connection may also be related to the vielbein arising from [[Lie group geometry]],
$$
\f{M} = \f{\xi_R^B} T_B = g_I^- \f{d} g_I = \f{e^B} T_B  
$$
In this way, the Maurer-Cartan connection is a frame as well as a connection for a Lie group.

To reiterate: The Maurer-Cartan connection is a connection for a $2n$ dimensional principal bundle. This is just something new to try playing with. It's not the [[Maurer-Cartan form]], which is a 1-form over the $n$ dimensional Lie group manifold.
The ''Maurer-Cartan form'' (//''M-C form''//) is a particular [[Lieform]] field, $\f{\cal I}(x) = \f{dx^i} {\cal I}_i{}^A T_A$, defined over a [[Lie group]] manifold. It arises in [[Lie group geometry]] from the equation for the right action vector field,
$$
\ve{\xi_A^R} \f{d} g = g T_A
$$
The inverse matrix of this vector field's components gives the components of the Maurer-Cartan form, ${\cal I}_i{}^A = \lp \xi^R_i \rp^A$, which are the same as the components of the natural Lie group geometry vielbein. However, unlike the vielbein, which is a set of 1-forms, the Maurer-Cartan form is Lie algebra valued. By playing with the above equation, at every manifold point, $x$, it equals the [[inverse]] of the group element corresponding to that point times the [[exterior derivative]] of the group element, 
$$
\f{\cal I} = g^- \f{d} g
$$
The components of the Maurer-Cartan form may be found by solving this equation, or equivalently by solving for the right action vector fields and inverting.  With a nondegenerate [[Killing form]], the calculation of the components is usually done via
$$
\lp \xi^R_i \rp^A = \lp \et^{AB} T_B, g^- \pa_i g(x) \rp
$$

The M-C form is a sort of identity map from vectors on the Lie group manifold to Lie algebra elements, $\f{\cal I} = \f{{\cal I}^A} T_A = \f{\xi_R^A} T_A$ -- specifically, it maps right acting vector fields at a point to their corresponding Lie algebra element, $\ve{\xi^R_A} \f{\cal I} = T_A$. The M-C form provides an explicit isomorphism from vector valued fields to Lie algebra valued fields, $\ve{v} \f{\cal I} = v \in {\rm Lie}(G)$, and thus acts as the anchor of a [[Lie algebroid]]. If the equivalence between Lie algebra elements and their corresponding right acting vector fields is taken seriously, then the resulting ''Ehresmann-Maurer-Cartan [[vector valued form]]'' (//''E-M-C VVF''//), $\f{\ve{\cal I}}$, is nothing but the [[identity projection|vector projection]] on the Lie group manifold:
$$
\f{\ve{\cal I}} = \f{\xi_R^A} \ve{\xi^R_A} = \f{dx^i} \ve{\pa_i}
$$ 
Looking at it in a weird way, the E-M-C VVF is the [[Ehresmann connection]] for a fiber bundle with the Lie group manifold as fiber and a single point as the base.

There is a manifold [[diffeomorphism]], $x \mapsto y = y_h(x)$, corresponding to any choice of right acting group element, $h \in G$, according to $g(y_h(x)) = R_h g(x) = g(x) h$. The [[pullback]] of the M-C form under this diffeomorphism is
$$
R_h^* \f{\cal I} = \f{dx^i} \fr{\pa y_h^j}{\pa x^i} {\cal I}_j{}^A(y_h(x)) T_A = \f{dx^i} h^- g^-(x) \pa_i g(x) h = h^- \f{\cal I}(x) h
$$
which is often taken as a defining property of the M-C form. Similarly, the pullbacks of the M-C form under the left and adjoint actions are $L_h^* \f{\cal I} = h \f{\cal I} h^-$ and $A_h^* \f{\cal I} = h \f{\cal I} h^-$.  Note that this relates to the fact that, as the identity projection, the E-M-C VVF is invariant under any diffeomorphism, $\phi^* \f{\ve{\cal I}} = \f{\ve{\cal I}}$. 

A formula for the exterior derivative of an inverse element, $\f{d} g^- = - g^- \lp \f{d} g \rp g^-$, comes from
$$
0 = \f{d} \lp g g^- \rp = g \f{d} g^- + \lp \f{d} g \rp g^-
$$
So the exterior derivative of the Maurer-Cartan form is
$$
\f{d} \f{\cal I} = \f{d} \lp g_I^- \lp \f{d} g_I \rp  \rp = \lp \f{d} g_I^- \rp \lp \f{d} g_I \rp = - \f{\cal I} \f{\cal I} = -\f{\cal I} \times \f{\cal I} = - \ha \lb \f{\cal I} , \f{\cal I} \rb
$$
This gives the ''Maurer-Cartan equation'',
$$
\begin{eqnarray}
0 &=& \f{d} \f{\cal I} + \ha \lb \f{\cal I}, \f{\cal I} \rb = \ff{\cal F} \\
0 &=& \f{d} \f{{\cal I}^C} + \ha \f{{\cal I}^A} \f{{\cal I}^B} C_{AB}{}^C
\end{eqnarray}
$$
which gives vanishing [[curvature]] for the M-C form. Of course, the [[FuN curvature]] of the E-M-C VVF (which is the identity projection) also vanishes, $\ff{\ve{\cal F}} = - \ha \lb \f{\ve{\cal I}}, \f{\ve{\cal I}} \rb_L = 0$.
[>img[images/person/Max Tegmark.jpg]]Homepage: http://space.mit.edu/home/tegmark/index.html
*Location: MIT

Selected work:
*[[The Mathematical Universe|papers/0704.0646v1.pdf]]
**Computable Universe Hypothesis
**Complexity based measure on space of possible mathematics
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Maximal tori inside <SPAN class="math">SU(3) \otimes SU(2)_L \otimes U(1)_Y \otimes Spin(1,3)</SPAN>
</td></tr>
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@@
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[>img[images/person/Michael Edwards.jpg]]
*Location: Santa Cruz

Persuaded me that the [[FuN derivative]] was worth thinking about in the context of [[Ehresmann connection]]s, and wrote a nice [[synopsis|papers/BRST2-6.pdf]].
The ''Minkowski metric'', $\et_{\mu \nu}$, is a $4 \times 4$ diagonal matrix with unit magnitude real entries.  The choice of ''signature'' is somewhat arbitrary, and mostly a matter of taste.  For positive time and negative space signature $(p=1, q=3)$, generally preferred by field theorists,
\[ \et_{\mu \nu}
= 
\lb \begin{matrix}
1& 0& 0& 0\\
0& -1& 0& 0\\
0& 0& -1& 0\\
0& 0& 0& -1
\end{matrix} \rb
 = \cases {
      1&\text{if $\mu=\nu=0$}\cr
      -1&\text{if $\mu=\nu>0$}}
= \et^{\mu \nu}
\]
while for negative time and positive space signature $(p=3,q=1)$, generally preferred by relativists,
\[ \eta_{\mu \nu}
= 
\lb \begin{matrix}
-1& 0& 0& 0\\
0& 1& 0& 0\\
0& 0& 1& 0\\
0& 0& 0& 1
\end{matrix} \rb
 = \cases {
      -1&\text{if $\mu=\nu=0$}\cr
      1&\text{if $\mu=\nu>0$}}
= \et^{\mu \nu}
\]
(This "matrix" notation is a bit sloppy, since $\et_{\mu \nu}$ is not really a matrix but just a collection of indexed coefficients.) All computations should be signature ambivalent.  When they're not, the signature may be accommodated by including $\et_{00}= \pm 1$ in expressions.

The ''generalized Minkowski metric'' is $n \times n$ &mdash; accommodating extra spatial dimensions (of signature $-\et_{00}$).

The Minkowski metric may be used to raise or lower label [[indices]], such as in "$\ga^\al = \et^{\al \be} \ga_\be$" and "$v_\mu = \et_{\mu \nu} v^\nu$".  The Minkowski metric with one index raised or lowered is just the Kronecker delta, $\et^\al_\be = \et^{\al \ga} \et_{\ga \be} = \de^\al_\be$.
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By putting "<html>{{{</html>" and "<html>}}}</html>" on their own lines.
For a [[division algebra]], such as the [[octonion]]s, the ''Moufang identities'' are
1. $z(x(zy)) = ((zx)z)y$
2. $x(z(yz)) = ((xz)y)z$
3. $(zx)(yz) = (z(xy))z$
4. $(zx)(yz) = z((xy)z)$
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The ''Nieh-Yan density'' is an invariant closed 4-form,
$$N = \f{d} \lp \f{e} \cdot \ff{T} \rp = \ff{T} \cdot \ff{T} - \ff{R} \cdot \f{e} \f{e}$$
This may make for an interesting KK action term...

Another invariant cloese 4-form is the ''Pontryagin density'',
$$P = \li \ff{R} \ff{R} \ri$$

mentioned in http://arxiv.org/abs/gr-qc/0603134
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*[[On a Covartiant Formulation of the Barbero-Immirzi Connection|papers/070134.pdf]]
**New paper by [[Carlo Rovelli]] et. al. on a cleaner way of getting a $su(2)$ gravity connection from a $spin(4)$ connection.
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<img SRC="talks/IfA11/images/Glyphtionary.png" height=300px>
</center></html> 
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</center></html>
<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
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</center></html>@@
<<tiddler HideTags>>''Cartan subalgebra'': $\;\;\;
C = C^a T_a = \ha \om^S \, T^\om_{12} + \ha \om^T \, T^\om_{34} + W \,T^W_3 + Y \,T^Y + g^3 \, T^g_3 + g^8 \, T^g_8 \vp{A_{\big(}}$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\subset \; spin(1,3) \,\oplus\, su(2)_L \,\oplus\, u(1)_Y \,\oplus\, su(3) \vp{A_{\big(}} $
(Spanned by a maximal commuting set of six gravitational and Standard Model generators.)
''Weight vectors'', $\ps_\al$ : $ \;\;\; C \, \ps_\al = i \al \, \ps_\al = i C^a \al_a \ps_\al \vp{A^{\big(}_{\big(}} \;\;\;\;\;\; $ ''Weights'': $\;\;\; \al_a = \{\om^S,\om^T,W,Y,g^3,g^8\} $
''Root vectors'', $V_\al$, are eigenvectors in the adjoint: $ \;\;\; [ C , V_\al ] = i \al \, V_\al = i C^a \al_a V_\al $

''Interactions''
$$
\begin{array}{rcl}
{\rm boson} \; + \; {\rm fermion} \!\!&\!\!=\!\!&\!\! {\rm fermion} \\
V_\al \, \ps_\be \!\!&\!\!=\!\!&\!\! \ps_\de  \quad \Leftrightarrow \quad \al + \be = \de \\[.5em]
{\rm pf:} \;\;\;\; C \, V_\al \, \ps_\be \!\!&\!\!=\!\!&\!\! [C, V_\al ] \ps_\be + V_\al C \, \ps_\be
 = i \al \, V_\al \ps_\be + i \be \, V_\al \ps_\be = i (\al + \be) V_\al \ps_\be \\[1em]
{\rm boson} \; + \; {\rm boson} \!\!&\!\!=\!\!&\!\! {\rm boson} \\
[ V_\al , V_\be ] \!\!&\!\!=\!\!&\!\! V_\de  \quad \Leftrightarrow \quad \al + \be = \de \\[.5em]
{\rm pf:} \;\;\;\; [C , [ V_\al , V_\be ] ] \!\!&\!\!=\!\!&\!\!
- [V_\al,[V_\be,C]] - [V_\be,[C,V_\al]]
= i (\al + \be) [ V_\al , V_\be ]
\end{array}
$$

<html><center>
<table class="gtable">
<tr>

<td>
<table class="gtable">
<tr><td>
weight vectors
</td></tr><tr><td>
weights
</td></tr>
</table>
</td>

<td><SPAN class="math">\;\; \longleftrightarrow \;\;</SPAN></td>

<td>
<table class="gtable">
<tr><td>
eigenvectors
</td></tr><tr><td>
eigenvalues
</td></tr>
</table>
</td>

<td><SPAN class="math">\;\; \longleftrightarrow \;\;</SPAN></td>

<td>
<table class="gtable">
<tr><td>
states
</td></tr><tr><td>
quantum numbers
</td></tr>
</table>
</td>

<td><SPAN class="math">\;\; \longleftrightarrow \;\;</SPAN></td>

<td>
<table class="gtable">
<tr><td>
particles
</td></tr><tr><td>
charges
</td></tr>
</table>
</td>

</tr>
</table>
</center></html>
authors: [[Laurent Freidel]], J. Kowalski--Glikman, A. Starodubtsev
arxiv: http://arxiv.org/abs/gr-qc/0607014
locally: [[0607014|papers/0607014.pdf]]
abstract:
    Since the work of Mac-Dowell-Mansouri it is well known that gravity can be written as a gauge theory for the de Sitter group. In this paper we consider the coupling of this theory to the simplest gauge invariant observables that is, Wilson lines. The dynamics of these Wilson lines is shown to reproduce exactly the dynamics of relativistic particles coupled to gravity, the gauge charges carried by Wilson lines being the mass and spin of the particles. Insertion of Wilson lines breaks in a controlled manner the diffeomorphism symmetry of the theory and the gauge degree of freedom are transmuted to particles degree of freedom. 

*Nice summation of BF.
*Three topological terms: Euler, Pontryagin, Nieh-Yan
*Interesting treatment of Imirzi parameter
*Ahh, their main idea seems to be putting in a term and identifying it as a spinning particle. I guesss that's nice, but what's the big deal? Anyway, it's a cute way of putting in point particle matter, rather than QFT matter fields. The matter action is an integral along the parameterized path of the particle.
*Path integral to Wilson line correspondence
*matter as gravitational singularity.

This looks like the result you should get if you include the Dirac action and insert an arbitrarily boosted point particle solution for the Dirac field.
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/Zuck09/images/Pati-Salam.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math">spin(4) \,\oplus\, spin(6) \,\,\oplus\,\, 4 \otimes 4</SPAN>
</td></tr>
</table>
</center></html>
<html>
<center>
<table class="gtable">

<tr>
<td COLSPAN="3">
<SPAN class="math">so(6) + so(4) + 4 \!\times\!4 + \bar{"}</SPAN>
</td>
</tr>

<tr>
<td>
<img SRC="talks/CSUF09/images/PSElectric.png">
</td>
</tr>

</table>
</center>
</html>
<<tiddler HideTags>>
<html>
<center>
<table class="gtable">

<tr>
<td COLSPAN="3">
<SPAN class="math">so(6) + so(4) + 4 \!\times\!4 + \bar{"}</SPAN>
</td>
</tr>

<tr>
<td>
<img SRC="talks/CSUF09/images/PSE6Cox.png">
</td>
</tr>

</table>
</center>
</html>
<<tiddler HideTags>>
<<tiddler HideTags>>@@display:block;text-align:center;[img[images/png/pati-salam table.png]]@@$$
\big( SO(3,1) + 4\times4 + SU(2)_L + SU(2)_R \big) + \big( U(1) + SU(3) \big) 
$$
The three [[trace]]less, Hermitian, ''Pauli matrices'', $\si^P_A$, are
$$
\begin{array}{ccc}
\sigma_{1}^{P}=\left[\begin{array}{cc}
0 & 1\\
1 & 0\end{array}\right] & \sigma_{2}^{P}=\left[\begin{array}{cc}
0 & -i\\
i & 0\end{array}\right] & \sigma_{3}^{P}=\left[\begin{array}{cc}
1 & 0\\
0 & -1\end{array}\right]\end{array}
$$
The [[cross product|antisymmetric bracket]] of any two gives
$$
\si^P_A \times \si^P_B = \ha \lp \si^P_A \si^P_B - \si^P_B \si^P_A \rp = i \ep_{ABC} \si^P_C
$$
with $\ep_{ABC}$ the [[permutation symbol]]. The symmetric product gives
$$
\si^P_A \cdot \si^P_B = \ha \lp \si^P_A \si^P_B + \si^P_B \si^P_A \rp = \de_{AB} 1
$$
The product of the three Pauli matrices is
$$
\si_1^P \si_2^P \si_3^P = \left[\begin{array}{cc}
i & 0\\
0 & i\end{array}\right]
= i 1
$$
with the identity, $1 = \si_0^P$, often referred to as the (non-traceless) ''zero-eth Pauli matrix''.
<<tiddler HideTags>>
@@display:block;text-align:center;[img[images/png/standard model and gravity 4s.png]]@@
[>img[images/person/Peter Michor.jpg]]Homepage: http://www.mat.univie.ac.at/~michor/
*Location: Austria
*arXiv papers: http://arxiv.org/find/gr-qc/1/au:+Michor_P/0/1/0/all/0/1

Big on [[natural]]ness.
I think I annoyed him with my questions.

Selected work:
*[[Topics in Differential Geometry|papers/Topics in Differential Geometry.pdf]]
**Great introductory book to the tough stuff, and seems to contain the best from his other work
**Haar measure on p124
**[[Hodge dual]] p209
**[[FuN derivative]] p215
**grab theorem from p226 on relationship between [[Lie algebra]] and holonomy algebra
**homogeneous space p230
**gauge transformations p240
**[[FuN curvature]] p245
**covariant derivative p248, p260
**[[holonomy]] p251
**characteristic classes p263 (Wow!)
**Hamiltonian mechanics p283
*[[The Frolicher-Nijenhuis Bracket|papers/The Frolicher-Nijenhuis Bracket.pdf]]
*[[Remarks on the Frolicher-Nijenhuis Bracket|papers/Remarks on the Frolicher-Nijenhuis Bracket.pdf]]
*[[Gauge Theory for Fiber Bundles|papers/Gauge Theory for Fiber Bundles.pdf]]
*[[Natural Operations in Differential Geometry|papers/Natural Operations in Differential Geometry.pdf]]
<<tiddler HideTags>>Pirated from GS&W, [[Superstring Theory|http://www.amazon.com/Superstring-Cambridge-Monographs-Mathematical-Physics/dp/0521357527/ref=pd_bbs_sr_3/104-9709999-3726336?ie=UTF8&s=books&qid=1179001057&sr=8-3]]:
\begin{eqnarray}
E &=& B + \Ps = \ha b^{\al\be} \ga^{\small (16)+}_{\al\be} + \ps^a Q^+_a \\
&\in& so(16) + S^{\small (16)+} = {\rm Lie}(E8)
\end{eqnarray}
[[Lie brackets|Lie algebra]] between generators (structure constants):
$$
{\small
\begin{array}{rcl}
\big[ \ga^{\small (16)+}_{\al \be}, \ga^{\small (16)+}_{\ga \de} \big] &=& 2 \, \big\{ - \et_{\al \ga} \ga^{\small (16)+}_{\be \de} + \et_{\al \de} \ga^{\small (16)+}_{\be \ga} + \et_{\be \ga} \ga^{\small (16)+}_{\al \de} - \et_{\be \de} \ga^{\small (16)+}_{\al \ga} \big\}^{\p{(}} \\
\big[ \ga^{\small (16)+}_{\al \be}, Q^+_a \big] &=& \big( \ga^{\small (16)+}_{\al \be} \big)^b{}_c \big( Q^+_a \big)^c Q^+_b = \ga^{\small (16)+}_{\al \be} Q^+_a \\
\big[ Q^+_a, Q^+_b \big] &=& - \big( {\ga^{\small (16)+}}^{\al \be} \big)_{ab} \ga^{\small (16)+}_{\al \be}
\end{array}
}
$$
${\rm Lie}(E8)$ brackets act as multiplication between $120$ dimensional [[Cl(16)]] [[Clifford|Clifford algebra]] [[bivector|Clifford basis elements]]s, $B$, and positive [[chiral]], $128$ dim column [[spinor]]s, $\Ps$:
$$
\begin{array}{rcll}
\lb B_1, B_2 \rb \!\!&\!\!=\!\!&\!\! B_1 B_2 - B_2 B_1 & \in \; so(16) \\
\lb B, \Ps \rb \!\!&\!\!=\!\!&\!\! B^+ \, \Ps & \in \; S^{\small (16)+} \\
\lb \Ps_1, \Ps_2 \rb \!\!&\!\!=\!\!&\!\! -\Ps_1^\dagger \Ga^+ \Ps_2 & \in \; so(16)_{{\p{\big(}}_{\p{(}}}
\end{array}
$$
<<tiddler HideTags>>Work forwards, guess the answer, then work backwards.

Work forwards towards unification:
#[[Gauge fields|principal bundle]], [[gravity|spacetime]] and Higgs in one [[connection]].
#Calculate its [[curvature]] to get the interactions.
#Join fermions as ([[Grassmann|Grassmann number]] valued) [[BRST ghosts|BRST technique]] of a larger connection.
#Correct [[standard model]] and gravitational interactions and charges from the curvature.

Guess the answer:
*Pure [[geometry of a principal bundle|Ehresmann principal bundle connection]] -- just vector fields.
*One very large [[Lie group]] is a match!

Work backwards:
#All interactions from the [[structure|Lie algebra]] of this group, after symmetry breaking.
#Explains exactly what and why [[spinor]]s are.
#Gives three generations.
#Calculating particle masses (CKM) is a possibility.
Format blocks of CSS definitions as:
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That way the code will be run without the wikitext messing it up, and it will still be displayed nicely.
<<tiddler HideTags>><html><center>
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<td border=none>
<table class="gtable">
<tr><td>
<img src="talks/RW12/images/GraviGUT E8.png" width="420" height="420">
</td></tr><tr><td>
<SPAN class="math">E_{8(-24)} = spin(12,4) \,\,\oplus\,\, 128^\mathbb{R}_{S+}</SPAN>
</td></tr>
</table>
</td>
<td>
<SPAN class="math">
\,
</SPAN></td>
<td>

<table class="gtable">
<tr><td>
<SPAN class="math">
\begin{array}{l}
\mbox{120 generators in } spin(12,4) \\[.25em]
\;\;\;\;\;\; \mbox{91 in } spin(11,3) \\[.25em]
\;\;\;\;\;\; \;\;\;\;\;\; \mbox{6 for } \om \mbox{ in } spin(1,3) \\[.25em]
\;\;\;\;\;\; \;\;\;\;\;\; \mbox{45 in } spin(10) \\[.25em]
\;\;\;\;\;\; \;\;\;\;\;\; \;\;\;\;\;\; \mbox{12 for } W, B, g \\[.25em]
\;\;\;\;\;\; \;\;\;\;\;\; \;\;\;\;\;\; \mbox{3 for } W', Z' \\[.25em]
\;\;\;\;\;\; \;\;\;\;\;\; \;\;\;\;\;\; \mbox{30 for colored } X \mbox{ bosons} \\[.25em]
\;\;\;\;\;\; \;\;\;\;\;\; \mbox{40 for } e\ph \mbox{ frame 4} \times \mbox{Higgs 10 } \\[.25em]
\;\;\;\;\;\; \mbox{1 for Peccei-Quinn } w \mbox{ in } spin(1,1) \\[.25em]
\;\;\;\;\;\; \mbox{8 for } e\th \mbox{ frame 4} \times \mbox{''axion'' Higgs 2}\\[.25em]
\;\;\;\;\;\; \mbox{20 for more } X \mbox{ bosons} \\[.25em]

\mbox{128 generators in } 128^\mathbb{R}_{S^+} \mbox{ of } spin(12,4) \\[.25em]
\;\;\;\;\;\; \mbox{64 for SM fermions in } 64^\mathbb{R}_{S^+} \mbox{ of } spin(11,3) \\[.25em]
\;\;\;\;\;\; \mbox{64 for ''mirror fermions'' in } 64^\mathbb{R}_{S^-} \\[.25em]
\end{array}
</SPAN>
</td></tr>
</table>

</td>
</tr>
</table>
</center>
</html>
http://arxiv.org/abs/quant-ph/0610204
author: Rafael Sorkin
*primacy of path integral history formulation
decent summary to look at for quantizing perturbed BF
http://arxiv.org/pdf/hep-th/0610194
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/RW12/images/Strong interaction.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math">su(3) \,\,\oplus\,\, 3 \,\,\oplus\,\, \bar{3} \vp{{\big(}^{\big(}}</SPAN>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>>$$
\begin{array}{rcl}
(g^3 T^g_3 + g^8 T^g_8) \; \ps_{q_r} \!\!&\!\!=\!\!&\!\! i ( g^3 \al^{q_r}_3 + g^8 \al^{q_r}_8 ) \; \ps_{q_r} \\[1em]

{\small
\lb \begin{array}{cccccc}
 & \!\! - \! \fr{1}{2} g^3 \!-\! \fr{1}{2 \sqrt{3}} g^8 \!\! & & & & \\[-.5em]
 \!\! \fr{1}{2} g^3 \!+\! \fr{1}{2 \sqrt{3}} g^8 \!\! & & & & & \\[-.5em]
 & & & \!\! \fr{1}{2} g^3 \!-\! \fr{1}{2 \sqrt{3}} g^8 \!\! & & \\[-.5em]
 & & \!\! - \! \fr{1}{2} g^3 \!+\! \fr{1}{2 \sqrt{3}} g^8 \!\! &  & & \\[-.5em]
 & & & & & \fr{1}{\sqrt{3}} g^8 \\[-.5em]
 & & & & - \fr{1}{\sqrt{3}} g^8 & 
\end{array} \rb
\lb
\begin{array}{c}
1 \\ -i \\ 0 \\ 0 \\ 0 \\ 0
\end{array}
\rb }
\!\!&\!\!=\!\!&\!\! 
{\small
i \lp g^3 \big( \fr{1}{2} \big) + g^8 \big( \fr{1}{2 \sqrt{3}} \big) \rp
\lb
\begin{array}{c}
1 \\ -i \\ 0 \\ 0 \\ 0 \\ 0
\end{array}
\rb
}  \\[-1.5em]
\end{array}
$$
<html><center>
<table class="gtable">
<tr border=none>
<td border=none>

<table class="gtable">
<tr><td>
<SPAN class="math">V_{g^{r \bar{g}}} \ps_{q^g} = \ps_{q^r}  \vp{A_{\big(}}</SPAN>
</td></tr>
<tr><td>
<SPAN class="math">\al^{g^{r \bar{g}}} + \al^{q^g} = \al^{q^r} \vp{A_{\Big(}}</SPAN>
</td></tr><tr><td>
<img SRC="images/png/quark gluon vertex.png" height=160px>
</td></tr>
</table>

</td>
<td>
<SPAN class="math">\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;</SPAN></td>
<td>

<table class="gtable">
<tr><td>
<img src="talks/RW12/images/Strong interaction.png" width="310" height="310">
</td></tr><tr><td>
<SPAN class="math">su(3) \,\,\oplus\,\, 3 \,\,\oplus\,\, \bar{3}</SPAN>
</td></tr>
</table>

</td>
</tr>
</table>
</center>
</html>
<<tiddler HideTags>>$$
\begin{array}{rcl}
(g^3 T^g_3 + g^8 T^g_8) \; \ps_{q_r} \!\!&\!\!=\!\!&\!\! i ( g^3 \al^{q_r}_3 + g^8 \al^{q_r}_8 ) \; \ps_{q_r} \\[1em]

{\small
\lb \begin{array}{ccc}
  \fr{i}{2} g^3 \!+\! \fr{i}{2 \sqrt{3}} g^8 \!\! & & \\[-.5em]
 & \!\! - \! \fr{i}{2} g^3 \!+\! \fr{i}{2 \sqrt{3}} g^8 \!\! & \\[-.5em]
 & & - \fr{i}{\sqrt{3}} g^8 
\end{array} \rb
\lb
\begin{array}{c}
1 \\ 0 \\ 0
\end{array}
\rb }
\!\!&\!\!=\!\!&\!\! 
{\small
i \lp g^3 \big( \fr{1}{2} \big) + g^8 \big( \fr{1}{2 \sqrt{3}} \big) \rp
\lb
\begin{array}{c}
1 \\ 0 \\ 0
\end{array}
\rb
}
\end{array}
$$<html><center>
<table class="gtable">
<tr border=none>
<td border=none>

<table class="gtable">
<tr><td>
<SPAN class="math">V_{g^{r \bar{g}}} \ps_{q^g} = \ps_{q^r}  \vp{A_{\big(}}</SPAN>
</td></tr>
<tr><td>
<SPAN class="math">\al^{g^{r \bar{g}}} + \al^{q^g} = \al^{q^r} \vp{A_{\Big(}}</SPAN>
</td></tr><tr><td>
<img SRC="images/png/quark gluon vertex.png" height=160px>
</td></tr>
</table>

</td>
<td>
<SPAN class="math">\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;</SPAN></td>
<td>

<table class="gtable">
<tr><td>
<img src="talks/IfA11/images/Strong interaction.png" width="310" height="310">
</td></tr><tr><td>
<SPAN class="math">su(3) \,\,\oplus\,\, 3 \,\,\oplus\,\, \bar{3}</SPAN>
</td></tr>
</table>

</td>
</tr>
</table>
</center>
</html>
http://arxiv.org/abs/gr-qc/0404088
*looks to be an excellent treatment of issues with path integral treatment of GR
*justifies 3+1 dimensions from algebraic topology
| !rank | !group | !a.k.a. | !dim | !name |
| $r$ | $A_r$ | $SU(r+1)$ | $r(r+2)$ | [[special unitary group]] |
| $r$ | $B_r$ | $SO(2r+1)$ | $r(2r+1)$ | odd [[special orthogonal group]] |
| $r$ | $C_r$ | $Sp(2r)$ | $r(2r+1)$ | symplectic group |
| $r>2$ | $D_r$ | $SO(2r)$ | $r(2r-1)$ | &nbsp; even [[special orthogonal group]] &nbsp; |
| $2$ | $G_2$ |  | $14$ | G2 |
| $4$ | $F_4$ |  | $52$ | F4 |
| $6$ | $E_6$ |  | $78$ | [[E6]] |
| $7$ | $E_7$ |  | $133$ | E7 |
| $8$ | $E_8$ |  | $248$ | [[E8]] |
<<tiddler HideTags>>
"E8 is perhaps the most beautiful structure in all of mathematics, but it's very complex."
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; -- Hermann Nicolai
/***
|Name|RearrangeTiddlersPlugin|
|Source|http://www.TiddlyTools.com/#RearrangeTiddlersPlugin|
|Version|2.0.0|
|Author|Eric Shulman|
|OriginalAuthor|Joe Raii|
|License|http://www.TiddlyTools.com/#LegalStatements|
|~CoreVersion|2.1|
|Type|plugin|
|Description|drag tiddlers by title to re-order story column display|

adapted from: http://www.cs.utexas.edu/~joeraii/dragn/#Draggable
changes by ELS:
* hijack refreshTiddler() instead of overridding createTiddler()
* find title element by className instead of elementID
* set cursor style via code instead of stylesheet
* set tooltip help text
* set tiddler "position:relative" when starting drag event, restore saved value when drag ends
* update 2006.08.07: use getElementsByTagName("*") to find title element, even when it is 'buried' deep in tiddler DOM elements (due to custom template usage)
* update 2007.03.01: use apply() to invoke hijacked core function
* update 2008.01.13: only hijack core function once.  (allows for dynamic loading of plugin via bookmarklet)
* update 2008.10.19: added onclick popup menu with 'move to top' and 'move to bottom' commands
* update 2010.11.30: use story.getTiddler()
***/
//{{{

if (Story.prototype.rearrangeTiddlersHijack_refreshTiddler===undefined) {
Story.prototype.rearrangeTiddlersHijack_refreshTiddler = Story.prototype.refreshTiddler;
Story.prototype.refreshTiddler = function(title,template)
{
	this.rearrangeTiddlersHijack_refreshTiddler.apply(this,arguments);
	var theTiddler = this.getTiddler(title); if (!theTiddler) return;
	var theHandle;
	var children=theTiddler.getElementsByTagName("*");
	for (var i=0; i<children.length; i++) if (hasClass(children[i],"title")) { theHandle=children[i]; break; }
	if (!theHandle) return theTiddler;

	Drag.init(theHandle, theTiddler, 0, 0, null, null);
	theHandle.style.cursor="move";
	theHandle.title="drag title to re-arrange tiddlers, click for more options..."
	theTiddler.onDrag = function(x,y,myElem) {
		if (this.style.position!="relative")
			{ this.savedstyle=this.style.position; this.style.position="relative"; }
		y = myElem.offsetTop;
		var next = myElem.nextSibling;
		var prev = myElem.previousSibling;
		if (next && y + myElem.offsetHeight > next.offsetTop + next.offsetHeight/2) { 
			myElem.parentNode.removeChild(myElem);
			next.parentNode.insertBefore(myElem, next.nextSibling);//elems[pos+1]);
			myElem.style["top"] = -next.offsetHeight/2+"px";
		}
		if (prev && y < prev.offsetTop + prev.offsetHeight/2) { 
			myElem.parentNode.removeChild(myElem);
			prev.parentNode.insertBefore(myElem, prev);
			myElem.style["top"] = prev.offsetHeight/2+"px";
		}
	};
	theTiddler.onDragEnd = function(x,y,myElem) {
		myElem.style["top"] = "0px";
		if (this.savedstyle!=undefined)
			this.style.position=this.savedstyle;
	};
	theHandle.onclick=function(ev) {
		ev=ev||window.event;
		var p=Popup.create(this); if (!p) return;
		var b=createTiddlyButton(createTiddlyElement(p,"li"),
			"\u25B2 move to top of column ","move this tiddler to the top of the story column",
			function() {
				var t=story.getTiddler(this.getAttribute("tid"));
				t.parentNode.insertBefore(t,t.parentNode.firstChild); // move to top of column
				window.scrollTo(0,ensureVisible(t));
				return false;
			});
		b.setAttribute("tid",title);
		var b=createTiddlyButton(createTiddlyElement(p,"li"),
			"\u25BC move to bottom of column ","move this tiddler to the bottom of the story column",
			function() {
				var t=story.getTiddler(this.getAttribute("tid"));
				t.parentNode.insertBefore(t,null); // move to bottom of column
				window.scrollTo(0,ensureVisible(t));
				return false;
			});
		b.setAttribute("tid",title);
		Popup.show();
		ev.cancelBubble=true; if (ev.stopPropagation) ev.stopPropagation(); return(false);
	};
	return theTiddler;
}
}

/**************************************************
 * dom-drag.js
 * 09.25.2001
 * www.youngpup.net
 **************************************************
 * 10.28.2001 - fixed minor bug where events
 * sometimes fired off the handle, not the root.
 **************************************************/

var Drag = {
	obj:null,

	init:
	function(o, oRoot, minX, maxX, minY, maxY) {
		o.onmousedown = Drag.start;
		o.root = oRoot && oRoot != null ? oRoot : o ;
		if (isNaN(parseInt(o.root.style.left))) o.root.style.left="0px";
		if (isNaN(parseInt(o.root.style.top))) o.root.style.top="0px";
		o.minX = typeof minX != 'undefined' ? minX : null;
		o.minY = typeof minY != 'undefined' ? minY : null;
		o.maxX = typeof maxX != 'undefined' ? maxX : null;
		o.maxY = typeof maxY != 'undefined' ? maxY : null;
		o.root.onDragStart = new Function();
		o.root.onDragEnd = new Function();
		o.root.onDrag = new Function();
	},

	start:
	function(e) {
		var o = Drag.obj = this;
		e = Drag.fixE(e);
		var y = parseInt(o.root.style.top);
		var x = parseInt(o.root.style.left);
		o.root.onDragStart(x, y, Drag.obj.root);
		o.lastMouseX = e.clientX;
		o.lastMouseY = e.clientY;
		if (o.minX != null) o.minMouseX = e.clientX - x + o.minX;
		if (o.maxX != null) o.maxMouseX = o.minMouseX + o.maxX - o.minX;
		if (o.minY != null) o.minMouseY = e.clientY - y + o.minY;
		if (o.maxY != null) o.maxMouseY = o.minMouseY + o.maxY - o.minY;
		document.onmousemove = Drag.drag;
		document.onmouseup = Drag.end;
		Drag.obj.root.style["z-index"] = "10";
		return false;
	},

	drag:
	function(e) {
		e = Drag.fixE(e);
		var o = Drag.obj;
		var ey = e.clientY;
		var ex = e.clientX;
		var y = parseInt(o.root.style.top);
		var x = parseInt(o.root.style.left);
		var nx, ny;
		if (o.minX != null) ex = Math.max(ex, o.minMouseX);
		if (o.maxX != null) ex = Math.min(ex, o.maxMouseX);
		if (o.minY != null) ey = Math.max(ey, o.minMouseY);
		if (o.maxY != null) ey = Math.min(ey, o.maxMouseY);
		nx = x + (ex - o.lastMouseX);
		ny = y + (ey - o.lastMouseY);
		Drag.obj.root.style["left"] = nx + "px";
		Drag.obj.root.style["top"] = ny + "px";
		Drag.obj.lastMouseX = ex;
		Drag.obj.lastMouseY = ey;
		Drag.obj.root.onDrag(nx, ny, Drag.obj.root);
		return false;
	},

	end:
	function() {
		document.onmousemove = null;
		document.onmouseup = null;
		Drag.obj.root.style["z-index"] = "0";
		Drag.obj.root.onDragEnd(parseInt(Drag.obj.root.style["left"]), parseInt(Drag.obj.root.style["top"]), Drag.obj.root);
		Drag.obj = null;
	},

	fixE:
	function(e) {
		if (typeof e == 'undefined') e = window.event;
		if (typeof e.layerX == 'undefined') e.layerX = e.offsetX;
		if (typeof e.layerY == 'undefined') e.layerY = e.offsetY;
		return e;
	}
};
//}}}
//''Shows DefaultTiddlers + most recently modified tiddlers as default when any TiddlyWiki or adaptation is first loaded.''//
//To use, copy this tiddler's contents to a new tiddler on your site and tag it "systemConfig".//

{{{
var num = 3;
var ignore_tags = ['systemConfig', 'systemTiddlers', 'plugin', 'system'];

function in_array(item, arr){for(var i=0;i<arr.length;i++)if(item==arr[i])return true};
function get_parent(tiddler){while(tiddler && in_array('comments', tiddler.tags)) tiddler=store.fetchTiddler(tiddler.tags[0]);return tiddler};
function unique_list(list){var l=[];for(i=0;i<list.length;i++)if(!in_array(list[i], l))l.push(list[i]);return l};
function get_recent_tiddlers(){
  var tiddlers = store.getTiddlers('modified');
  var names = store.getTiddlerText("DefaultTiddlers").readBracketedList();
  var ignore_tiddlers = [];
  for(var i=0; i<ignore_tags.length; i++)
    ignore_tiddlers=ignore_tiddlers.concat(store.getTaggedTiddlers(ignore_tags[i]));
  for(var i=tiddlers.length-1; i>=0; i--) {
    if(in_array('comments', tiddlers[i].tags)) {
      var t = get_parent(tiddlers[i]);
      if(t)names.push(t.title)
    }
    else if(!in_array(tiddlers[i], ignore_tiddlers))
      names.push(tiddlers[i].title);
  }
  return unique_list(names).slice(0, num);
}
var names = get_recent_tiddlers();
_restart = restart
restart = function() {
  if(window.location.hash) _restart();
  else story.displayTiddlers(null,names);
}
}}}
<<tiddler HideTags>>
One particularly interesting way $e8$ can be broken down:

\begin{eqnarray}
e8 &=& e6 + su(3) + 54 \! \times \! 3 \\
 &=& so(1,9) + u(1) + 32 + su(3) + 54 \! \times \! 3\\
 &=& so(1,3) + su(2) + su(2) + u(1) + 4 \! \times \! 8 + u(1) + 32 + su(3) + 54 \! \times \! 3 \\
&\to& {\scriptsize \frac{1}{2}} \om + W + B + {\scriptsize \frac{1}{4}} e \ph + G + 3 \! \times \! \ps + X? \p{{}^{\big(}}
\end{eqnarray}

How does this $e8$ breakdown relate to [[e8 triality decomposition]]?

\begin{eqnarray}
e8 &=& so(1,7) + so(8) + 3 \! \times \! 8 \! \times \! 8 \\
 &=& so(1,3) + so(4) + 4 \! \times \! 4 + so(6) + so(2) + 6 \! \times \! 2  + 3 \! \times \! 8 \! \times \! 8 \\
 &=& so(1,3) + su(2) + su(2) + 4 \! \times \! 4 + su(4) + u(1) + 6 \! \times \! 2  + 3 \! \times \! 8 \! \times \! 8 \p{{}_{\big(}}
\end{eqnarray}
/***
''Name:'' ReferencesPlugin
''Author:'' Garrett Lisi
''Description:'' Places a comma separated list of referring tiddlers at the bottom of each tiddler -- replacing the "references" command bar button.
''Installation:'' Copy this tiddler, change the [[StyleSheet]] to set the references class style, and add a line in the [[ViewTemplate]].

''Code:''
***/
/*{{{*/
config.macros.references = {};
config.macros.references.handler = function(place,macroName,params,wikifier,paramString,tiddler)
{
	var references = store.getReferringTiddlers(tiddler.title);
	if(references.length>0)
		{
//		createTiddlyText(place,"\xAB ");	
		createTiddlyLink(place,references[0].title,true);
		}
	for(var r=1; r<references.length; r++)
		if(references[r].title != tiddler.title)
			{
			createTiddlyText(place,", ");
			createTiddlyLink(place,references[r].title,true);
			}
}
/*}}}*/
<<tiddler HideTags>>''Cartan subalgebra'': $\quad C=C^a T_a \;\; \subset \; {\rm Lie}(G) \vp{|_(}$
Built from a maximal commuting set of $R$ generators,
$$
\big[ T_a, T_b \big] = T_a T_b - T_b T_a = 0 \qquad \forall \quad 1 \le a,b \le R
$$
''Root vectors'', $V_\be$, are eigenvectors of $C$ in the Lie bracket,
$$
[ C , V_\be ] = \al_\be V_\be = \sum_a i C^a\al_{a\be} V_\be
$$
''Roots'', $\al_{a\be}$, are the eigenvalue coefficients. The pattern of roots in $R$ dimensions corresponds to the Lie algebra,
$$
[ V_\be , V_\ga ] = V_\de \quad \Leftrightarrow \quad \al_{\be} + \al_{\ga} = \al_{\de}
$$
''Weight  vectors'' and ''weights'' are eigenvectors and eigenvalue coefficients of $C$ acting on some representation space,
$$
C \, V_\be = \al_\be V_\be
$$
Weight vectors are particles, weights are their quantum numbers.
[[Contracting|vector-form algebra]] the [[coordinate basis vectors]] with the [[Riemann curvature]] gives the ''Ricci curvature'',
$$
\f{R}{}_m = \ve{\pa_k} \ff{R}^k{}_m = \ve{\pa_k} \f{dx^i} \f{dx}^j \ha R_{ij}{}^k{}_m
 = \f{dx}^j R_{ij}{}^i{}_m
$$
with the components of the ''Ricci curvature tensor'' equaling a partial contraction of the Riemann curvature tensor,
$$
R_{jm} = R_{ij}{}^i{}_m = 2 \pa_{\lb i \rd} \Ga^i{}_{\ld j \rb m} + 2 \Ga^i{}_{\lb i \rd l} \Ga^l{}_{\ld j \rb m}
$$
This tensor is symmetric if the [[torsion]] vanishes, $R_{jm}=R_{mj}$. In terms of the [[tangent bundle spin connection|tangent bundle connection]], the Ricci curvature is
$$
\f{R}{}_\al = \f{e^\de} R_{\de \al} = \ve{e_\be} \ff{R}^\be{}_\al
= \f{dx^j} 2 \lp e_\be \rp^i \lp \pa_{\lb i \rd} w_{\ld j \rb}{}^\be{}_\al + w_{\lb i \rd}{}^\be{}_\ga w_{\ld j \rb}{}^\ga{}_\al \rp
$$
with coefficients $R_{\de \al} = R_{\be \al}{}^\be{}_\de$. If the spin connection is torsionless, the Ricci curvature tensor may also be written as
\begin{eqnarray}
R_{\ga \al} &=& R_{\be \ga}{}^\be{}_\al = 2 \pa_{\lb \be \rd} w_{\ld \ga \rb}{}^\be{}_\al + 2 w_{\lb \be \ga \rb}{}^\ep w_\ep{}^\be{}_\al - 2 w_{\lb \be \rd}{}^{\ep \be} w_{\ld \ga \rb}{}_{\ep \al} \\
&=& 2 \pa_{\lb \be \rd} w_{\ld \ga \rb}{}^\be{}_\al + w_{\be \ga}{}^\ep w_\ep{}^\be{}_\al - w_\be{}^{\ep \be} w_{\ga \ep \al}
\end{eqnarray}
The [[vector bundle curvature]] for a [[tangent bundle]] describes the local geometry of the base manifold. Applying the [[tangent bundle covariant derivative|tangent bundle connection]] twice, and taking the [[antisymmetric|index bracket]] part, gives the tangent bundle curvature,
$$
\na_{\lb i \rd} \na_{\ld j \rb} \ve{v} = \na_{\lb i \rd} \lp \pa_{\ld j \rb} v^k + \Ga^k{}_{\ld j \rb l} v^l \rp \ve{\pa_k} 
= \lp \pa_{\lb i \rd} \Ga^k{}_{\ld j \rb l} + \Ga^k{}_{\lb i \rd m} \Ga^m{}_{\ld j \rb l} \rp v^l \ve{\pa_k}
= \ha R_{ij}{}^k{}_l v^l \ve{\pa_k}
$$
The components of the ''Riemann curvature'' (//''tangent bundle curvature''//), $\ff{R}^k{}_l = \ha \f{dx^i} \f{dx^j} R_{ij}{}^k{}_l$, are the components of the conventional Riemann curvature tensor after rearrangement, $R_{ij}{}^k{}_l \leftrightarrow R^k{}_{lij}$. (//The non-conventional Riemann index placement used here instead follows the conventional index placement for curvature tensors.//) The components are:
$$
R_{ij}{}^k{}_l = 2 \pa_{\lb i \rd} \Ga^k{}_{\ld j \rb l} + 2 \Ga^k{}_{\lb i \rd m} \Ga^m{}_{\ld j \rb l}
$$
Written with fewer indices, this is:
$$
\ff{R}^k{}_l = \f{d} \f{\Ga}^k{}_l + \f{\Ga}^k{}_m \f{\Ga}^m{}_l
$$
A different expression for the tangent bundle curvature, $\ff{R}^\be{}_\al = \ha \f{dx^i} \f{dx^j} R_{ij}{}^\be{}_\al$, may also be written in terms of the [[tangent bundle spin connection|tangent bundle connection]],
$$
\na_{\lb i \rd} \na_{\ld j \rb} \ve{v} = \na_{\lb i \rd} \lp \pa_{\ld j \rb} v^\be + w_{\ld j \rb}{}^\be{}_\al v^\al \rp \ve{e_\be} 
= \lp \pa_{\lb i \rd} w_{\ld j \rb}{}^\be{}_\al + w_{\lb i \rd}{}^\be{}_\ga w_{\ld j \rb}{}^\ga{}_\al \rp v^\al \ve{e_\be}
= \ha R_{ij}{}^\be{}_\al v^\al \ve{e_\be}
$$
with components:
$$
R_{ij}{}^\be{}_\al = 2 \pa_{\lb i \rd} w_{\ld j \rb}{}^\be{}_\al + 2 w_{\lb i \rd}{}^\be{}_\ga w_{\ld j \rb}{}^\ga{}_\al
$$
Or, with fewer indices:
$$
\ff{R}^\be{}_\al  = \ff{F}^\be{}_\al = \f{d} \f{w}^\be{}_\al + \f{w}^\be{}_\ga \f{w}^\ga{}_\al
$$
If the spin connection is [[torsion]]less, the Riemann curvature tensor may also be written, using the [[frame]], as
$$
R_{ \de \ga}{}^\be{}_\al = \lp e_\de\rp^i \lp e_\ga\rp^j R_{ij}{}^\be{}_\al
= 2 \pa_{\lb \de \rd} w_{\ld \ga \rb}{}^\be{}_\al + 2 w_{\lb \de \ga \rb}{}^\ep w_\ep{}^\be{}_\al - 2 w_{\lb \de \rd}{}^{\ep \be} w_{\ld \ga \rb}{}_{\ep \al}
$$
in which $\pa_\al = \lp e_\al \rp^i \pa_i$ and $w_\al{}^\ga{}_\be = \lp e_\al \rp^i w_i{}^\ga{}_\be$.

The Riemann curvature may alternatively be obtained from the [[tangent bundle holonomy]].
<<tiddler HideTags>>''Cartan subalgebra'': $\quad C=C^a T_a \;\; \subset \; {\rm Lie}(G) \vp{|_(}$
Built from a maximal commuting set of $R$ generators,
$$
\big[ T_a, T_b \big] = T_a T_b - T_b T_a = 0 \qquad \forall \quad 1 \le a,b \le R
$$
''Root vectors'', $T_\be$, are eigenvectors of $C$ in the Lie bracket,
$$
[ C , T_\be ] = \al_\be T_\be = \sum_a i C^a\al_{a\be} T_\be
$$
''Roots'', $\al_{a\be}$, are the eigenvalue coefficients. The pattern of roots in $R$ dimensions corresponds to the Lie algebra,
$$
[ T_\be , T_\ga ] = T_\de \quad \Leftrightarrow \quad \al_{\be} + \al_{\ga} = \al_{\de}
$$
''Weight  vectors'' and ''weights'' are eigenvectors and eigenvalue coefficients of $C$ acting on some representation space,
$$
C \, \ps_\be = \al_\be \ps_\be
$$
Weight vectors are particle states, weights are their charge quantum numbers.
GUT unification can be done using the $SU(5)$ subalgebra of $SO(10)$, but there is a (probably) better way. $SO(10)$ has $SU(2) \times SU(2) \times SU(4)$ as a maximal subalgebra. The $SU(2) \times SU(2)$ is $SO(4)$ and the $SU(4)$ is $SO(6)$, so the $SO(4) \times SO(6)$ are diagonal blocks of the $SO(10)$. The Dynkin diagram surgery for this reduction is the removal of the central $SU(2)$ node.

Ref:
*Howard Georgi's book, p283:
**http://www.amazon.com/gp/reader/0738202339/ref=sib_dp_pop_toc/104-9709999-3726336?ie=UTF8&p=S00E#
**Also see p169 of his recent talk on GUT's for the 16 complex dim spinor rep:
***[[GUTs|papers/yt100sym_georgi.pdf]]
<html>
<center>
<table class="gtable">

<tr>
<td COLSPAN="3">
<SPAN class="math">so(10) + 16 + \bar{16}</SPAN>
</td>
</tr>

<tr>
<td>
<img SRC="talks/CSUF09/images/E6Electric.png">
</td>
</tr>

</table>
</center>
</html>
<<tiddler HideTags>>
<html>
<center>
<table class="gtable">

<tr>
<td COLSPAN="3">
<SPAN class="math">so(10) + 16 + \bar{16}</SPAN>
</td>
</tr>

<tr>
<td>
<img SRC="talks/CSUF09/images/E6E6Cox.png">
</td>
</tr>

</table>
</center>
</html>
<<tiddler HideTags>>
The ''three dimensional [[special unitary group]]'' (//''special unitary group of order two''//), $G = SU(2)$, is the [[Lie group]] of [[unitary]] $2 \times 2$ complex matrices with unit [[determinant]]. Its elements, $g \in G$, may be parameterized and obtained by [[exponentiating|exponentiation]] the [[su(2)]], $T_A = \fr{i}{2} \si_A^P$, [[Lie algebra]] generators,
$$
g(x) = e^{x^i T_i} = e^X
$$
with $X=x^i T_i \in su(2)$. It is possible to carry out this exponentiation explicitly, and do calculations in these coordinates. However, it is more instructive to convert to [[spherical coordinates]], with
$$
X = 2 a^1 \sin(a^2) \cos(a^3) T_1 + 2 a^1 \sin(a^2) \sin(a^3) T_2 + 2 a^1 \cos(a^2) T_3 
= \left[\begin{array}{cc}
i a^1 \cos(a^2) & i a^1 e^{-i a^3} \sin(a^2)\\
i a^1 e^{i a^3} \sin(a^2) & -i a^1 \cos(a^2)\end{array}\right]
$$
and perform the [[spectral decomposition|eigen]],
$$
X = U \La U^-
= \left[\begin{array}{cc}
- e^{-i a^3} \sin(\fr{a^2}{2}) & e^{-i a^3} \cos(\fr{a^2}{2}) \\
\cos(\fr{a^2}{2}) & \sin(\fr{a^2}{2}) \end{array}\right]
\left[\begin{array}{cc}
- i a^1 & 0 \\
0 & i a^1 \end{array}\right]
\left[\begin{array}{cc}
- e^{i a^3} \sin(\fr{a^2}{2}) & e^{i a^3} \cos(\fr{a^2}{2}) \\
\cos(\fr{a^2}{2}) & \sin(\fr{a^2}{2}) \end{array}\right]
 $$
in order to exponentiate and get:
\begin{eqnarray}
g(a) &=& e^X = U e^\La U^-
= \left[\begin{array}{cc}
- e^{-i a^3} \sin(\fr{a^2}{2}) & e^{-i a^3} \cos(\fr{a^2}{2}) \\
\cos(\fr{a^2}{2}) & \sin(\fr{a^2}{2}) \end{array}\right]
\left[\begin{array}{cc}
e^{- i a^1} & 0 \\
0 & e^{i a^1} \end{array}\right]
\left[\begin{array}{cc}
- e^{i a^3} \sin(\fr{a^2}{2}) & e^{i a^3} \cos(\fr{a^2}{2}) \\
\cos(\fr{a^2}{2}) & \sin(\fr{a^2}{2}) \end{array}\right] \\
&=&
\left[\begin{array}{cc}
\cos(a^1) + i \sin(a^1) \cos(a^2) & i \sin(a^1) e^{-i a^3} \sin(a^2) \\
i \sin(a^1) e^{i a^3} \sin(a^2) & \cos(a^1) - i \sin(a^1) \cos(a^2) \end{array}\right] \\
&=& \cos(a^1) 1 + \fr{\sin(a^1)}{a^1} X
\end{eqnarray}
This could have been found more easily by noting that $XX = - (a^1)^2$ -- but this won't be true for general Lie groups, while the above method generalizes nicely.

The [[Lie group geometry]] is described by the left and right acting (right and left invariant) vector fields, and their dual 1-form fields. Over most of the group manifold, the [[Maurer-Cartan form]],
$$
\f{\cal I}(a) = g^-(a) \f{d} g(a) = \f{da^i} \lp\xi^R_i\rp^A T_A = \f{da^i} \lp e_i\rp^A T_A 
$$
has components (best computed using Mathematica or something):
\begin{eqnarray}
\lp e_i\rp^A &=& \lp T^A, g^-(a) \pa_i g(a) \rp \\
&=& \ha
\left[\begin{array}{ccc}
\sin(a^2) \cos(a^3) & \sin(a^2) \sin(a^3) & \cos(a^2) \\
\sin(a^1) \lp \cos(a^1) \cos(a^2) \cos(a^3) - \sin(a^1) \sin(a^3) \rp & \sin(a^1) \lp \sin(a^1) \cos(a^3) + \cos(a^1) \cos(a^2) \sin(a^3) \rp & - \ha \sin(2 a^1) \sin(a^2) \\
\ha \lp - \sin^2(a^1) \sin(2 a^2) \cos(a^3) - \sin(2 a^1) \sin(a^2) \sin(a^3) \rp & \sin(a^1) \sin(a^2) \lp \cos(a^1) \cos(a^3) - \sin(a^1) \cos(a^2) \sin(a^3) \rp & \sin^2(a^1) \sin^2(a^2)
\end{array}\right]
\end{eqnarray}
Identifying these as the [[frame]] components for the [[Lie group tangent bundle geometry]], using the su(2) [[Killing form]], $g_{AB} = -2 \de_{AB}$, gives the metric for the Lie group geometry,
$$
g_{ij}(a) = \lp e_i\rp^A g_{AB} \lp e_j \rp^B
=
\left[\begin{array}{ccc}
-8 & 0 & 0 \\
0 & -8 \sin^2(a^1) & 0 \\
0 & 0 & -8 \sin^2(a^1) \sin^2(a^2)
\end{array}\right]
$$
 
$SU(2) = Spin(3)$ may also be thought of as the group generated by the bivectors of the three dimensional Clifford algebra, [[Cl(3)]]. Under this representation, each group element, $g$, is a $Cl(3)$ scalar plus a bivector. This is also equivalent to representation by [[quaternion]]s. The ''unitary group'', $\left\{ U\in GL(C)\mid UU^{\dagger}=1\right\} $, corresponds to the unitary [[subgroup]] of the Clifford Algebra, $\left\{ U\in Cl\mid U\gamma_{0}\widetilde{U}=\gamma_{0}\right\}$, with $\widetilde{U}$ the [[Clifford reverse|Clifford conjugate]].
The ''eight dimensional [[special unitary group]]'' (//''special unitary group of order three''//), $G = SU(3)$, is the [[Lie group]] of [[unitary]] $3 \times 3$ complex matrices with unit [[determinant]]. Its elements, $g \in G$, may be parameterized and obtained by [[exponentiating|exponentiation]] the [[su(3)]] [[Lie algebra]] generators,
$$
g(x) = e^{x^i T_i}
$$
For [[loops|vector-form algebra]] and higher grade multivectors.

http://www.mimuw.edu.pl/~pwit/TOK/sem4/online/node9.html
*[[John Baez]] has a nice recent writeup from his course on quantization:
**[[path integrals|papers/w07week08a.pdf]]
The ''Schwarzschild solution'' gives the unique geometry of [[spacetime]] in the vicinity of an uncharged, non-rotating, spherically symmetric mass, $M$.  This approximately describes spacetime around the sun, earth, or black holes.  The solution is most concisely expressed by the [[frame]],
$$
\f{e} = \f{dt} \lp 1 - \fr{R_s}{r} \rp^\ha \ga_0 + \f{dr} \fr{1}{c} \lp 1 - \fr{R_s}{r} \rp^{-\ha} \ga_1
+ \f{d\th} \fr{r}{c} \ga_2 + \f{d\ph} \fr{r \sin{\th}}{c} \ga_3
$$
having diagonal frame matrix.  The coordinates are $(x^0,x^1,x^2,x^3)=(t,r,\th,\ph)$ and have [[units]] $(T,L,0,0)$.  The solution has a coordinate singularity at $r=R_S=\fr{2GM}{c^2}$, corresponding to the ''Schwarzchild radius'' -- the horizon beyond which light cannot escape.

A coordinate singularity can be avoided by using freely falling coordinates, such as those of Gullstrand-Painleve, in which the frame is
$$
\f{e} = \f{dt} \ga_0 + \f{dt} \sqrt{\fr{R_S}{r}} \ga_1 + \f{dr} \fr{1}{c} \ga_1 + \f{d\th} \fr{r}{c} \ga_2 + \f{d\ph} \fr{r \sin{\th}}{c} \ga_3
$$
The angular [[spherical coordinates]], $\th$ and $\ph$, range from $0$ to $\pi$ and from $0$ to $2\pi$.  The radial coordinate, $r$, is scaled so the area of the 2D surface at $r=R$ is
$$
A = c^2 \int_{r=R} \f{e^2} \f{e^3} = \int \f{d\th}\f{d\ph} R^2 \sin{\th}=4\pi R^2
$$
for any time, $t$. The coframe is
$$
\ve{e} = \ga^0 \ve{\pa_t} - \ga^0 c \sqrt{\fr{R_S}{r}} \ve{\pa_r} + \ga^1 c \ve{\pa_r} + \ga^2 \fr{c}{r} \ve{\pa_\th} + \ga^3 \fr{c}{r \sin(\th)} \ve{\pa_\ph}
$$
The [[torsion]]less [[spin connection]], found by solving [[Cartan's equation]], $0=\f{d} \f{e} + \f{\om} \times \f{e}$, is
\begin{eqnarray}
\f{\om} &=& - \ve{e} \times \f{d} \f{e} + \fr{1}{4} \lp \ve{e} \times \ve{e} \rp \lp \f{e} \cdot \f{d} \f{e} \rp \\
&=& - \f{d t} \fr{c R_S}{2 r^2} \ga_{01} - \f{d r} \fr{1}{2 r} \sqrt{\fr{R_S}{r}} \ga_{01}
+ \f{d \th} \sqrt{\fr{R_S}{r}} \ga_{02} + \f{d \th} \ga_{12}
+ \f{d \ph} \sqrt{\fr{R_S}{r}} \sin(\th) \ga_{03} + \f{d \ph} \sin(\th) \ga_{13} + \f{d \ph} \cos(\th) \ga_{23} 
\end{eqnarray}
The [[Clifford-Riemann curvature]] is
\begin{eqnarray}
\ff{R} &=& \f{d} \f{\om} + \ha \f{\om} \f{\om} \\
&=& - \f{d t} \f{d r} \fr{c R_S}{r^3} \ga_{01} 
+ \f{d t} \f{d \th} \fr{c R_S}{2 r^2} \ga_{02}
+ \f{d t} \f{d \th} \fr{c R_S}{2r^2} \sqrt{\fr{R_S}{r}} \ga_{12}
+ \f{d t} \f{d \ph} \fr{c R_S \sin{\th}}{2r^2} \ga_{03} \\
&+& \f{d t} \f{d \ph} \fr{c R_S \sin{\th}}{2r^2} \sqrt{\fr{R_S}{r}} \ga_{13}
+ \f{d r} \f{d \th} \fr{R_S}{2r^2} \ga_{12}
+ \f{d r} \f{d \ph} \fr{R_S \sin{\th}}{2r^2} \ga_{13}
- \f{d \th} \f{d \ph} \fr{R_S \sin{\th}}{r} \ga_{23}
\end{eqnarray}
The [[Clifford-Ricci curvature]] is
\begin{eqnarray}
\f{R} &=& \ve{e} \times \ff{R} = 0
\end{eqnarray}
showing that the Schwarzschild solution satisfies the vacuum [[Einstein's equation]] away from the curvature singularity at $r=0$.

Ref:
*http://en.wikipedia.org/wiki/Gullstrand%E2%80%93Painlev%C3%A9_coordinates
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<table class="gtable">
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</td></tr><tr><td>
<SPAN class="math"></SPAN>
</td></tr>
</table>
</center></html>
!@@display:block;text-align:center;Open Source Science@@

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@@display:block;text-align:center;Cameron Neylon@@

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<<search>><<newTiddler>><<permaview>><<collapseAll>><<closeAll>><<saveChanges>><<slider chkSliderOptionsPanel OptionsPanel '>' 'More options'>>
<<tabs txtMainTab Contents 'Hierarchy of tags and content' TabContents Latest 'Recently modified tiddlers' TabTimeline Tags 'List all tags' TabTags All 'List all tiddlers' TabAll>>

Deferential Geometry
http://deferentialgeometry.org
$$
\begin{array}{rcl}
\udf{A} \!\!&\!\!=\!\!&\!\!  \f{H}{}_1 + \f{H}{}_2 + \ud{\Ps} = {\small \frac{1}{2}} \f{\om} + {\small \frac{1}{4}} \f{e} \ph + \f{W} + \f{B}{}_1 + \f{w} + \f{B}{}_2 + \f{x} \Ph + \f{g} + \ud{\nu^e} + \ud{e} + \ud{u} + \ud{d}
\; \in \; \udf{\rm Lie}(E8) = \udf{e8}
\\[.5em]

\!\!&\!\!=\!\!&\!\!
{\small
\lb \begin{array}{cccccccc}
\frac{1}{2} \f{\om}{}_L \!+\! i \f{W}{}^3 \!&\! i \f{W}{}^1 \!+\! \f{W}{}^2 \!&\! - \! \frac{1}{4} \f{e}{}_R \ph_1 \!&\! \frac{1}{4} \f{e}{}_R \ph_+ \!&
\; \ud{\nu}{}_L &\!\! \ud{u}{}_L^r \!\!&\!\! \ud{u}{}_L^g \!\!&\!\! \ud{u}{}_L^b \\

i \f{W}{}^1 \!-\! \f{W}{}^2 \!&\! \frac{1}{2} \f{\om}{}_L \!-\! i \f{W}{}^3 \!&\! \p{-} \frac{1}{4} \f{e}{}_R \ph_- \!&\! \frac{1}{4} \f{e}{}_R \ph_0 \!&
\; \ud{e}{}_L &\!\! \ud{d}{}_L^r \!\!&\!\! \ud{d}{}_L^g \!\!&\!\! \ud{d}{}_L^b \\

-\frac{1}{4} \f{e}{}_L \ph_0 & \frac{1}{4} \f{e}{}_L \ph_+ & \! \frac{1}{2} \f{\om}{}_R \!+\! i \f{B}{}_1^3 \! \!&\! i \f{B}{}_1^1 \!+\! \f{B}{}_1^2 \!&
\; \ud{\nu}{}_R &\!\! \ud{u}{}_R^r \!\!&\!\! \ud{u}{}_R^g \!\!&\!\! \ud{u}{}_R^b \\

\p{-}\frac{1}{4} \f{e}{}_L \ph_- & \frac{1}{4} \f{e}{}_L \ph_1 &\! i \f{B}{}_1^1 \!-\! \f{B}{}_1^2 \!&\! \! \frac{1}{2} \f{\om}{}_R \!-\! i \f{B}{}_1^3 \! &
\; \ud{e}{}_R &\!\! \ud{d}{}_R^r \!\!&\!\! \ud{d}{}_R^g \!\!&\!\! \ud{d}{}_R^b \\

& & & & \; i \f{B}{}_2 &\!\! \!\!&\!\! \!\!&\!\!  \\
& & & &  &\!\!\! \frac{-i}{3} \! \f{B}{}_2 \!+\! i \f{g}{}^{3+8} \!\!\!&\!\!\! i\f{g}{}^1 \!-\! \f{g}{}^2 \!\!\!&\!\!\! i\f{g}{}^4 \!-\! \f{g}{}^5 \\
& & & &  &\!\!\! i\f{g}{}^1 \!+\! \f{g}{}^2 \!\!\!&\!\!\! \frac{-i}{3} \! \f{B}{}_2 \!-\! i \f{g}{}^{3+8} \!\!\!&\!\!\! i\f{g}{}^6 \!-\! \f{g}{}^7 \\
& & & &  &\!\!\! i\f{g}{}^4 \!+\! \f{g}{}^5 \!\!\!&\!\!\! i\f{g}{}^6 \!+\! \f{g}{}^7 \!\!\!&\!\!\! \frac{-i}{3} \! \f{B}{}_2 \!-\!\! \frac{2i}{\sqrt{3}}\f{g}{}^8
\end{array} \rb
}
\\[1.5em]
\udff{F} \!\!&\!\!=\!\!&\!\! \f{d} \udf{A} + \udf{A} \udf{A} = 
( \f{d} \f{H}{}_1 + \f{H}{}_1 \f{H}{}_1 ) + ( \f{d} \f{H}{}_2 + \f{H}{}_2 \f{H}{}_2 ) + ( \f{d} \ud{\Ps} + \f{H}{}_1 \ud{\Ps} - \ud{\Ps} \f{H}{}_2 )  \; \in \; \udff{e8} \\[.5em]

\!\!&\!\!=\!\!&\!\!
\ha \big( (\f{d} \f{\om} + \ha \f{\om} \f{\om}) - \fr{1}{8} \f{e} \f{e} \ph^2 \big)
+ \fr{1}{4} \big( ( \f{d} \f{e} \!+\! \ha [ \f{\om}, \f{e} ] ) \ph - \f{e} ( \f{d} \ph \!+\! [ \f{W} \!+\! \f{B}{}_1, \ph ] ) \big)
+ (\f{d} \f{W} + \f{W} \f{W})
\\
&&
\!\!+\, (\f{d} \f{B}{}_1 + \f{B}{}_1 \f{B}{}_1) + \f{d} \f{w} + \f{d} \f{B}{}_2 + \f{x}\Ph\f{x}\Ph
+ \big( ( \f{d} \f{x} \!+\! [ \f{w} \!+\! \f{B}{}_2, \! \f{x} ] ) \Ph \!-\! \f{x} ( \f{d} \Ph \!+\! [ \f{g}, \! \Ph ] ) \big)
+ (\f{d} \f{g} + \f{g} \f{g}) \\
&&
\!\!+\, \big( ( \f{d} + {\scriptsize \frac{1}{2}} \f{\om} + {\scriptsize \frac{1}{4}} \f{e}\ph ) \ud{\Ps}
+ \f{W} \ud{\Ps}{}_L + \f{B}{}_1 \ud{\Ps}{}_R - \ud{\Ps} ( \f{w} + \f{B}{}_2 + \f{x} \Ph ) - \ud{\Ps}{}_q \, \f{g} \big) \\[.5em]

\!\!&\!\!=\!\!&\!\!
\ha \big( \ff{R} - \fr{1}{8} \f{e} \f{e} \ph^2 \big)
+ \fr{1}{4} \big( \ff{T} \ph - \f{e} \f{D} \ph \big)
+ \ff{F}{}_W + \ff{F}{}_{B_1} 
+ \ff{F}{}_{w} + \ff{F}{}_{B_2} + \f{x}\Ph\f{x}\Ph
+ \big( (\f{D} \f{x}) \Ph - \f{x} \f{D} \Ph \big)
+ \ff{F}{}_{g}
+ \f{D} \ud{\Psi}
\end{array}
$$
$$
S \,= \int \big< \ff{\od{B}} \udff{F}
+ {\scriptsize \frac{\pi G}{4}} \ff{B}{}_G \ff{B}{}_G \ga - \ff{B'} \ff{*B'} \big>
= \int \big< \fff{\od{B}} \f{D} \ud{\Ps}
+ \nf{e} {\scriptsize \frac{1}{16 \pi G}} \ph^2  \big( R - \fr{3}{2} \ph^2 \big) - \fr{1}{4} \ff{F'} \ff{*F'} \big>
\vp{{\Big(}_{\Big(}^{\Big(}}
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
$$
$$
Z = \int D A \, e^{\fr{i}{\hbar} S[A]} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; p[A] = \frac{1}{Z} \, e^{\fr{i}{\hbar} S[A]} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
$$
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/RW12/images/spin.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math">spin(1,3) \,\,\oplus\,\, 4_S^\mathbb{C} \,\,\oplus\,\, 4_V</SPAN>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/Zuck09/images/Spin.png" width="500" height="500">
</td></tr><tr><td>
<SPAN class="math">spin(1,3) \,\,\oplus\,\, 4_S^\mathbb{C} \,\,\oplus\,\, 4_V</SPAN>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/RW12/images/Spin(10).png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math">spin(10) \,\,\oplus\,\, 16^\mathbb{C}_{S+}</SPAN>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/Zuck09/images/GraviGUTD7.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math">spin(11,3) \,\oplus\, 64^\mathbb{R}_{S+}</SPAN>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/RW12/images/GraviGUTD7.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math">spin(11,3) \,\oplus\, 64^\mathbb{R}_{S+}</SPAN>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/Zuck09/images/GraviGUTD7.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math">spin(11,3) \,\oplus\, 64^\mathbb{R}_{S+}</SPAN>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>>$Cl^1(12,4)$ basis vector matrices, $\Ga'_x$, using the $\Ga_i$ from $Cl^1(11,3)$ :
$$
\Ga'_i = \si_1 \otimes \Ga_i \;\;\;\;\;\;\;\; \Ga'_{15} = \si_1 \otimes \Ga \;\;\;\;\;\;\;\; \Ga'_{16} = - i \, \si_2 \otimes 1
$$
$Cl^2(12,4)$ basis bivectors, $\Ga'_{xy}$ :
$$
\Ga'_{ij} = 1 \otimes \Ga_{ij} = 
\lb \begin{array}{cc}
\Ga'^+_{ij} & 0 \cr
0 & \Ga'^-_{ij} \cr
\end{array} \rb
\;\;\;\;\;\;\;\; 
\Ga'_{i \, 15} = 1 \otimes \Ga_i \Ga
\;\;\;\;\;\;\;\; 
\Ga'_{i \, 16} = \si_3 \otimes \Ga_i
\;\;\;\;\;\;\;\; 
\Ga'_{15 \, 16} = \si_3 \otimes \Ga
$$
$spin(12,4)$ positive chiral basis bivector matrices:
$$
\Ga'^+_{ij} = \Ga_{ij}
\;\;\;\;\;\;\;\;
\Ga'^+_{i \, 15} = \Ga_i \Ga
\;\;\;\;\;\;\;\; 
\Ga'^+_{i \, 16} = \Ga_i
\;\;\;\;\;\;\;\; 
\Ga'^+_{15 \, 16} = \Ga
$$
A $G = \ha G^{xy} \Ga'^+_{xy} \in spin(12,4)$ acting on a $128^{\mathbb{R}}_{S+}$ spinor, comprised of $spin(11,3)$ spinors:
$$
G \, \ps^{128}_{S+} =
\lb \begin{array}{cc}
\ha G^{ij} \Ga^+_{ij} + G^{15 \, 16} &  -G^{i \, 15} \Ga^+_i + G^{i \, 16} \Ga^+_i \cr
 G^{i \, 15} \Ga^+_ i + G^{i \, 16} \Ga^+_i & \ha G^{ij} \Ga^-_{ij} - G^{15 \, 16} \cr
\end{array} \rb
\lb \begin{array}{c}
\ps^{64}_{S+} \cr
\ps^{64}_{S-} \cr
\end{array} \rb
$$
Embedding of $spin(11,3) \,\oplus\, 64^\mathbb{R}_{S+}$ GraviGUT in $spin(12,4) \,\oplus\, 128^\mathbb{R}_{S+}$ GraviGUT:
$$
\Ga^+_{ij} = \Ga'^{++}_{ij} \;\;\;\;\;\;\;\; Q_{\io} = Q'^+_{\io}
$$
Invariant bilinear form for spinors:
$$
\ch^\dagger A \ps = (\ch, \ps) = (g \ch, g \ps) \;\; \Rightarrow \;\; (\ch, G \ps) = - (G \ch, \ps) \;\; \Leftrightarrow \;\;
\ch^\dagger A G \ps = - \ch^\dagger G^\dagger A \ps   
$$
$$
A G = - G^\dagger A  \;\; \forall \;\; G = \ha G^{xy} \Ga'^+_{xy} \;\;\; \Leftarrow \;\;\; A = (\Ga'_1 \Ga'_2 \Ga'_3 \Ga'_{16})^+ = - \si_1 \otimes 1 \otimes 1 \otimes \si_2 \otimes \si_2 \otimes 1 \otimes 1
$$
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/IfA11/images/Standard Model Q.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math">su(2)_L \,\oplus\, u(1)_Y \,\oplus\, su(3) \,\,\oplus\,\, (2_L \!\,\oplus\,\! 2_R) \!\otimes\! (1\!\,\oplus\,\!3)</SPAN>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/RW12/images/Standard Model Q.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math">su(2)_L \,\oplus\, u(1)_Y \,\oplus\, su(3) \,\,\oplus\,\, (2_L \!\,\oplus\,\! 2_R) \!\otimes\! (1\!\,\oplus\,\!3)</SPAN>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>>
\begin{equation}
-\frac{1}{2}\partial_{\nu}g^{a}_{\mu}\partial_{\nu}g^{a}_{\mu}
-g_{s}f^{abc}\partial_{\mu}g^{a}_{\nu}g^{b}_{\mu}g^{c}_{\nu}
-\frac{1}{4}g^{2}_{s}f^{abc}f^{ade}g^{b}_{\mu}g^{c}_{\nu}g^{d}_{\mu}g^{e}_{\nu}
+\frac{1}{2}ig^{2}_{s}(\bar{q}^{\sigma}_{i}\gamma^{\mu}q^{\sigma}_{j})g^{a}_{\mu}
+\bar{G}^{a}\partial^{2}G^{a}+g_{s}f^{abc}\partial_{\mu}\bar{G}^{a}G^{b}g^{c}_{\mu}
-\partial_{\nu}W^{+}_{\mu}\partial_{\nu}W^{-}_{\mu}-M^{2}W^{+}_{\mu}W^{-}_{\mu}
-\frac{1}{2}\partial_{\nu}Z^{0}_{\mu}\partial_{\nu}Z^{0}_{\mu}-\frac{1}{2c^{2}_{w}}
M^{2}Z^{0}_{\mu}Z^{0}_{\mu}
-\frac{1}{2}\partial_{\mu}A_{\nu}\partial_{\mu}A_{\nu}
-\frac{1}{2}\partial_{\mu}H\partial_{\mu}H-\frac{1}{2}m^{2}_{h}H^{2}
-\partial_{\mu}\phi^{+}\partial_{\mu}\phi^{-}-M^{2}\phi^{+}\phi^{-}
-\frac{1}{2}\partial_{\mu}\phi^{0}\partial_{\mu}\phi^{0}-\frac{1}{2c^{2}_{w}}M\phi^{0}\phi^{0}
-\beta_{h}[\frac{2M^{2}}{g^{2}}+\frac{2M}{g}H+\frac{1}{2}(H^{2}+\phi^{0}\phi^{0}+2\phi^{+}\phi^{-%%@
})]+\frac{2M^{4}}{g^{2}}\alpha_{h}
-igc_{w}[\partial_{\nu}Z^{0}_{\mu}(W^{+}_{\mu}W^{-}_{\nu}-W^{+}_{\nu}W^{-}_{\mu})
-Z^{0}_{\nu}(W^{+}_{\mu}\partial_{\nu}W^{-}_{\mu}-W^{-}_{\mu}\partial_{\nu}W^{+}_{\mu})
+Z^{0}_{\mu}(W^{+}_{\nu}\partial_{\nu}W^{-}_{\mu}-W^{-}_{\nu}\partial_{\nu}W^{+}_{\mu})]
-igs_{w}[\partial_{\nu}A_{\mu}(W^{+}_{\mu}W^{-}_{\nu}-W^{+}_{\nu}W^{-}_{\mu})
-A_{\nu}(W^{+}_{\mu}\partial_{\nu}W^{-}_{\mu}-W^{-}_{\mu}\partial_{\nu}W^{+}_{\mu})
+A_{\mu}(W^{+}_{\nu}\partial_{\nu}W^{-}_{\mu}-W^{-}_{\nu}\partial_{\nu}W^{+}_{\mu})]
-\frac{1}{2}g^{2}W^{+}_{\mu}W^{-}_{\mu}W^{+}_{\nu}W^{-}_{\nu}+\frac{1}{2}g^{2}
W^{+}_{\mu}W^{-}_{\nu}W^{+}_{\mu}W^{-}_{\nu}
+g^2c^{2}_{w}(Z^{0}_{\mu}W^{+}_{\mu}Z^{0}_{\nu}W^{-}_{\nu}-Z^{0}_{\mu}Z^{0}_{\mu}W^{+}_{\nu}
W^{-}_{\nu})
+g^2s^{2}_{w}(A_{\mu}W^{+}_{\mu}A_{\nu}W^{-}_{\nu}-A_{\mu}A_{\mu}W^{+}_{\nu}
W^{-}_{\nu})
+g^{2}s_{w}c_{w}[A_{\mu}Z^{0}_{\nu}(W^{+}_{\mu}W^{-}_{\nu}-W^{+}_{\nu}W^{-}_{\mu})-%%@
2A_{\mu}Z^{0}_{\mu}W^{+}_{\nu}W^{-}_{\nu}]
-g\alpha[H^3+H\phi^{0}\phi^{0}+2H\phi^{+}\phi^{-}]
-\frac{1}{8}g^{2}\alpha_{h}[H^4+(\phi^{0})^{4}+4(\phi^{+}\phi^{-})^{2}+4(\phi^{0})^{2}
\phi^{+}\phi^{-}+4H^{2}\phi^{+}\phi^{-}+2(\phi^{0})^{2}H^{2}]
-gMW^{+}_{\mu}W^{-}_{\mu}H-\frac{1}{2}g\frac{M}{c^{2}_{w}}Z^{0}_{\mu}Z^{0}_{\mu}H
-\frac{1}{2}ig[W^{+}_{\mu}(\phi^{0}\partial_{\mu}\phi^{-}-\phi^{-}\partial_{\mu}\phi^{0})
-W^{-}_{\mu}(\phi^{0}\partial_{\mu}\phi^{+}-\phi^{+}\partial_{\mu}\phi^{0})]
+\frac{1}{2}g[W^{+}_{\mu}(H\partial_{\mu}\phi^{-}-\phi^{-}\partial_{\mu}H)
-W^{-}_{\mu}(H\partial_{\mu}\phi^{+}-\phi^{+}\partial_{\mu}H)]
+\frac{1}{2}g\frac{1}{c_{w}}(Z^{0}_{\mu}(H\partial_{\mu}\phi^{0}-\phi^{0}\partial_{\mu}H)
-ig\frac{s^{2}_{w}}{c_{w}}MZ^{0}_{\mu}(W^{+}_{\mu}\phi^{-}-W^{-}_{\mu}\phi^{+})
+igs_{w}MA_{\mu}(W^{+}_{\mu}\phi^{-}-W^{-}_{\mu}\phi^{+})
-ig\frac{1-2c^{2}_{w}}{2c_{w}}Z^{0}_{\mu}(\phi^{+}\partial_{\mu}\phi^{-}-\phi^{-%%@
}\partial_{\mu}\phi^{+})
+igs_{w}A_{\mu}(\phi^{+}\partial_{\mu}\phi^{-}-\phi^{-}\partial_{\mu}\phi^{+})
-\frac{1}{4}g^{2}W^{+}_{\mu}W^{-}_{\mu}[H^{2}+(\phi^{0})^{2}+2\phi^{+}\phi^{-}]
-\frac{1}{4}g^{2}\frac{1}{c^{2}_{w}}Z^{0}_{\mu}Z^{0}_{\mu}[H^{2}+(\phi^{0})^{2}+2(2s^{2}_{w}-%%@
1)^{2}\phi^{+}\phi^{-}]
-\frac{1}{2}g^{2}\frac{s^{2}_{w}}{c_{w}}Z^{0}_{\mu}\phi^{0}(W^{+}_{\mu}\phi^{-}+W^{-%%@
}_{\mu}\phi^{+})
-\frac{1}{2}ig^{2}\frac{s^{2}_{w}}{c_{w}}Z^{0}_{\mu}H(W^{+}_{\mu}\phi^{-}-W^{-}_{\mu}\phi^{+})
+\frac{1}{2}g^{2}s_{w}A_{\mu}\phi^{0}(W^{+}_{\mu}\phi^{-}+W^{-}_{\mu}\phi^{+})
+\frac{1}{2}ig^{2}s_{w}A_{\mu}H(W^{+}_{\mu}\phi^{-}-W^{-}_{\mu}\phi^{+})
-g^{2}\frac{s_{w}}{c_{w}}(2c^{2}_{w}-1)Z^{0}_{\mu}A_{\mu}\phi^{+}\phi^{-}-%%@
g^{1}s^{2}_{w}A_{\mu}A_{\mu}\phi^{+}\phi^{-}
-\bar{e}^{\lambda}(\gamma\partial+m^{\lambda}_{e})e^{\lambda}
-\bar{\nu}^{\lambda}\gamma\partial\nu^{\lambda}
-\bar{u}^{\lambda}_{j}(\gamma\partial+m^{\lambda}_{u})u^{\lambda}_{j}
-\bar{d}^{\lambda}_{j}(\gamma\partial+m^{\lambda}_{d})d^{\lambda}_{j}
+igs_{w}A_{\mu}[-(\bar{e}^{\lambda}\gamma^{\mu}
e^{\lambda})+\frac{2}{3}(\bar{u}^{\lambda}_{j}\gamma^{\mu} %%@
u^{\lambda}_{j})-\frac{1}{3}(\bar{d}^{\lambda}_{j}\gamma^{\mu} 
d^{\lambda}_{j})]
+\frac{ig}{4c_{w}}Z^{0}_{\mu}
[(\bar{\nu}^{\lambda}\gamma^{\mu}(1+\gamma^{5})\nu^{\lambda})+
(\bar{e}^{\lambda}\gamma^{\mu}(4s^{2}_{w}-1-\gamma^{5})e^{\lambda})+
(\bar{u}^{\lambda}_{j}\gamma^{\mu}(\frac{4}{3}s^{2}_{w}-1-\gamma^{5})u^{\lambda}_{j})+
(\bar{d}^{\lambda}_{j}\gamma^{\mu}(1-\frac{8}{3}s^{2}_{w}-\gamma^{5})d^{\lambda}_{j})]
+\frac{ig}{2\sqrt{2}}W^{+}_{\mu}[(\bar{\nu}^{\lambda}\gamma^{\mu}(1+\gamma^{5})e^{\lambda})
+(\bar{u}^{\lambda}_{j}\gamma^{\mu}(1+\gamma^{5})C_{\lambda\kappa}d^{\kappa}_{j})]
+\frac{ig}{2\sqrt{2}}W^{-}_{\mu}[(\bar{e}^{\lambda}\gamma^{\mu}(1+\gamma^{5})\nu^{\lambda})
+(\bar{d}^{\kappa}_{j}C^{\dagger}_{\lambda\kappa}\gamma^{\mu}(1+\gamma^{5})u^{\lambda}_{j})]
+\frac{ig}{2\sqrt{2}}\frac{m^{\lambda}_{e}}{M}
[-\phi^{+}(\bar{\nu}^{\lambda}(1-\gamma^{5})e^{\lambda})
+\phi^{-}(\bar{e}^{\lambda}(1+\gamma^{5})\nu^{\lambda})]
-\frac{g}{2}\frac{m^{\lambda}_{e}}{M}[H(\bar{e}^{\lambda}e^{\lambda})
+i\phi^{0}(\bar{e}^{\lambda}\gamma^{5}e^{\lambda})]
+\frac{ig}{2M\sqrt{2}}\phi^{+}
[-m^{\kappa}_{d}(\bar{u}^{\lambda}_{j}C_{\lambda\kappa}(1-\gamma^{5})d^{\kappa}_{j})
+m^{\lambda}_{u}(\bar{u}^{\lambda}_{j}C_{\lambda\kappa}(1+\gamma^{5})d^{\kappa}_{j}]
+\frac{ig}{2M\sqrt{2}}\phi^{-}
[m^{\lambda}_{d}(\bar{d}^{\lambda}_{j}C^{\dagger}_{\lambda\kappa}(1+\gamma^{5})u^{\kappa}_{j})
-m^{\kappa}_{u}(\bar{d}^{\lambda}_{j}C^{\dagger}_{\lambda\kappa}(1-\gamma^{5})u^{\kappa}_{j}]
-\frac{g}{2}\frac{m^{\lambda}_{u}}{M}H(\bar{u}^{\lambda}_{j}u^{\lambda}_{j})
-\frac{g}{2}\frac{m^{\lambda}_{d}}{M}H(\bar{d}^{\lambda}_{j}d^{\lambda}_{j})
+\frac{ig}{2}\frac{m^{\lambda}_{u}}{M}\phi^{0}(\bar{u}^{\lambda}_{j}\gamma^{5}u^{\lambda}_{j})
-\frac{ig}{2}\frac{m^{\lambda}_{d}}{M}\phi^{0}(\bar{d}^{\lambda}_{j}\gamma^{5}d^{\lambda}_{j})
+\bar{X}^{+}(\partial^{2}-M^{2})X^{+}+\bar{X}^{-}(\partial^{2}-M^{2})X^{-}
+\bar{X}^{0}(\partial^{2}-\frac{M^{2}}{c^{2}_{w}})X^{0}+\bar{Y}\partial^{2}Y
+igc_{w}W^{+}_{\mu}(\partial_{\mu}\bar{X}^{0}X^{-}-\partial_{\mu}\bar{X}^{+}X^{0})
+igs_{w}W^{+}_{\mu}(\partial_{\mu}\bar{Y}X^{-}-\partial_{\mu}\bar{X}^{+}Y)
+igc_{w}W^{-}_{\mu}(\partial_{\mu}\bar{X}^{-}X^{0}-\partial_{\mu}\bar{X}^{0}X^{+})
+igs_{w}W^{-}_{\mu}(\partial_{\mu}\bar{X}^{-}Y-\partial_{\mu}\bar{Y}X^{+})
+igc_{w}Z^{0}_{\mu}(\partial_{\mu}\bar{X}^{+}X^{+}-\partial_{\mu}\bar{X}^{-}X^{-})
+igs_{w}A_{\mu}(\partial_{\mu}\bar{X}^{+}X^{+}-\partial_{\mu}\bar{X}^{-}X^{-})
-\frac{1}{2}gM[\bar{X}^{+}X^{+}H+\bar{X}^{-}X^{-}H+\frac{1}{c^{2}_{w}}\bar{X}^{0}X^{0}H]
+\frac{1-2c^{2}_{w}}{2c_{w}}igM[\bar{X}^{+}X^{0}\phi^{+}-\bar{X}^{-}X^{0}\phi^{-}]
+\frac{1}{2c_{w}}igM[\bar{X}^{0}X^{-}\phi^{+}-\bar{X}^{0}X^{+}\phi^{-}]
+igMs_{w}[\bar{X}^{0}X^{-}\phi^{+}-\bar{X}^{0}X^{+}\phi^{-}]
+\frac{1}{2}igM[\bar{X}^{+}X^{+}\phi^{0}-\bar{X}^{-}X^{-}\phi^{0}]
\end{equation}
<<tiddler HideTags>>$${\small \begin{array}{rcrccccccccccc}
T^\om_{23} \!\!&\!\! = \!\!&\!\! i \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
T^\om_{13} \!\!&\!\! = \!\!&\!\! i \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
T^\om_{12} \!\!&\!\! = \!\!&\!\! i \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
T^\om_{14} \!\!&\!\! = \!\!&\!\!   \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
T^\om_{24} \!\!&\!\! = \!\!&\!\! - \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
T^\om_{34} \!\!&\!\! = \!\!&\!\!   \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
\vp{A^{\Big(}} T^W_1 \!\!&\!\! = \!\!&\!\! -\fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
            \!\!&\!\!    \!\!&\!\! -\fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
T^W_2 \!\!&\!\! = \!\!&\!\! \fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
            \!\!&\!\!    \!\!&\!\! \fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
T^W_3 \!\!&\!\! = \!\!&\!\! -\fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
            \!\!&\!\!    \!\!&\!\! -\fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\vp{A^{\Big(}}  T^Y \!\!&\!\! = \!\!&\!\! \fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
           \!\!&\!\!    \!\!&\!\! -\fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
           \!\!&\!\!    \!\!&\!\! +\fr{i}{6} \!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
           \!\!&\!\!    \!\!&\!\! +\fr{i}{6} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
           \!\!&\!\!    \!\!&\!\! +\fr{i}{6} \!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\end{array}
\;\;\;\;\;\;\;\;\;\;\;
\begin{array}{rcrccccccccccc}
\vp{A^{\Big(}} T^g_1 \!\!&\!\! = \!\!&\!\! -\fr{i}{4} \!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
      \!\!&\!\!    \!\!&\!\! -\fr{i}{4} \!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
T^g_2 \!\!&\!\! = \!\!&\!\! -\fr{i}{4} \!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
      \!\!&\!\!    \!\!&\!\! +\fr{i}{4} \!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
T^g_3 \!\!&\!\! = \!\!&\!\! -\fr{i}{4} \!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
      \!\!&\!\!    \!\!&\!\! +\fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
T^g_4 \!\!&\!\! = \!\!&\!\! \fr{i}{4} \!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
           \!\!&\!\!    \!\!&\!\! -\fr{i}{4} \!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
T^g_5 \!\!&\!\! = \!\!&\!\! \fr{i}{4} \!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
           \!\!&\!\!    \!\!&\!\! -\fr{i}{4} \!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
T^g_6 \!\!&\!\! = \!\!&\!\! -\fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
           \!\!&\!\!    \!\!&\!\! +\fr{i}{4} \!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
T^g_7 \!\!&\!\! = \!\!&\!\! -\fr{i}{4} \!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
           \!\!&\!\!    \!\!&\!\! +\fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
T^g_8 \!\!&\!\! = \!\!&\!\! -\fr{i}{4 \sqrt{3}} \!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
           \!\!&\!\!    \!\!&\!\! -\fr{i}{4 \sqrt{3}} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
           \!\!&\!\!    \!\!&\!\! +\fr{i}{2 \sqrt{3}} \!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2
\end{array}
}
\;\;\;\;\;\;\;\;\;
\ps=
\lb \!\!
\begin{array}{c}
\nu \cr e \cr u^r \cr d^r \cr u^g \cr d^g \cr u^b \cr d^b \cr 
\end{array}
\!\! \rb
\begin{array}{c} 
\left\{
\;
\lb \!\!
\begin{array}{c}
e_{Lr}^{\wedge} \cr e_{Li}^{\wedge} \cr e_{Lr}^{\vee} \cr e_{Li}^{\vee} \cr
e_{Rr}^{\wedge} \cr e_{Ri}^{\wedge} \cr e_{Rr}^{\vee} \cr e_{Ri}^{\vee} \\
\end{array}
\!\! \rb
\right.
\\[6.5em]
\end{array}
$$
<<tiddler HideTags>>$${\small \begin{array}{rcrccccccccccc}
T^\om_{23} \!\!&\!\! = \!\!&\!\! i \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
T^\om_{13} \!\!&\!\! = \!\!&\!\! i \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
T^\om_{12} \!\!&\!\! = \!\!&\!\! i \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
T^\om_{14} \!\!&\!\! = \!\!&\!\!   \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
T^\om_{24} \!\!&\!\! = \!\!&\!\! - \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
T^\om_{34} \!\!&\!\! = \!\!&\!\!   \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
\vp{A^{\Big(}} T^W_1 \!\!&\!\! = \!\!&\!\! -\fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
            \!\!&\!\!    \!\!&\!\! -\fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
T^W_2 \!\!&\!\! = \!\!&\!\! \fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
            \!\!&\!\!    \!\!&\!\! \fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
T^W_3 \!\!&\!\! = \!\!&\!\! -\fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
            \!\!&\!\!    \!\!&\!\! -\fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\vp{A^{\Big(}}  T^Y \!\!&\!\! = \!\!&\!\! \fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
           \!\!&\!\!    \!\!&\!\! -\fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
           \!\!&\!\!    \!\!&\!\! +\fr{i}{6} \!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
           \!\!&\!\!    \!\!&\!\! +\fr{i}{6} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
           \!\!&\!\!    \!\!&\!\! +\fr{i}{6} \!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\end{array}
\;\;\;\;\;\;\;\;\;\;\;
\begin{array}{rcrccccccccccc}
\vp{A^{\Big(}} T^g_1 \!\!&\!\! = \!\!&\!\! -\fr{i}{4} \!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
      \!\!&\!\!    \!\!&\!\! -\fr{i}{4} \!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
T^g_2 \!\!&\!\! = \!\!&\!\! -\fr{i}{4} \!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
      \!\!&\!\!    \!\!&\!\! +\fr{i}{4} \!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
T^g_3 \!\!&\!\! = \!\!&\!\! -\fr{i}{4} \!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
      \!\!&\!\!    \!\!&\!\! +\fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
T^g_4 \!\!&\!\! = \!\!&\!\! \fr{i}{4} \!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
           \!\!&\!\!    \!\!&\!\! -\fr{i}{4} \!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
T^g_5 \!\!&\!\! = \!\!&\!\! \fr{i}{4} \!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
           \!\!&\!\!    \!\!&\!\! -\fr{i}{4} \!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
T^g_6 \!\!&\!\! = \!\!&\!\! -\fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
           \!\!&\!\!    \!\!&\!\! +\fr{i}{4} \!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
T^g_7 \!\!&\!\! = \!\!&\!\! -\fr{i}{4} \!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
           \!\!&\!\!    \!\!&\!\! +\fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
T^g_8 \!\!&\!\! = \!\!&\!\! -\fr{i}{4 \sqrt{3}} \!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
           \!\!&\!\!    \!\!&\!\! -\fr{i}{4 \sqrt{3}} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
           \!\!&\!\!    \!\!&\!\! +\fr{i}{2 \sqrt{3}} \!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2
\end{array}
}
\;\;\;\;\;\;\;\;\;
\ps=
\lb \!\!
\begin{array}{c}
\nu \cr e \cr u^r \cr d^r \cr u^g \cr d^g \cr u^b \cr d^b \cr 
\end{array}
\!\! \rb
\begin{array}{c} 
\left\{
\;
\lb \!\!
\begin{array}{c}
e_{Lr}^{\wedge} \cr e_{Li}^{\wedge} \cr e_{Lr}^{\vee} \cr e_{Li}^{\vee} \cr
e_{Rr}^{\wedge} \cr e_{Ri}^{\wedge} \cr e_{Rr}^{\vee} \cr e_{Ri}^{\vee} \\
\end{array}
\!\! \rb
\right.
\\[6.5em]
\end{array}
$$
<html>
<center>
<table class="gtable">

<tr>
<td COLSPAN="3">
<SPAN class="math">su(3) + su(2)_L + u(1)_Y \,+\, (1\!+\!3) \!\times\! 2_L + (1\!+\!3) \!\times\! (1\!+\!1)_R + \bar{"}</SPAN>
</td>
</tr>

<tr>
<td COLSPAN="3">
<img SRC="talks/CSUF09/images/SMElectric.png">
</td>
</tr>

</table>
</center>
</html>
<<tiddler HideTags>>
<html>
<center>
<table class="gtable">

<tr>
<td COLSPAN="3">
<SPAN class="math">\big( so(1,3) + su(2)_L + u(1)_R + 4 \!\times\! (2\!+\!2) + u(1)_B + su(3) \big) + 8 \!\times\! 8</SPAN>
</td>
</tr>

<tr>
<td>
<img SRC="talks/CSUF09/images/SMGPSsG.png">
</td>
</tr>

</table>
</center>
</html>
<<tiddler HideTags>>
<html>
<center>
<table class="gtable">

<tr>
<td COLSPAN="3">
<SPAN class="math">E8 = \big( so(1,7) + so(7,1) \big) + 8_+ \!\times\! 8_+ + 8_v \!\times\! 8_v + 8_- \!\times\! 8_-</SPAN>
</td>
</tr>

<tr>
<td>
<img SRC="talks/CSUF09/images/E8PSsG.png">
</td>
</tr>

</table>
</center>
</html>
<<tiddler HideTags>>
<<tiddler HideTags>>$$
\udf{A}  = \f{H} + \f{G} + \ud{\ps}
=
{\small
\lb \begin{array}{cc}
\f{H^+} & \ud{\ps}^- \\
& \f{G^-}
\end{array}  \rb
}
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \in \;\; \f{so}(1,7) + \f{so}(8) + \ud{\mathbb{C}}(8 \times 8)
$$
$$
{\small
\!\! = \!\! \lb \begin{array}{cccccccc}
\frac{1}{2} \f{\om_L} \!+\! i \f{W^3} \!&\! i \f{W^1} \!+\! \f{W^2} \!&\! - \! \frac{1}{4} \f{e_R} \ph_0^* \!&\! \frac{1}{4} \f{e_R} \ph_+ \!&
\; \ud{\nu}{}_L &\!\! \ud{u}{}_L^r \!\!&\!\! \ud{u}{}_L^g \!\!&\!\! \ud{u}{}_L^b \\

i \f{W^1} \!-\! \f{W^2} \!&\! \frac{1}{2} \f{\om_L} \!-\! i \f{W^3} \!&\! \p{-} \frac{1}{4} \f{e_R} \ph_+^* \!&\! \frac{1}{4} \f{e_R} \ph_0 \!&
\; \ud{e}{}_L &\!\! \ud{d}{}_L^r \!\!&\!\! \ud{d}{}_L^g \!\!&\!\! \ud{d}{}_L^b \\

-\frac{1}{4} \f{e_L} \ph_0 & \frac{1}{4} \f{e_L} \ph_+ & \! \frac{1}{2} \f{\om_R} \!+\! i \f{B} \! \!& &
\; \ud{\nu}{}_R &\!\! \ud{u}{}_R^r \!\!&\!\! \ud{u}{}_R^g \!\!&\!\! \ud{u}{}_R^b \\

\p{-}\frac{1}{4} \f{e_L} \ph_+^* & \frac{1}{4} \f{e_L} \ph_0^* & &\! \! \frac{1}{2} \f{\om_R} \!-\! i \f{B} \! &
\; \ud{e}{}_R &\!\! \ud{d}{}_R^r \!\!&\!\! \ud{d}{}_R^g \!\!&\!\! \ud{d}{}_R^b \\

& & & & \; i \f{B} &\!\! \!\!&\!\! \!\!&\!\!  \\
& & & &  &\!\!\! \frac{-i}{3} \! \f{B} \!+\! i \f{G^{3+8}} \!\!\!&\!\!\! i\f{G^1} \!-\! \f{G^2} \!\!\!&\!\!\! i\f{G^4} \!-\! \f{G^5} \\
& & & &  &\!\!\! i\f{G^1} \!+\! \f{G^2} \!\!\!&\!\!\! \frac{-i}{3} \! \f{B} \!-\! i \f{G^{3+8}} \!\!\!&\!\!\! i\f{G^6} \!-\! \f{G^7} \\
& & & &  &\!\!\! i\f{G^4} \!+\! \f{G^5} \!\!\!&\!\!\! i\f{G^6} \!+\! \f{G^7} \!\!\!&\!\!\! \frac{-i}{3} \! \f{B} \!-\!\! \frac{2i}{\sqrt{3}}\f{G^8}
\end{array} \rb
}
$$
Correct interactions and charges from [[curvature]]:
$$\begin{array}{rcl}
\udff{F} \!\!&\!\!=\!\!&\!\! \f{d} \udf{A} + \udf{A} \udf{A} \\
\!\!&\!\!=\!\!&\!\! ( \f{d} \f{H} + \f{H} \f{H} ) + ( \f{d} \f{G} + \f{G} \f{G} ) + ( \f{d} \ud{\ps} + \f{H} \ud{\ps} + \ud{\ps} \f{G} )
\end{array}$$
<html>
<center>
<table class="gtable">
<tr border=none>

<td border=none>
<table class="ptable">
<tr>
<th COLSPAN="2"><SPAN class="math">G2</SPAN></th>
<th></th>
<th><SPAN class="math">V_\be</SPAN></th>
<th></th>
<th><SPAN class="math">g^3</SPAN></th>
<th><SPAN class="math">g^8</SPAN></th>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{r\bar{g}}</SPAN></td>
<td></td>
<td><SPAN class="math">(T_2 - i T_1)</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{r}g}</SPAN></td>
<td></td>
<td><SPAN class="math">(-T_2 - i T_1)</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{r\bar{b}}</SPAN></td>
<td></td>
<td><SPAN class="math">(T_5-i T_4)</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">\fr{\sqrt{3}}{2}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{r}b}</SPAN></td>
<td></td>
<td><SPAN class="math">(-T_5-i T_4)</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{\sqrt{3}}{2}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{g}b}</SPAN></td>
<td></td>
<td><SPAN class="math">(-T_7-i T_6)</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">-\fr{\sqrt{3}}{2}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{g\bar{b}}</SPAN></td>
<td></td>
<td><SPAN class="math">(T_7-i T_6)</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{\sqrt{3}}{2}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#D90000} </SPAN></td>
<td><SPAN class="math">q^r</SPAN></td>
<td></td>
<td><SPAN class="math">[1,0,0]</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">\smash{\fr{1}{2\sqrt{3}}}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#00BF00} </SPAN></td>
<td><SPAN class="math">q^g</SPAN></td>
<td></td>
<td><SPAN class="math">[0,1,0]</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{\fr{1}{2\sqrt{3}}}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#0000F7} </SPAN></td>
<td><SPAN class="math">q^b</SPAN></td>
<td></td>
<td><SPAN class="math">[0,0,1]</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{\sqrt{3}}}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\butr{#D90000} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^r</SPAN></td>
<td></td>
<td><SPAN class="math">[1,0,0]</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{2\sqrt{3}}}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\butr{#00BF00} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^g</SPAN></td>
<td></td>
<td><SPAN class="math">[0,1,0]</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{2\sqrt{3}}}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\butr{#0000F7} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^b</SPAN></td>
<td></td>
<td><SPAN class="math">[0,0,1]</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">{\fr{1}{\sqrt{3}}}</SPAN></td>
</tr>
</table>
</td>

<td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>

<td>
<embed src="images/svg/g2.svg" type="image/svg+xml" width="450px" height="450px" />
<br><br>
<img SRC="images/png/dynkin g2.png">
</td>

</tr>
</table>
</center>
</html>
<<tiddler HideTags>>
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/IfA11/images/Strong interaction.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math">su(3) \,\,\oplus\,\, 3 \,\,\oplus\,\, \bar{3} \vp{{\big(}^{\big(}}</SPAN>
</td></tr>
</table>
</center></html>
<html>
<center>
<table class="gtable">

<tr>
<td COLSPAN="3">
<SPAN class="math">su(3) + 3 + \bar{3}</SPAN>
</td>
</tr>

<tr>
<td>
<SPAN class="math">g^{r \bar{g}}+q^g \to q^r</SPAN>
<br><br>
<img SRC="images/png/quark gluon vertex.png" height=160px>
<br><br>
<SPAN class="math">T_{g^{r \bar{g}}} \ps_{q^g} = \ps_{q^r}</SPAN>
<br><br>
<SPAN class="math">\al_{g^{r \bar{g}}} + \al_{q^g} = \al_{q^r}</SPAN>
</td>
<td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>
<td>
<img SRC="talks/CSUF09/images/InteractionG2.png">
</td>
</tr>

</table>
</center>
</html>
<<tiddler HideTags>>
<<tiddler HideTags>>$$
\begin{array}{rclclc}
\f{\om} \!\!&\!\!=\!\!&\!\! \f{dx^k} \om_k^{\p{k}\mu\nu}  \ha \ga_{\mu\nu} \!&\!\! \in \!\!&\! \f{spin}(3,1) = \f{Cl}^2(3,1)
&

\f{e} = \f{dx^k} (e_k)^\mu \ga_\mu \, \in \, \f{Cl}^1(3,1) \vp{|_{(}} \\

\f{W} \!\!&\!\!=\!\!&\!\! \f{dx^k} W_k^{\p{i}\pi} \fr{i}{2} \si_\pi  \!&\!\! \in \!\!&\! \f{su}(2)_L
\qquad
\frac{i}{2}
\!
\lb
\matrix{
\f{W}^3 & \!\!\! \f{W}^1 \!\!-\! i \f{W}^2 \! \\
\! \f{W}^1 \!\!+\! i \f{W}^2 \!\!\! & -\f{W}^3
}
\rb

&
\quad
\lb \matrix{
\ph_+ \\ \ph_0
} \rb
\qquad
\lb \matrix{
u_L \\ d_L
} \rb
\\

\f{B} \!\!&\!\!=\!\!&\!\! \f{dx^k} B_k i \!&\!\! \in \!\!&\! \f{u}(1)_Y
&
\quad
 \\

\f{g} \!\!&\!\!=\!\!&\!\! \f{dx^k} g_k^{\p{k}A} \fr{i}{2} \la_A  \!&\!\! \in \!\!&\! \f{su}(3)
&
\quad
\lb u^r, u^g, u^b \rb \vp{|^{(^(}_{(}}
\end{array}
\begin{array}{c}
\quad
\lb \matrix{
u_L^\wedge \\ u_L^\vee \\ u_R^\wedge \\ u_R^\vee
} \rb
\; \\
\; \\
\end{array}
$$
<html><center>
<img src="talks/Zuck09/images/Periodic table.png" width="440" height="340">
</center></html>
<<tiddler HideTags>>$$
\begin{array}{lrclcl}
\mbox{grav:} \!\!&\!\! \f{\om} \!\!&\!\!=\!\!&\!\! \f{dx^\mu} \om_\mu^{\p{\mu}ab}  \ha \ga_{ab} \!\!&\!\! \in \!\!\!&\!\! spin(1,3) \\[.25em]

\mbox{weak:} \!\!&\!\! \f{W} \!\!&\!\!=\!\!&\!\! \f{dx^\mu} W_\mu^{\p{\mu}I} \fr{i}{2} \si_I  \!\!&\!\! \in \!\!\!&\!\! su(2)_L \\[.25em]

\mbox{hyper:} \!\!&\!\! \f{B} \!\!&\!\!=\!\!&\!\! \f{dx^\mu} B^Y_\mu \,\, i \!\!&\!\! \in \!\!\!&\!\! u(1)_Y \\[.25em]

\mbox{strong:} \!\!&\!\! \f{g} \!\!&\!\!=\!\!&\!\! \f{dx^\mu} g_\mu^{\p{\mu}A} \fr{i}{2} \la_A  \!\!&\!\! \in \!\!\!&\!\! su(3)
\end{array}
\;\;\;
\begin{array}{ll}
\mbox{frame:} & \f{e} = \f{dx^\mu} (e_\mu)^a \ga_a \, \in 4 \\[.5em]

\mbox{Higgs:} & \ph = 
\lb \matrix{
\ph_+ \\ \ph_0
} \rb
\,
\in 2^\mathbb{C} \\

\p{A} & \p{B{{\Big(}^(}}

\end{array}
\;\;\;
\begin{array}{c}
\mbox{fermions:} \\

\lb \matrix{
u_L^\wedge \\ u_L^\vee \\ u_R^\wedge \\ u_R^\vee
} \rb
,
\lb \matrix{
u_L \\ d_L
} \rb
,
\lb \matrix{
u^r \\ u^g \\ u^b
} \rb
\\

\in 4{}_S^\mathbb{C},\vp{A^{\big(}} \;\;\;\;\;\; 2^\mathbb{C}, \;\;\;\;\;\; 
3^\mathbb{C}
\end{array}
$$
<html><center>
<img src="talks/Zuck09/images/Periodic table.png" width="414" height="320">
</center></html>
<<tiddler HideTags>>$$
\begin{array}{lrclcl}
\mbox{grav:} \!\!&\!\! \f{\om} \!\!&\!\!=\!\!&\!\! \f{dx^\mu} \om_\mu^{\p{\mu}ab}  \ha \ga_{ab} \!\!&\!\! \in \!\!\!&\!\! spin(1,3) \\[.25em]

\mbox{weak:} \!\!&\!\! \f{W} \!\!&\!\!=\!\!&\!\! \f{dx^\mu} W_\mu^{\p{i}B} \fr{i}{2} \si_B  \!\!&\!\! \in \!\!\!&\!\! su(2)_L \\[.25em]

\mbox{hyper:} \!\!&\!\! \f{B} \!\!&\!\!=\!\!&\!\! \f{dx^\mu} B_\mu i \!\!&\!\! \in \!\!\!&\!\! u(1)_Y \\[.25em]

\mbox{strong:} \!\!&\!\! \f{g} \!\!&\!\!=\!\!&\!\! \f{dx^\mu} g_\mu^{\p{k}A} \fr{i}{2} \la_A  \!\!&\!\! \in \!\!\!&\!\! su(3)
\end{array}
\;\;\;
\begin{array}{ll}
\mbox{frame:} & \f{e} = \f{dx^\mu} (e_\mu)^a \ga_a \, \in 4 \\[.5em]

\mbox{Higgs:} & \ph = 
\lb \matrix{
\ph_+ \\ \ph_0
} \rb
\,
\in 2^\mathbb{C} \\

\p{A} & \p{B{{\Big(}^(}}

\end{array}
\;\;\;
\begin{array}{c}
\mbox{fermions:} \\

\lb \matrix{
u_L^\wedge \\ u_L^\vee \\ u_R^\wedge \\ u_R^\vee
} \rb
,
\lb \matrix{
u_L \\ d_L
} \rb
,
\lb \matrix{
u^r \\ u^g \\ u^b
} \rb
\\

\in 4{}_S^\mathbb{C},\vp{A^{\big(}} \;\;\;\;\;\; 2^\mathbb{C}, \;\;\;\;\;\; 
3^\mathbb{C}
\end{array}
$$
<html><center>
<img src="images/png/standard model and gravity 3.png" width="414" height="320">
</center></html>
<<tiddler HideTags>>$$
S_\ps = \int \nf{e} \bar{\ps} \ve{e} \f{D} \ps =
\int \nf{d^4x} |e| \left\{ \bar{\ps} \ga^a \lp e_a \rp^\mu \big(
\pa_\mu 
+ {\tiny \frac{1}{4}} \om_\mu^{\p{\mu}bc} \ga_{bc}
+ W_\mu^{\p{\mu}\pi} T^W_\pi
+ B_\mu T^Y
+ g_\mu^{\p{\mu}A} T^g_A
\big) \ud{\ps}
+ \bar{\ps} \ph \ud{\ps} \right\}
$$ $$
\begin{array}{lrclcl}
\mbox{grav:} \!\!&\!\!\! \f{\om} \!\!&\!\!\!=\!\!\!&\!\! \f{dx^\mu} \om_\mu^{\p{\mu}ab}  \ha \ga_{ab} \!\!&\!\! \in \!\!\!&\!\! spin(1,3) \\[.25em]

\mbox{weak:} \!\!&\!\!\! \f{W} \!\!&\!\!\!=\!\!\!&\!\! \f{dx^\mu} W_\mu^{\p{i}B} \fr{i}{2} \si_B  \!\!&\!\! \in \!\!\!&\!\! su(2)_L \\[.25em]

\mbox{hyper:} \!\!&\!\!\! \f{B} \!\!&\!\!\!=\!\!\!&\!\! \f{dx^\mu} B_\mu i \!\!&\!\! \in \!\!\!&\!\! u(1)_Y \\[.25em]

\mbox{strong:} \!\!&\!\!\! \f{g} \!\!&\!\!\!=\!\!\!&\!\! \f{dx^\mu} g_\mu^{\p{k}A} \fr{i}{2} \la_A  \!\!&\!\! \in \!\!\!&\!\! su(3)
\end{array}
\;\;\;\;\;
\begin{array}{ll}
\mbox{frame:} \!\!&\!\! \f{e} = \f{dx^\mu} (e_\mu)^a \ga_a \, \in 4 \\[.5em]

\mbox{Higgs:} \!\!&\!\! \ph = 
\lb \matrix{
\ph_+ \\ \ph_0
} \rb
\,
\in 2^\mathbb{C} \;\;\;\;\;\;\;\; \ud{\ps} = \!\!\! \\

\p{A} \!\!&\!\! \p{B{{\Big(}^(}} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \mbox{fermions:} \!\!\!\!

\end{array}

\begin{array}{c}
\p{a}
\lb \matrix{
u_L^\wedge \\ u_L^\vee \\ u_R^\wedge \\ u_R^\vee
} \rb
,
\matrix{
\lb \matrix{
u_L \\ d_L
} \rb
\\[.5em]
\lb \matrix{
\nu_{L} \\ e_L
} \rb
}
,
\lb \matrix{
u^r \\ u^g \\ u^b
} \rb
\\

\in \, 4{}_S^\mathbb{C} \;\; , \vp{A^{\big(}} \;\;\;\; 2^\mathbb{C} \;\;\; , \;\;\; 3^\mathbb{C}
\end{array}
$$
<html>
<table class="gtable">

<tr>
<td>

<table class="gtable">
<tr><td>
<img src="images/png/Wvertex.png" width="181" height="149">
</tr></td>
</table>

</td>
<td>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>
<td>
<img src="images/png/standard model and gravity 3.png" width="414" height="320">
</td>
</tr>
</table>
</html>
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!Header 1
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{{{
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A carriage return ends a paragraph, so a slightly larger (this should be made bigger) space appears between paragraphs.  But the beginning of a new paragraph is not indented, even if tabs or spaces are inserted.  So, for now, use extra carriage returns to separate paragraphs.

And maybe implement tab indentation in the future.
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="images/png/subatomic.png" height="450">
</td></tr><tr><td>
<SPAN class="math"></SPAN>
</td></tr>
</table>
</center></html>
http://arxiv.org/abs/hep-th/0610039
Super coset spaces play an important role in the formulation of supersymmetric theories. The aim of this paper is to review and discuss the geometry of super coset spaces with particular focus on the way the geometrical structures of the super coset space G/H are inherited from the super Lie group G. The isometries of the super coset space are discussed and a definition of Killing supervectors - the supervectors associated with infinitesimal isometries - is given that can be easily extended to spaces other than coset spaces.
<<tiddler HideTags>>Bosons and fermions in separate parts of one Lie algebra (or Lie superalgebra),
$$
\big( so(1,7) + so(7,1) \big) + S
$$
''Superconnection'':
\begin{eqnarray}
\udf{A} &=& \f{H} + \ud{\Ps} \\
&& \f{H} = \f{dx^i} H_i^{\p{i}A} T_A \\
&& \ud{\Ps} = \ud{\Ps^\al} T_\al
\end{eqnarray}
''Supercurvature'':
\begin{eqnarray}
\udff{F} &=& \f{d} \udf{A} + \ha \big[ \udf{A}, \udf{A} \big] \\
&=& \big( \f{d} \f{H} + \f{H} \f{H} \big) + \big( \f{d} \ud{\Ps} + \ha \big[ \f{H}, \ud{\Ps} \big] \big)
+ \ha \big[ \ud{\Ps}, \ud{\Ps} \big] \\
&=& \ff{F^H} + \f{D} \ud{\Ps} + \ud{\Ps} \ud{\Ps}
\end{eqnarray}

Geometric description of fermions: Lie algebra valued Grassmann fields.
<<tiddler HideTags>>Geometric description of fermion fibers as Lie algebra valued anticommuting fields.
<html><center><table class="gtable">
<tr>
<td><img src="talks/Zuck09/images/Twist.png" width="250" height="250"></td>
<td><SPAN class="math">\;\;\;\;\;\;\;\; \longleftrightarrow \;\;\;\;\;\;\;\;</SPAN></td>
<td><img src="talks/Zuck09/images/Circle twist.png" width="250" height="250"></td>
</tr>
</table></center></html>For a principal $G$-bundle, with $H$ a reductive subalgebra in $G=H \oplus K$, define the ''superconnection'' to be the direct sum of an $H$ connection and a $K$ valued anticommuting field:
$$
\begin{array}{rclcrclcrcl}
\udf{A} \!\!&\!\!=\!\!&\!\! \f{H} + \ud{\ps} & \;\; &  \in \!\!&\!\! G \!\!& \; & [H,H] \!\!&\!\! \subset \!\!&\!\! H \\
& &\!\! \f{H} = \f{dx^\mu} H_\mu^{\p{\mu}A} T_A \!\!& \;\; & \in \!\!&\!\! H \!\!& \;\;\;\;\;\;\;\;\;\; & [H,K] \!\!&\!\! \subset \!\!&\!\! K \\
& &\!\! \ud{\ps} =  \ud{\ps}^{\, \io} T_\io \!\!& \;\; &  \in \!\!&\!\! K \!\!& \; & [K,K] \!\!&\!\! \subset \!\!&\!\! G
\end{array}
$$
''Supercurvature'':
$$
\udff{F} = \f{d} \udf{A} + {\textstyle \ha} [\udf{A},\udf{A}]
= (\f{d} \f{H} + {\textstyle \ha} [\f{H},\f{H}] ) + (\f{d} \ud{\ps} + [\f{H},\ud{\ps}] ) + {\textstyle \ha} [\ud{\ps},\ud{\ps}]
= \ff{F}{}_H + \f{D} \ud{\ps} + \ud{\ps}\ud{\ps}
$$
<<tiddler HideTags>>

\begin{eqnarray}
\udf{A} &=& \f{H} + \ud{\ps} \;\;\;\;\;\;\;\;\; \in \; spin(11,3) \,\oplus\, 64^\mathbb{R}_{S+}\\
&& \f{H} = \f{dx^i} H_i^{\p{i}A} T_A \\
&& \ud{\ps} = \ud{\ps^\io} Q_\io
\end{eqnarray}
''Supercurvature'':
\begin{eqnarray}
\udff{F} &=& \f{d} \udf{A} + \ha \big[ \udf{A}, \udf{A} \big] \\
&=& \big( \f{d} \f{H} + \f{H} \f{H} \big) + \big( \f{d} \ud{\ps} + {\textstyle \ha} \big[ \f{H}, \ud{\ps} \big] \big)
+ {\textstyle \ha} \big[ \ud{\ps}, \ud{\ps} \big] \\
&=& \ff{F^H} + \f{D} \ud{\ps} + \ud{\ps} \ud{\ps}
\end{eqnarray}
|!Symbol|![[LaTeX]]|!Use|
| $\mathbb{R} \;\; \mathbb{C} \;\; n \;\; \ud{a}$ | {{{ \mathbb{R} \mathbb{C} n \ud{a} }}} |[[real numbers|http://en.wikipedia.org/wiki/Real_numbers]], complex numbers, dimension, [[Grassmann number]] |
| $M \; \; T_p M \; \; T_p^* M$ | {{{M T_p M T_p^* M }}} |[[manifold]], [[tangent space to M at point p|coordinate basis vectors]], [[cotangent space to M at point p|coordinate basis 1-forms]] |
| $x^i \; \; \ve{\pa_i} \; \; \ve{v} \; \; \vv{l}$ | {{{x^i \ve{\pa_i} \ve{v} \vv{l} }}} |[[coordinates|manifold]] and [[coordinate basis vectors]] with coordinate [[indices]], [[tangent vector]], [[loop|vector-form algebra]] |
| $t \; \; \ta$ | {{{ t \ta }}} |parameter time, [[proper time]] |
| $\f{dx^i} \;\; \f{a} \;\; \ff{b} \;\; \fff{c} \;\; \nf{f}$ | {{{\f{dx^i} \f{a} \ff{b} \fff{c} \nf{f} }}} |[[coordinate basis 1-forms]], [[1-form]], [[2-form|differential form]], 3-form, [[differential form]] of high or unspecified form grade |
| $\pa_i \;\; \f{\pa} \;\; \f{d}$ | {{{\pa_i \f{\pa} \f{d} }}} |[[partial derivative]], partial derivative, [[exterior derivative]] |
| $\ph \; \; \ph^* \; \; \ph_*$ | {{{\ph \ph^* \ph_* }}} |[[diffeomorphism]], [[pullback]], pushforward |
| ${\cal L}_{\ve{v}} \;\; \lb\ve{v},\ve{u}\rb_L \;\; \ve{\De}$ | {{{{\cal L}_{\ve{v}} \lb\ve{v},\ve{u}\rb_L \ve{\De} }}} |[[Lie derivative]], [[Lie bracket|Lie derivative]] of two [[vector fields|tangent bundle]], [[distribution]] |
| $\nf{\ve{A}} \;\; {\cal L}_{\nf{\ve{K}}} \;\; \lb\nf{\ve{K}},\nf{\ve{L}}\rb_L \;\; \f{\ve{P}}$ | {{{ \f{\ve{A}} {\cal L}_{ \nf{\ve{K}} } \lb\nf{\ve{K}},\nf{\ve{L}}\rb_L \f{\ve{P}} }}} |[[vector valued form]], [[FuN derivative]], FuN bracket, [[vector projection]] |
| $\f{\ve{\cal A}} \;\; \ff{\ve{\cal F}} \;\; \f{\cal D}$ | {{{ \f{\ve{\cal A}} \ff{\ve{\cal F}} \f{\cal D} }}} |[[Ehresmann connection]], [[FuN curvature]], [[Ehresmann covariant derivative]] |
| $\de_i^j \;\; \et_{\al \be} \; \; \ep_{\al \dots \be} \; \; \otimes$ | {{{ \de_i^j \et_{\al \be} \ep_{\al \dots \be} \otimes }}} |[[Kronecker delta|http://en.wikipedia.org/wiki/Kronecker_delta]], [[Minkowski metric]], [[permutation symbol]], [[Kronecker product]] |
| $G \;\; g^- \;\; T_A \;\; \lb{T_A,T_B}\rb$ | {{{G g^- T_A \lb{T_A,T_B}\rb }}} |[[Lie group]], [[inverse]] of a group element, [[Lie algebra]] generators, [[commutator]] bracket |
| $\f{\na} \;\; \f{A} \;\; \ff{F}$ | {{{\f{\na} \f{A} \ff{F} }}} |[[covariant derivative]], [[connection]], [[curvature]] |
| $\f{\cal I} \;\; \f{\ve{\cal I}} \;\; \ve{\xi^L_A} \;\; \ve{\xi^R_A}$ | {{{\f{\cal I} \f{\ve{\cal I}} \ve{\xi^L_A} \ve{\xi^R_A} }}} |[[Maurer-Cartan form]], Ehresmann-Maurer-Cartan form, [[left and right action vector fields|Lie group geometry]] |
| $Cl \; \; Cl^*$ | {{{Cl Cl^* }}} |[[Clifford algebra]], [[Clifford group]] |
| $\ga_\al \; \; \ga_{\al \dots \be} \; \; \ga$ | {{{\ga_\al \ga_{\al \dots \be} \ga }}} |[[Clifford basis vectors]], [[Clifford basis elements]], Clifford [[pseudoscalar]] |
| $\hat{A} \; \; \tilde{A} \; \; \bar{A} \; \; A^\dagger \; \; \overline{A}$ | {{{\hat{ \tilde{ \bar{ \overline{A}^\dagger } } } }}} |[[Clifford involution, reverse, conjugate, Hermitian conjugate, Dirac conjugate|Clifford conjugate]]  |
| $\cdot \; \; \times$ | {{{\cdot \times }}} |symmetric and antisymmetric [[Clifford algebra]] product  |
| $\lb{A,\dots,B}\rb_A \;\; a_{\lb{\al\dots\be}\rb}$ | {{{ \lb{A,\dots,B}\rb_A a_{\lb{\al\dots\be}\rb} }}} |[[antisymmetric bracket]], [[index bracket]] |
| $\li{A}\ri_q \; \; \li{A}\ri$ | {{{ \li{A}\ri_q \li{A}\ri }}} |[[Clifford grade]] $q$ part, [[scalar part|Clifford grade]] |
| $\f{A} \; \; \ff{b} \; \; \ve{e}$ | {{{ \f{A} \ff{b} \ve{e} }}} |[[Lieform]]s or [[Clifform]]s |
| $\lp{e_i}\rp^\al \;\; \lp{e_\al}\rp^i \;\; g_{ij} \;\; \lp\ve{u},\ve{v}\rp$ | {{{ \lp{e_i}\rp^\al \lp{e_\al}\rp^i g_{ij} \lp\ve{u},\ve{v}\rp }}} |co[[frame]] matrix, frame matrix, [[metric]], scalar product |
| $\f{e^\al} \;\; \ve{e_\al}$ | {{{ \f{e^\al} \ve{e_\al} }}} |co[[frame]] 1-forms, orthonormal basis vectors |
| $\f{e} \;\; \ve{e}$ | {{{ \f{e} \ve{e} }}} |co[[frame]], frame |
| $\nf{e} \;\; \ll{e}\rl$ | {{{ \nf{e} \ll{e}\rl }}} |[[volume form]], frame [[determinant]] |
| $\nf{*f} \;\; \ff{\vv{\ep}}$ | {{{ \nf{*f} \ff{ \vv{\ep} } }}} |[[Hodge dual]], Hodge dual projector |
| $\f{e^s} \;\; \lp{e^s_i}\rp^\al \;\; s$ | {{{ \f{e^s} \;\; \lp{e^s_i}\rp^\al \;\; s }}} |[[special frame]], special coframe matrix, conformal scalar |
| $TM \;\; T^*M$ | {{{ TM T^*M }}} |[[tangent bundle]], [[cotangent bundle]] |
| $\Ga^k{}_{ij} \;\; \f{\Ga}^k{}_j \;\; \ff{R}^k{}_j$ | {{{ \Ga^k{}_{ij} \f{\Ga}^k{}_j \ff{R}^k{}_j }}} |[[Christoffel symbols]], [[tangent bundle connection]], [[Riemann curvature]] |
| $\f{R}{}_j \;\; R$ | {{{ \f{R}{}_j R }}} |[[Ricci curvature]], [[curvature scalar]] |
| $L^\be{}_\al \;\; \f{w}^\be{}_\al \;\; \ff{F}^\be{}_\al$ | {{{ L^\be{}_\al \f{w}^\be{}_\al \ff{F}^\be{}_\al }}} |[[Lorentz rotation]], [[tangent bundle spin connection|tangent bundle connection]], [[Riemann curvature]] |
| $ ClM \;\; Cl^1M$ | {{{ ClM Cl^1M }}} |[[Clifford bundle]], [[Clifford vector bundle]] |
| $\f{A} \;\; \f{\om} \;\; \ff{R}$ | {{{ \f{A} \f{\om} \ff{R} }}} |[[Clifford connection]], [[spin connection]], [[Clifford-Riemann curvature]] |
| $\f{R} \;\; R$ | {{{ \f{R} R }}} |[[Clifford-Ricci curvature]], [[Clifford curvature scalar]] |
| $\ff{T} \;\; \f{\ka}$ | {{{ \ff{T} \f{\ka} }}} |[[torsion]], contorsion |
| $\ud{C} \;\; \nf{\od{B}} \;\; \udf{A} \;\; \udff{F}$ | {{{ \ud{C} \nf{\od{B}} \udf{A} \udff{F} }}} |[[BRST|BRST technique]] ghost, anti-ghost, extended connection, extended curvature |
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<<list all>>
*<<slider chkSliderphysicsF physicsF 'physics >' 'physics stuff, what this site is all about'>>
*<<slider chkSlidermetaF metaF 'meta >' 'describe the operation of this site'>>
<<ListTagged none>>
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{{{
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*<<tag physics>>- physics stuff, what this site is all about
**<<tag sym>>- symmetries, groups
***<<tag pb>>- principal bundles
****<<tag ss>>- homogeneous spaces
****<<tag cartan>>- Cartan geometry
***<<tag brst>>- BRST formalism
***<<tag kk>>-	Kaluza-Klein theory, Killing vector fields
****<<tag ss>>- homogeneous spaces
****<<tag cartan>>- Cartan geometry
***<<tag lie>>- Lie algebras, Lie groups
***<<tag sm>>- the standard model of particles
****<<tag gut>>- SU(5), SO(10), TSmith, etc.
****<<tag higgs>>- Higgs scalar and symmetry breaking
***<<tag clifford>>- Clifford algebra
****<<tag dirac>>- Dirac operators, Dirac equation
*****<<tag spin>>- spin odds and ends (more theoretical then dirac)
**<<tag gr>>- General Relativity
***<<tag nat>>- natural operators, vectors, forms
***<<tag kk>>-	Kaluza-Klein theory, Killing vector fields
****<<tag ss>>- homogeneous spaces
*****<<tag cp2>>- complex projective space, Kahler manifolds
****<<tag cartan>>- Cartan geometry
***<<tag cosmo>>- cosmology
***<<tag grscal>>- gr plus a scalar field, Brans-Dicke theories, conformal transformations
***<<tag lqg>>- loop quantum gravity, loops, spin foams, spin networks
***<<tag tors>>- torsion, teleparallel gravity
**<<tag ham>>- Hamiltonian dynamics, symplectic geometry
**<<tag qm>>- quantum mechanics
***<<tag qft>>- Quantum Field Theory
***<<tag qmi>>- interpretations and minor modifications of quantum mechanics
****<<tag bohm>>- Bohmian quantum mechanics
**<<tag math>>- stuff of a more abstract mathematical nature
***<<tag dg>>- basics of differential geometry
****<<tag fib>>- fiber bundles
*****<<tag nat>>- natural operators, vectors, forms
*****<<tag pb>>- principal bundles
******<<tag ss>>- homogeneous spaces
*******<<tag cp2>>- complex projective space, Kahler manifolds
******<<tag cartan>>- Cartan geometry
***<<tag lie>>- Lie algebras, Lie groups
**<<tag other>>- AI, mech, fluids, thermodynamics, entropy
**<<tag speculative>>- wild speculation, rather than solid stuff
**<<tag paper>>- notes about or containing links to a paper
***<<tag person>>- people with interesting physics, usually with links to papers
*<<tag meta>>- describes the operation of this site
**<<tag editing>>- tips on editing and authoring notes, including all sorts of tools
**<<tag 0>>- a note that's linked to but is empty or needs editing
**<<tag system>>- control how the site operates and is layed out
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***<<tag xsystemConfig>>- deactivated code
***<<tag systemTiddlers>>- system control notes loaded at startup, used to control content
***<<tag plugin>>- code snippets enhancing functionality, and containing descriptions of what they do
***<<tag template>>- custom css page template, used to describe layout
***<<tag folder>>- a folder is a tag is a note
**<<tag illus>>- tiddlers containing illustrations
**<<tag slide>>- presentation slide (start note title with ".")

The tags group collections of tiddlers into sets -- they're adjectives, directories, or folders.  By selecting a tag, visible in the upper right of each tiddler, you can jump to any tiddler labeled with that tag -- a navigational convenience.  It's efficacious to build a flexible hierarchy of tagged content.  Each tiddler should be labeled by its most appropriate, "lowest" leaf tags -- more than one as appropriate.  Click on the tag to see a popup menu of tiddlers with that tag.  Alternatively, these folders and their contents may be selectively displayed in the "Contents" tab to the left.

//implementing hierarchical tagging seems to be a bit of a mess right now... wait and it will get better.//
[>img[images/person/Thanu Padmanabhan.jpg]]Homepage: http://www.iucaa.ernet.in/~paddy/
*Location: Pune, India
*arXiv papers: http://arxiv.org/find/gr-qc/1/au:+Padmanabhan_T/0/1/0/all/0/1

Selected work:
*[[Holographic Gravity and the Surface term in the Einstein-Hilbert Action|http://arxiv.org/abs/gr-qc/0412068]]
**Einstein's equation from the extrinsic curvature surface term alone!
**related paper by Sotiriou: http://arxiv.org/abs/gr-qc/0603096
*http://arxiv.org/abs/gr-qc/0309053
**Horizons, for collections (congruences) of observers, at coordinate singularities for the corresponding coords.
**Observers only have access to fields and dynamics inside this region, bounded by horizons.
**Upon a Wick rotation, the horizon and region beyond disappears into the origin of Euclidean space -- a conical singularity.
***The resulting Euclidean space region is naturally periodic in $t$ (in example, it resembles angular coord on a cone).
**Boundary area quantization in order to avoid quantum entanglement across the boundary.
**Action/Entropy is related to the degrees of freedom hidden behind the boundary/horizon. 
**Lots of good stuff in appendices.  explicit calculations.
***extrinsic curvature and boundary term
***derivation of unique EH action
*http://arxiv.org/abs/gr-qc/0204019
**"action is the free energy of the horizon"
*http://arxiv.org/abs/gr-qc/0311036
**How to relate stat mech, QM, and GR
**Introductory level, good survey
**conical singulartity regularization (again)
<<tiddler HideTags>>@@display:block;text-align:center;
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$\p{{}_{\small (}^{(}}$&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
[[Garrett Lisi]] &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Singularity University 2010@@
<<tiddler HideTags>>@@display:block;text-align:center;
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$\p{{}_{\small (}^{(}}$&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
[[Garrett Lisi]]&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Maui, 2011@@
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$\p{{}_{\small (}^{(}}$&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
[[Garrett Lisi]]&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Mindshare, 2011@@
Wow, I'm [[This Week's Find in Mathematical Physics|http://math.ucr.edu/home/baez/week253.html]]! Here's a quick tiddler I wrote that week when I found out:

[[John Baez]] discusses my work on describing all fields of the standard model and gravity as parts of a [[superconnection]] for the [[E8]] [[principal bundle]] over a four dimensional base manifold. And he discusses a LOT of other, related mathematics that will be keeping me busy over the next year, at least.

On 6/25/08 I delivered my [[talk for Loops 07]] in Morelia. (That links to a wiki tiddler including links to all my talk slides (printed out from this web page), as well as supporting links, AND the audio files for the talk as I practiced it (high bandwidth recommended). The slides are also available as [[this pdf file|talks/Loops07/Loops2007.pdf]]. The talk as I actually gave it is available at the [[Loops '07|http://www.matmor.unam.mx/eventos/loops07/cont_abs.html]] site under my name. It includes many excellent questions asked by Lee Smolin and others afterwards. (An excellent question being one a speaker has already thought through and wants to talk about anyway.) This talk was VERY well received -- I've had so many physics conversations with LQG people in the past two days that my head is ready to explode. They are such great and friendly people! Sadly, on the morning of the third day I started having flu symptoms, and pulled myself away from the discussion and into hotel room quarantine -- I didn't want to infect new friends with viruses, just ideas. I'm here in my room trying to get better, and being sad, when up pops a link to This Week's Finds.... happy and sick is a bizarre combination.

!Summary of the proposed [[E8]] [[T.o.E.|theory of everything]] for physicists, from the top down:
The universe is described by a Yang-Mills theory, with [[E8]] as the gauge group over a four dimensional base [[manifold]]. The field is a non-compact [[e8]] valued [[connection]] [[1-form]] which breaks up into different parts of the [[Lie algebra]],
\begin{eqnarray}
\f{A} &=& \f{H} + \f{G} + \f{\Ps}{}_I + \f{\Ps}{}_{II} + \f{\Ps}{}_{III} \\
&& \in \f{so}(1,7) \oplus \f{so}(1,7) \oplus \f{End}(V^{(1,7)}) \oplus \f{End}(S^{(1,7)+}) \oplus \f{End}(S^{(1,7)-}) 
\end{eqnarray}
The action for this connection is presumed to be invariant under gauge transformations taking the $\Ps$'s to zero -- we gauge away the $\Ps$'s. The [[BRST technique]] (Faddeev-Popov) of standard QFT is used to replace the $\Ps$'s with blocks of [[Grassmann|Grassmann number]] valued ghost fields in the same part of the Lie agebra, giving the BRST extended connection,
$$
\udf{A} = \f{H} + \f{G} + \ud{\Ps}{}_I + \ud{\Ps}{}_{II} + \ud{\Ps}{}_{III} 
$$
Computing the [[curvature]] of this extended connection gives
$$
\udff{F} = \big( \f{d} \f{H} + \f{H} \f{H} \big)  + \big( \f{d} \f{G} + \f{G} \f{G} \big) + \big( \f{d} \ud{\Ps}{}_{I-III} + \f{H} \ud{\Ps}{}_{I-III} + \ud{\Ps}{}_{I-III} \f{G} \big)
$$
Here, in the beautiful structure of $E8$, the $so(1,7)$ parts of the connection act as [[Clifford bivectors|Clifford algebra]] multiplying a set of three blocks of Grassmann valued [[spinor]]s from the left and from the right. These ghost fields have precisely the transformations and charges of the fermions -- so I go ahead and interpret them as the physical fermions. (If you don't like this derivation, you're welcome to just start with a superconnection and go from there.) The first $so(1,7)$ part of the connection, $\f{H}$, is broken up into the gravitational [[spin connection]], $\f{\om} \in \f{\rm so}(1,3)$, the electroweak fields, $\f{W} + \f{B} \in \f{\rm su}(2) \oplus \f{\rm u}(1) \subset \f{\rm su}(2) \oplus \f{\rm su}(2) = \f{\rm so}(4)$, and the combined [[frame]] and Higgs are assigned to the rest of $\f{H}$, giving
$$
\f{H} = \ha \f{\om} + \fr{1}{4}\f{e}\ph + \f{B} + \f{W}
$$
The $\f{G_{\rm strong}} \in \f{\rm su}(3) \subset \f{\rm su}(4) = \f{\rm so}(6)$ gluons and part of the $\f{B}$ go in the second $\f{\rm so}(1,7)$ part of the connection, $\f{G}$, with $19$ of $28$ generators left unused. (We could just as well have started with $\f{\rm so}(8)$ for this part of $E8$ -- I'm not sure which is better yet.) The curvature, $\udff{F}$, gives all the correct interactions of the standard model and [[modified BF gravity]]. This is very beautiful -- the entire structure of the standard model and gravity, including interactions, fits snugly in $E8$, considered by some to be the most beautiful structure in mathematics. Exactly what you want for a T.O.E. But it doesn't work perfectly yet -- I haven't figured out if the Higgs can correctly mix the generations; ideally, I want to get the CKMPMNS (mass) matrix out of $E8$. This will be what I work on from here until I get it or find it doesn't work -- and it will very distinctly be one or the other. Since we have room (unassigned generators) in $\f{G}$, we can steal some, reducing that block to $\f{G} \in \f{\rm so}(6)$ while still fitting the gluons, and using those stolen generators to have more Higgs fields in a larger $\f{H} \in \f{\rm so}(1,9)$. These Higgs terms mix between the generations, and I will be trying to use this new gauge field reassignment to fit the fermions in what's left over and get the mass matrix out. Depite how well this picture has come together so far, it may just not work, but it's what I'm after.

From my perspective, fitting the standard model into the structure of $E8$ came as a complete shock: I did not initially build this theory from the top down, but from the bottom up -- by spending years massaging the standard model and gravity into the most elegant (but weird!) and minimalistic mathematical framework possible. If you look at [[this paper|http://arxiv.org/abs/gr-qc/0511120]], you will find this strange Lie algebra block structure of $\udf{A} = \f{H} + \f{G} + \ud{\Ps}$, with a big empty space I couldn't explain. (Note that E8 is not mentioned in this paper!) When I bumped into $E8$, and found this same algebraic block structure but with two more generations of fermions... what can I say, you get only one or two moments like that in life, if you're very lucky.

!!Summary of the summary:
Everything is described by a broken [[e8]] valued [[superconnection]] over our four dimensional base manifold,
\begin{eqnarray}
\udf{A} &=& {\small \frac{1}{2}} \f{\om} + {\small \frac{1}{4}} \f{e} \ph + \f{B} + \f{W} + \f{G} + \ud{\nu^e} + \ud{e} + \ud{u} + \ud{d}  \\
&& + \ud{\nu^\mu} + \ud{\mu} + \ud{c} + \ud{s}
+ \ud{\nu^\ta} + \ud{\ta} + \ud{t} + \ud{b}
\end{eqnarray}
All standard model interactions (and gravity) come from the curvature of this connection.

For the current version of how this work is playing out, check out [[the big picture]].
To hide text within a tiddler so that it is not displayed you can wrap it in {{{/%}}} and {{{%/}}}. It can be a useful trick for hiding drafts or annotating complex markup. Edit this tiddler to see an example.
/%This text is not displayed
until you try to edit %/
What to do next.

New tiddlers and changes:
*[[Cl(3,1)]]
*[[energy-momentum tensor]]
*If $so(8)$ and [[su(3)]] are embedded in [[E8]] as in [[the big picture]], then the coupling constants for GR and EW at that ToE scale should be the same, and the [[su(3)]] coupling constant should be larger by a factor of $\sqrt{2}$ because of how the $su(3)$ root hexagon is scaled compared to the [[Gell-Mann matrices]].
**Maybe pull running SM coupling constants from Frank Wilczek's [[paper|http://arxiv.org/abs/hep-th/9803075]].
*clean up [[the big picture]]
**improve Higgs and torsion part in action
*[[SO(1,7)]] or [[SO(7,1)]] modeled on [[SO(8)]]
*[[Cl(1,15)]] modeled on [[Cl(16)]]
*[[CP2]] from [[broken SU(3)]]
*[[doubly homogeneous space]] -- [[normalizer]] $H \triangleleft N_G(H) \subset G$
**Baez TWF on double coset space
*[[calculus of variations]]
*[[Hamiltonian]]

New [[Tags]] and hierarchy adjustment:
*[[e8]] under sym
*[[toe]] under gr and sm
*[[conf]] conferences and talks, under meta

New Illustrations:
*[[submanifold]]
*[[Killing vector]]
*[[Ehresmann Cartan geometry]]

Papers to read:
*Frederick Witt, on [[triality]] (includes spinor valued 1-forms)
**[[Special metrics and Triality|papers/0602414.pdf]]
**[[Special metric structures and closed forms|papers/0502443.pdf]]
***thesis

Change referenced paper files to "author - title|author - title.pdf"?

New features:
*Have Google searches navigate to search results?

And<<slider chkSliderTDM [[To Do Maybe]] 'Maybe do these >' 'things I maybe want to do'>> ([[To Do Maybe]])
New notes:
*might need to change signs for all [[Clifford rotation]]s, to make counter-clockwise positive instead of negative, and change coefficient order in the [[spin connection]].

New [[Tags]]
*nat is awfully full

Content to add:
*from FQXi proposal
*from BF paper
*from physics notes
*summary/presentation collection of notes
*from slides to content

Features to add:
*have search return a stack of collapsed notes
*remove system tags and tiddlers from [[TabContents]] and [[Tags]]
*Contents lists hierarchical tags + descriptions (folder slider tooltips)
**any way to do this automatically?
***maybe with Udo's data in folders
*web public
**Halo Scan comments?
**"Comments" for collecting comments. solicit "new note" requests
**"Sandbox" is one public note for people to fool around in
**Ziddlywiki http://ziddlywiki.org/forum/ pretty cool discussion format.
**comments via a form at the end of a note?  some plugin allows this... Udo?
***http://tiddlywiki.abego-software.de/
***won't currently work with pytw. also tidddler->tiddler problem...
*papers under their author, and independently under their tags (this way, don't need to include all people)
**tiddlers like "Clifford papers" to collect refs
**ref papers or outside links as needed
*venn tag grouping... intersections, exclusions, etc...
*tagged templates? journal, paper, comment. http://www.gensoft.revhost.net/TaggedTemplating.html
**nah, just copy existing table layout
*script to edit in LyX
**needs to translate back and forth via intermediate files
*add editing command toolbar, http://aiddlywiki.sourceforge.net/wikibar_demo_2.html
**or tinyMCE wysiwyg editor
**tool Palettes
***LaTeX palettes in edit view that let you click on buttons to insert (latex) text.  These should be customizable.
*pwd tags meta/system/plugin
*more/change [[Keyboard Shortcuts]]
**keyboard commands to edit note
<<tiddler HideTags>>

1) Figure out everything (math!)

2) Lie group cosmology (paper writing)

3) Pacific Science Institute (Maui, construction)

4) Science Hostel (reservation website, python, django, github)

5) Elementary Particle Explorer (html5 (currently flash))

6) Deferential Geometry (tiddlywiki, javascript)

7) Surf

Gar@Li.si
http://arxiv.org/abs/gr-qc/0608135
Authors: Etera Livine
We review the canonical analysis of the Palatini action without going to the time gauge as in the standard derivation of Loop Quantum Gravity. This allows to keep track of the Lorentz gauge symmetry and leads to a theory of Covariant Loop Quantum Gravity. This new formulation does not suffer from the Immirzi ambiguity, it has a continuous area spectrum and uses spin networks for the Lorentz group. Finally, its dynamics can easily be related to Barrett-Crane like spin foam models.

*looks like a good intro to recent developments
<<tiddler HideTags>>[[John Baez]] in [[TWF90|http://math.ucr.edu/home/baez/week90.html]]:
<<<
we now look at the vector space
$$
so(8) + so(8) + end(V) + end(S^+) + end(S^-)
$$
...Since $so(8)$ has a representation as linear transformations of $V$, it has two representations on $end(V)$, corresponding to ''left and right matrix multiplication''; glomming these two together we get a representation of $so(8) + so(8)$ on $end(V)$. Similarly we have representations of $so(8) + so(8)$ on $end(S^+)$ and $end(S^-)$. Putting all this stuff together we get a Lie algebra, if we do it right - and it's $E8$.
<<<
$$
E = H + G + \Ps_I + \Ps_{II} + \Ps_{III} \;\;\;\; \in {\rm Lie}(E8)\p{{}_{(}}
$$
$$
\begin{array}{rclcrcl}
[ H , \Ps_I ] \!\!&\!\!=\!\!&\!\! H \, \Ps_I & & [ H , \Ps_{II} ] \!\!&\!\!=\!\!&\!\! H^+ \, \Ps_{II} & & [ H , \Ps_{III} ] \!\!&\!\!=\!\!&\!\! H^- \, \Ps_{III} \\
[ G , \Ps_I ] \!\!&\!\!=\!\!&\!\! \Ps_I \, G & & [ G , \Ps_{II} ] \!\!&\!\!=\!\!&\!\! \Ps_{II} \, G^+ & & [ G , \Ps_{III} ] \!\!&\!\!=\!\!&\!\! \Ps_{III} \, G^-\p{{}_{(}}
\end{array}
$$
<<tiddler HideTags>>

<html><center>

<img src="talks/Zuck09/images/Twist.png" width="300" height="300">
</center></html>
easy to invert

Use a complex frame, Kronecker delta instead of Minkowski, and the Cholesky decomposition of the metric.
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$\p{{}_{\small (}^{(}}$
[[Garrett Lisi]] &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; [[Structure and representations of exceptional groups|http://www.birs.ca/birspages.php?task=displayevent&event_id=10w5039]] &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;  Banff 2010@@
<<tiddler HideTags>>Work forwards, guess the answer, then work backwards.

Work forwards:
#[[Gauge fields|principal bundle]], [[gravity|spacetime]] and Higgs in one [[connection]].
#Calculate its [[curvature]] to get the interactions.
#Join fermions as ([[Grassmann|Grassmann number]] valued) [[BRST ghosts|BRST technique]] of a larger connection.
#Correct [[standard model]] and gravitational interactions and charges from the curvature.

Guess the answer:
*Pure [[geometry of a principal bundle|Ehresmann principal bundle connection]] -- just vector fields.
*One very large [[Lie group]] is a match!

Work backwards:
#All interactions from the group [[structure|Lie algebra]], after symmetry breaking.
#Explains exactly what and why [[spinor]]s are.
#Gives three generations.
#Lots still to do, but do-able.~~&nbsp;~~
<<tiddler HideTags>>$$\begin{array}{rcl}
L_D \!\!&\!\!=\!\!&\!\! \bar{\ps} \ga^\mu \lp e_\mu\rp^i \big(
\pa_i 
+ {\small \frac{1}{4}} \om_i^{\p{i}\nu\rh} \ga_{\nu\rh}
+ G_i^{\p{i}A} T_A
\big) \ps
+ \bar{\ps} \ph \ps \\
\!\!&\!\!=\!\!&\!\!
\bar{\ps} \ga^\mu \lp e_\mu\rp^i \big(
\pa_i 
+ {\small \frac{1}{4}} \om_i^{\p{i}\nu\rh} \ga_{\nu\rh}
+ G_i^{\p{i}A} T_A
+ {\small \frac{1}{4}} (e_i)^\nu \ga_\nu \ph
\big) \ps \\
\!\!&\!\!=\!\!&\!\!
\bar{\ps} \ve{e} \big( \f{\pa} + \f{H} \big) \ps \vp{|_{\big(}} \\
\!\!&\!\!\p{=}\!\!&\!\! \;\;\;\;\;\;\;\;\;\;\;\;\; \f{H} = \ha \f{\om} + \f{G} + {\small \frac{1}{4}} \f{e} \ph
\end{array}$$
| $\; \f{e} = \f{dx^i} (e_i)^\mu \ga_\mu \;$ |$\in \f{Cl}^1(1,3)$ |gravitational [[frame]] |
| $\; \ve{e} = \ga^\mu (e_\mu)^i \ve{\pa_i} \;$ |$\in \ve{Cl}{}^1(1,3)$ |inverse [[frame]] |
| $\; \f{\om} = \f{dx^i} \ha \om_i^{\p{i}\mu\nu} \ga_{\mu\nu} \;$ |$\in \f{Cl}^2(1,3)$ |[[spin connection]] |
| $\; \f{G} = \f{dx^i }G_i^{\p{i}A} T_A \;$ |$\in \f{su}(3) \!+\! \f{su}(2) \!+\! \f{u}(1)$ |[[gauge fields|principal bundle]] |
$$
\begin{array}{rcl}
\ff{F} \!\!&\!\!=\!\!&\!\! \f{d} \f{H} + \f{H} \f{H} \\
\!\!&\!\!=\!\!&\!\! \ha ( \f{d} \f{\om} + \ha \f{\om} \f{\om} ) + \fr{1}{16} m^2 \f{e} \f{e} 
+ ( \f{d} \f{G} + \f{G} \f{G} ) \\
\!\!&\!\!\p{=}\!\!&\!\! + \fr{1}{4} ( \f{d} \f{e} + \ha [ \f{\om}, \f{e} ] ) \ph - \fr{1}{4} \f{e} ( \f{d} \ph + [ \f{G}, \ph ] ) \\
\!\!&\!\!=\!\!&\!\! \ha \big( \ff{R} + \fr{1}{8} m^2 \f{e} \f{e} \big)
+ \ff{F^G}
+ \fr{1}{4} \big( \ff{T} \ph - \f{e} \f{D} \ph \big)
\end{array}
$$
<<tiddler HideTags>>\begin{eqnarray}
\lp D \!\!\!\! / + \ph \rp \ud{\ps} &=& \ve{e} \, \f{\na} \ud{\ps}
= \ve{e} \lp \f{d} + \f{H} \rp \ud{\ps} \\
\f{H} &=& \ha \f{\om} + \fr{1}{4}\f{e}\ph + \f{B} + \f{W} \\
&\in& \f{\rm Lie}(H) = \f{Cl}^2(1,7) = \f{so}(1,7) \subset \f{\mathbb{C}}(8\times8)
\end{eqnarray}

$$
\begin{array}{ccc}
\begin{array}{rcl}
\ve{e} \!\! &\!=\!& \!\! \ga^\mu \lp e_\mu \rp^a \ve{\pa_a} \\
\f{e} \!\! &\!=\!& \!\! \f{dx^a} \lp e_a \rp^\mu \ga_\mu \\
\f{\om} \!\! &\!=\!& \!\! \f{dx^a} \ha \om_a^{\p{a}\nu \rh} \ga_{\nu \rh} \\
\f{B},\f{W} \!\! &\!=\!& \!\! \f{dx^a} \ha W_a^{\p{a}\ph\ps} \ga_{\ph\ps} \\
\ph \!\! &\!=\!& \!\! \ph^\ph \ga_\ph \\
\fr{1}{4} \f{e} \ph \!\! &\!=\!& \!\! \fr{1}{4} \f{dx^a} \lp e_a \rp^\mu \ph^\ph \ga_{\mu \ph}
\end{array}
&
\;\;\;\;
&
\begin{array}{l}
{\rm My \; funny \; notation \! :} \\
\ve{\pa_a} \, \f{dx^b} = {\bf i}_{\pa_a} dx^b = \de_a^b \\
\ve{e} \f{e} = \ga^\mu \lp e_\mu \rp^a \ve{\pa_a} \, \f{dx^b} \lp e_b \rp^\nu \ga_\nu = 4 \\
\; \\
{\rm Higgs \; "vector" \; constrained \! :} \\
\ph \cdot \ph = \ph^\ph \ph^\ps \et_{\ph\ps} = -M^2
\end{array}
\end{array}
$$
@@display:block;text-align:center;[[indices]]: ^^&nbsp;^^spacetime coordinates: $0 \le a,b \le 3$
Clifford labels: (lower, spacetime) $0 \le \mu,\nu,\rh \le 3$, (higher) $5 \le \ph,\ps \le 8$@@
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<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/RW12/images/Weak interaction.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math">d_L \to W^- + u_L</SPAN>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/RW12/images/Weak interaction 2.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math">W^- \to e_L + \bar{\nu}_R</SPAN>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/RW12/images/Weak interaction n.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math">n = u_R^r + d_L^g + d_R^b</SPAN>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>>$$
\begin{array}{rcl}
(g^3 T^g_3 + g^8 T^g_8) \; \ps_{q_r} \!\!&\!\!=\!\!&\!\! i ( g^3 \al^{q_r}_3 + g^8 \al^{q_r}_8 ) \; \ps_{q_r} \\

{\small
\lb \begin{array}{cccccc}
 & \!\! - \! \fr{1}{2} g^3 \!-\! \fr{1}{2 \sqrt{3}} g^8 \!\! & & & & \\[-.5em]
 \!\! \fr{1}{2} g^3 \!+\! \fr{1}{2 \sqrt{3}} g^8 \!\! & & & & & \\[-.5em]
 & & & \!\! \fr{1}{2} g^3 \!-\! \fr{1}{2 \sqrt{3}} g^8 \!\! & & \\[-.5em]
 & & \!\! - \! \fr{1}{2} g^3 \!+\! \fr{1}{2 \sqrt{3}} g^8 \!\! &  & & \\[-.5em]
 & & & & & \fr{1}{\sqrt{3}} g^8 \\[-.5em]
 & & & & - \fr{1}{\sqrt{3}} g^8 & 
\end{array} \rb
\lb
\begin{array}{c}
1 \\ -i \\ 0 \\ 0 \\ 0 \\ 0
\end{array}
\rb }
\!\!&\!\!=\!\!&\!\! 
{\small
i \lp g^3 \big( \fr{1}{2} \big) + g^8 \big( \fr{1}{2 \sqrt{3}} \big) \rp
\lb
\begin{array}{c}
1 \\ -i \\ 0 \\ 0 \\ 0 \\ 0
\end{array}
\rb
}
\end{array}
$$

<html><center>
<table class="gtable">
<tr border=none>
<td border=none>

<table class="gtable">
<tr><td>
<SPAN class="math">V_{g^{r \bar{g}}} \ps_{q^g} = \ps_{q^r}  \vp{A_{\big(}}</SPAN>
</td></tr>
<tr><td>
<SPAN class="math">\al^{g^{r \bar{g}}} + \al^{q^g} = \al^{q^r} \vp{A_{\Big(}}</SPAN>
</td></tr><tr><td>
<img SRC="images/png/quark gluon vertex.png" height=160px>
</td></tr>
</table>


</td>
<td>
<SPAN class="math">\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;</SPAN></td>
<td>

<table class="gtable">
<tr><td>
<img src="talks/Zuck09/images/Strong interaction.png" width="324" height="280">
</td></tr><tr><td>
<SPAN class="math">su(3) + 3 + \bar{3} \vp{A^{\big(}}</SPAN>
</td></tr>
</table>

</td>
</tr>
</table>
</center>
</html>
<<tiddler HideTags>>6D Cartan subalgebra: $\;\; C = \om_S \, T^\om_{12} + \om_T \, T^\om_{34} + W \, T^W_3 + Y \, T^Y + g^3 \, T^g_3 + g^8 \, T^g_8 \vp{A_{\big(}}$
$$
{\small
\lb \begin{array}{cccccc}
 & \!\! - \! \fr{1}{2} g^3 \!-\! \fr{1}{2 \sqrt{3}} g^8 \!\! & & & & \\[-.5em]
 \!\! \fr{1}{2} g^3 \!+\! \fr{1}{2 \sqrt{3}} g^8 \!\! & & & & & \\[-.5em]
 & & & \!\! \fr{1}{2} g^3 \!-\! \fr{1}{2 \sqrt{3}} g^8 \!\! & & \\[-.5em]
 & & \!\! - \! \fr{1}{2} g^3 \!+\! \fr{1}{2 \sqrt{3}} g^8 \!\! &  & & \\[-.5em]
 & & & & & \fr{1}{\sqrt{3}} g^8 \\[-.5em]
 & & & & - \fr{1}{\sqrt{3}} g^8 & 
\end{array} \rb

\lb
\begin{array}{c}
1 \\ -i \\ 0 \\ 0 \\ 0 \\ 0
\end{array}
\rb
=
i \lp \big( \fr{1}{2} \big) g^3 + \big( \fr{1}{2 \sqrt{3}} \big) g^8 \rp
\lb
\begin{array}{c}
1 \\ -i \\ 0 \\ 0 \\ 0 \\ 0
\end{array}
\rb
}
$$
<html><center>
<table class="gtable">
<tr border=none>
<td border=none>

<table class="gtable">
<tr><td>

<table class="gtable">
<tr><td>
weight vectors
</td></tr><tr><td>
weights
</td></tr>
</table>

</td><td><SPAN class="math">\longleftrightarrow</SPAN></td><td>

<table class="gtable">
<tr><td>
states
</td></tr><tr><td>
quantum numbers
</td></tr>
</table>

</td></tr>
<tr><td><SPAN class="math">\updownarrow</SPAN></td><td></td><td><SPAN class="math">\updownarrow</SPAN></td></tr>
<tr><td>

<table class="gtable">
<tr><td>
eigenvectors
</td></tr><tr><td>
eigenvalues
</td></tr>
</table>

</td><td><SPAN class="math">\longleftrightarrow</SPAN></td><td>

<table class="gtable">
<tr><td>
particles
</td></tr><tr><td>
charges
</td></tr>
</table>

</td></tr>
</table>

</td>
<td>
<SPAN class="math">\;\;\;\;\;\;\;\;\;\;\;\;</SPAN></td>
<td>

<table class="gtable">
<tr><td>
<img src="talks/Zuck09/images/Strong interaction.png" width="324" height="280">
</td></tr><tr><td>
<SPAN class="math">su(3) + 3 + \bar{3} \vp{A^{\big(}}</SPAN>
</td></tr>
</table>

</td>
</tr>
</table>
</center>
</html>
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</script>This is [[Garrett Lisi]]'s personal wiki notebook in theoretical physics.

Each note describes some bit of mathematical physics, while linking it to everything else. The notes are organized by [[Tags]] in an expandable list to the left of this window. Or you can look up mathematical objects by [[symbol|Symbols]], or type any phrase to search for in the field to the left.  From any note you can follow links to others or click in the bottom list of notes that link to it. (These are wiki-links, so tabbed browsing won't work in the usual way.) You can also see which notes have been edited recently, under "[[Latest|TabTimeline]]." That's about it for basic orientation &mdash; you can read more [[About]] what you're looking at, or pick it up as you go.

Welcome to my brain, have fun looking around.

The last two notes I edited are below.

<<tiddler HideTags>>
$$
L_D = \bar{\ps} \lp \ga^\mu \pa_\mu + i m \rp \ps
$$

$$
\ga^\mu \pa_\mu =
\lb \begin{array}{cccc}
0 & 0 & \pa_0+\pa_3 & \pa_1-i\pa_2 \\
0 & 0 & \pa_1+i\pa_2 & \pa_0-\pa_3 \\
\pa_0-\pa_3 & -\pa_1+i\pa_2 & 0 & 0 \\
-\pa_1-i\pa_2 & \pa_0+\pa_3 & 0 & 0
\end{array} \rb
$$

$$
\ps = 
\lb \matrix{
e_L^\wedge \\ e_L^\vee \\ e_R^\wedge \\ e_R^\vee
} \rb
\in \ud{\mathbb{C}}^4
$$
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<b>abstruse goose</b>
</td></tr>
<tr><td><SPAN class="math">\p{a}</SPAN></td></tr>
<tr><td>
<img SRC="talks/IfA11/images/wis.png" height=480px>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>>$$
\udf{A}  = \f{H} + \f{G} + \ud{\ps}
=
{\small
\lb \begin{array}{cc}
\f{H^+} & \ud{\ps}^- \\
& \f{G^-}
\end{array}  \rb
}
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\;\;\;\;\;\;\;\;\;
$$
$$
{\small
\!\! = \!\! \lb \begin{array}{cccccccc}
\frac{1}{2} \f{\om_L} \!+\! i \f{W^3} \!&\! i \f{W^1} \!+\! \f{W^2} \!&\! - \! \frac{1}{4} \f{e_R} \ph_0^* \!&\! \frac{1}{4} \f{e_R} \ph_+ \!&
\; \ud{\nu}{}_L &\!\! \ud{u}{}_L^r \!\!&\!\! \ud{u}{}_L^g \!\!&\!\! \ud{u}{}_L^b \\

i \f{W^1} \!-\! \f{W^2} \!&\! \frac{1}{2} \f{\om_L} \!-\! i \f{W^3} \!&\! \p{-} \frac{1}{4} \f{e_R} \ph_+^* \!&\! \frac{1}{4} \f{e_R} \ph_0 \!&
\; \ud{e}{}_L &\!\! \ud{d}{}_L^r \!\!&\!\! \ud{d}{}_L^g \!\!&\!\! \ud{d}{}_L^b \\

-\frac{1}{4} \f{e_L} \ph_0 & \frac{1}{4} \f{e_L} \ph_+ & \! \frac{1}{2} \f{\om_R} \!+\! i \f{B} \! \!& &
\; \ud{\nu}{}_R &\!\! \ud{u}{}_R^r \!\!&\!\! \ud{u}{}_R^g \!\!&\!\! \ud{u}{}_R^b \\

\p{-}\frac{1}{4} \f{e_L} \ph_+^* & \frac{1}{4} \f{e_L} \ph_0^* & &\! \! \frac{1}{2} \f{\om_R} \!-\! i \f{B} \! &
\; \ud{e}{}_R &\!\! \ud{d}{}_R^r \!\!&\!\! \ud{d}{}_R^g \!\!&\!\! \ud{d}{}_R^b \\

& & & & \; i \f{B} &\!\! \!\!&\!\! \!\!&\!\!  \\
& & & &  &\!\!\! \frac{-i}{3} \! \f{B} \!+\! i \f{G^{3+8}} \!\!\!&\!\!\! i\f{G^1} \!-\! \f{G^2} \!\!\!&\!\!\! i\f{G^4} \!-\! \f{G^5} \\
& & & &  &\!\!\! i\f{G^1} \!+\! \f{G^2} \!\!\!&\!\!\! \frac{-i}{3} \! \f{B} \!-\! i \f{G^{3+8}} \!\!\!&\!\!\! i\f{G^6} \!-\! \f{G^7} \\
& & & &  &\!\!\! i\f{G^4} \!+\! \f{G^5} \!\!\!&\!\!\! i\f{G^6} \!+\! \f{G^7} \!\!\!&\!\!\! \frac{-i}{3} \! \f{B} \!-\!\! \frac{2i}{\sqrt{3}}\f{G^8}
\end{array} \rb
}
$$
Note: Only one generation, and fermion masses not quite right.${\p{\big(}}_{\p{(}}$
For three generations: &nbsp; $\udf{A} \; \in \; \f{so}(1,7) + \f{so}(8) + 3 * \ud{\mathbb{R}}(8 \times 8) \; = \; \;\;\; ?{\p{\Big(}}$  
BIG Lie algebra: &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; $n \;\, = \;\;\;\;\;\, 28 \;\;\, + \;\; 28 \;\;\, + \;\;3 \; * \; 64 \;\;\;\;\;\; = \; 248{\p{\big(}}$
<<tiddler SiteTitle>>
#[[Real simple compact Lie groups]]
#[[The Coleman-Mandula theorem]]
#[[BRST gauge fixing]]
##[[BRST technique]]
#[[BRST extended connection]]
#[[E8 connection]]
#[[E8 curvature]]
#[[Action for everything]]
#[[Pati-Salam model plus gravity]]
#[[Gravitational SO(3,1)]]
#[[Electroweak SU(2) and U(1)]]
#[[Graviweak SO(7,1)]]
#[[Graviweak F4]]
#[[E8 periodic table]]
#[[E8 Theory summary]]
#[[E8 Theory discussion]]
##[[principal bundle]]
##[[vector-form algebra]]
##[[spin connection]]
##[[frame]]
##[[su(2)]]
##[[su(3)]]
##[[indices]]
##[[Clifford algebra]]
##[[Cl(1,7)|Cl(8)]]
##[[connection]]
##[[Bosonic part of the connection]]
##[[spacetime frame]]
##[[spacetime spin connection]]
##[[modified BF gravity]]
##[[volume form]]
##[[Clifford curvature scalar]]
##[[curvature]]
##[[Clifford-Riemann curvature]]
##[[Gravitational part of the action]]
##[[Fermionic part of the action]]
#[[Gauge theory geometry]]
#[[Real simple compact Lie groups]]
#[[The Coleman-Mandula theorem]]
#[[BRST gauge fixing]]
##[[BRST technique]]
#[[BRST extended connection]]
#[[E8 Theory discussion]]
##[[principal bundle]]
##[[vector-form algebra]]
##[[spin connection]]
##[[frame]]
##[[su(2)]]
##[[su(3)]]
##[[indices]]
##[[Clifford algebra]]
##[[connection]]
##[[Bosonic part of the connection]]
##[[spacetime frame]]
##[[spacetime spin connection]]
##[[modified BF gravity]]
##[[volume form]]
##[[Clifford curvature scalar]]
##[[curvature]]
##[[Clifford-Riemann curvature]]
##[[Gravitational part of the action]]
##[[Fermionic part of the action]]
<<ListTagged alg>>
An ''almost complex structure'', $\f{\ve{J}}$, on a [[manifold]] is a [[vector projection]] that satisfies,
$$
\f{\ve{J}} \f{\ve{J}} = - \f{\ve{I}}
$$
The product of an almost complex structure with vectors is equivalent to multiplication by $i=\sqrt{-1}$. The manifold is ''complex'', admitting a complex coordinatization, iff the [[FuN curvature]] of the almost complex structure vanishes,
$$
\lb \f{\ve{J}} , \f{\ve{J}} \rb_L = 0
$$
and $\f{\ve{J}}$ is then called a ''complex structure''.
Refs:
*Jeffrey A. Harvey
**[[TASI 2004 Lectures on Anomalies|papers/0509097.pdf]]
***anomalies from the particle physics point of view
The ''antisymmetric bracket'' is an operation on a list of arbitrary [[Lie algebra]] generators or [[Clifford element]]s. The antisymmetric bracket of two elements,
\[ \lb A, B \rb_A = A \times B = \ha \lb A, B \rb = \ha \lp A B - B A \rp \]
equivalent to the ''cross product'', is equivalent to the [[commutator]] bracket with a multiplier of $\ha$.  The antisymmetric bracket of three elements is
\[ \lb A, B, C \rb_A = \fr{1}{3!} \lp ABC + BCA + CAB - ACB - CBA - BAC \rp \]
and so on for more elements. An antisymmetric bracket changes sign under the exchange of any two neighboring elements.

The antisymmetric bracket does not commute [[coordinate basis 1-forms]] -- these must be taken out of the bracket first. For example, for the cross product of two [[Clifform]]s,
$$
\f{A} \ti \f{B} = \lb \f{A},\f{B} \rb_A = \f{dx^i} \f{dx^j} \lb A_i, B_j \rb_A
= \f{dx^i} \f{dx^j} \ha \lp A_i B_j - B_j A_i \rp = \ha \lp \f{A} \f{B} + \f{B} \f{A} \rp
$$
In general, for two $p$ and $k$ forms,
$$
\nf{A} \ti \nf{B} = \lb \nf{A}, \nf{B} \rb_A = \ha \lp \nf{A} \nf{B} - \lp -1 \rp^{pk} \nf{B} \nf{A} \rp 
$$
Two [[fiber bundle]]s are ''associated'' if they have the same structure group and the same transition functions.
An ''automorphism'' is a structure preserving map from an object to itself.

If the object is a [[group]], with elements satisfying $g_1 g_2 = g_3$, then after a ''group automorphism'', $Aut:G \to G, g \mapsto g' = \ph(g)$, the new group elements must satisfy $g'_1 g'_2 = g'_3$ -- that's what's meant by "structure preserving". The group of all automorphisms of a group, $G$, is called the ''automorphism group'' of $G$, $Aut(G)$. The typical group automorphism is an ''inner automorphism'',
$$
g' = \ph_h(g) = A_h g = h g h^-
$$
with an element $h \in G$ acting on $G$ itself through conjugation (the [[adjoint action|group]]). These form the ''inner automorphism group'', $Inn(G)$. In some cases there may be group automorphisms that are not inner automorphisms. The automorphisms of $G$ which are not inner are called ''outer automorphism''s, and the [[coset]] is labeled $Out(G)=Aut(G)/Inn(G)$.
An ''automorphism bundle'' is a [[fiber bundle]] with a [[Lie group]], $G$, or algebra as the typical fiber and the [[automorphism]] group, $Aut(G)$, acting on $G$ as the structure group.

For most Lie groups the automorphism group is the same as the group itself, $Aut(G)=Inn(G)=G$, with all automorphisms represented by inner automorphisms, and the group action given by the [[adjoint action|group]] of $G$ on the $G$ fiber:
$$
g' = A_h g = h g h^-
$$
for any $h \in G$ in the structure group and $g \in G$ in the fiber. When not all automorphisms of $G$ are inner automorphisms things can get more interesting! But we will first handle the cases for when they are. Note that this bundle is different than a [[principal bundle]], for which the structure group action is the left action -- but there are many similar expressions.

For a section, $g(x)$, transforming under the adjoint action [[gauge transformation]], $g \mapsto g'=h g h^-$, the [[covariant derivative]] is
$$
\f{\na} g = \f{d} g + \f{A} g - g \f{A} = \f{d} g + \lb \f{A} , g \rb
$$
with the ''automorphism bundle [[connection]]'',  $\f{A} = \f{dx^i} A_i{}^B T_B$, a 1-form over M valued in the [[Lie algebra]] of $G$.

Any fiber element, $g$, at $t=0$ may be [[parallel transport]]ed to $g(t)=h(t)gh^-$ along a path on the base by a parameter dependent element, $h \in G$, the path holonomy, $h(t) = Pe^{-\int_0^t \f{A}}$, satisfying the [[path holonomy]] equation,
$$
\fr{d}{dt} h(t) = - \ve{v} \f{A} h
$$

Applying the covariant derivative twice (being careful with the signs of 1-forms hopping sides),
\begin{eqnarray}
\f{\na} \f{\na} g &=& \f{d} \lp \f{d} g + \f{A} g - g \f{A} \rp + \f{A} \lp \f{d} g + \f{A} g - g \f{A} \rp  + \lp \f{d} g + \f{A} g - g \f{A} \rp \f{A} \\
&=& \lp \f{d} \f{A} \rp g - \f{A} \f{d} g - \lp \f{d} g \rp \f{A} - g \f{d} \f{A} 
 + \f{A} \lp \f{d} g + \f{A} g - g \f{A} \rp  + \lp \f{d} g + \f{A} g - g \f{A} \rp \f{A} \\
&=& \lb \ff{F} , g \rb
\end{eqnarray}
gives the ''automorphism bundle [[curvature]]'',
$$
\ff{F} = \f{d} \f{A} + \f{A} \f{A}
$$
a Lie algebra valued 2-form. This expression for the curvature may alternatively be obtained from the [[holonomy]].

Under a gauge transformation, $g(x) \mapsto g'(x) = h(x) g(x) h^-(x)$, the covariant derivative changes to
\begin{eqnarray}
\f{\na'} g' &=& h \lp \f{\na} g \rp h^-\\
\f{d} \lp h g h^- \rp + \f{A'} h g h^- - h g h^- \f{A'} &=& h \lp \f{d} g \rp h^- + h \f{A} g h^- - h g \f{A} h^-
\end{eqnarray}
giving the transformation law for the connection,
$$
\f{A'} = h \f{A} h^- - \lp \f{d} h \rp h^- = h \f{A} h^- + h \lp \f{d} h^- \rp 
$$
An infinitesimal transformation, $h \simeq 1 + H$, changes the connection to
$$
\f{A'} \simeq \f{A} - \f{d} H - \f{A} H + H \f{A} = \f{A} - \f{\na} H
$$
The curvature consequently transforms under a gauge transformation to
$$
\ff{F'} = \f{d} \f{A'} + \f{A'} \f{A'} = h \ff{F} h^- \simeq \ff{F} + \lb H , \ff{F} \rb
$$

The covariant derivative acting on a Lie algebra valued [[differential form]] such as the curvature, transforming under the adjoint action, $\ff{F'} = h \ff{F} h^-$, is still
$$
\f{\na} \ff{F} = \f{d} \ff{F} + \lb \f{A} , \ff{F} \rb 
$$

It is worth repeating that when the automorphism group for $G$ includes automorphisms that are not inner, things are going to get more complicated...

//I'm starting to think that constructing this bundle just doesn't work. For one thing, I don't think I can make an [[Ehresmann connection]] for it since I haven't been able to build automorphism invariant vector fields. For another thing, automorphisms leave the identity point in the same place -- and singling out a point in the fiber like that would be an odd thing to do. I'll shelve the idea for now.//
review:
http://arxiv.org/abs/hep-ph/9512245
As a nice warm up, before tackling the standard model, it's instructive to see how the [[SU(3)]] [[Lie group geometry]] might go wobbly and become a [[Cartan geometry]]. It has a U(2) [[subgroup]], constructed from the [[simple]] [[SU(2)]] and U(1) Lie groups. The [[homogeneous space]], [[CP2]], is
$$
CP2 = \fr{SU(3)}{U(2)} = \fr{SU(3)}{SU(2) \times U(1)}
$$
<<ListTagged brst>>
<<ListTagged cartan>>
The ''center'', $Z(G) \triangleleft G$, of a [[group]], $G$, is an abelian [[normal subgroup]] consisting of all elements of $G$ that commute with all other elements,
$$
Z(G) = \lc z \in G \; | \; gz = zg \; \forall \, g \in G \rc
$$ 
The ''centralizer'', $C_G(a) \subset G$, of an element $a$ of a [[group]], $G$, is a [[subgroup]] consisting of all elements, $c \in G$, that commute with $a$,
$$
C_G(a) = \lc c \in G \; | \; ca = ac  \rc 
$$
The centralizer is the largest subgroup of $G$ having $a$ in its [[center]], $a \in Z(C_G(a))$. The centralizer of a subset, $S$, in $G$ is the subgroup consisting of all elements commuting with the elements of $S$,
$$
C_G(S) = \lc c \in G \; | \; cs = sc \; \forall s \in S \rc
$$
The centralizer of $G$ in $G$ is the center, $C_G(G) = Z(G)$. The centralizer of a subset in a subgroup, $H \subset G$, is
$$
C_H(S) = \lc c \in H \; | \; cs = sc \; \forall s \in S \rc
$$
Refs:
*Jeffrey A. Harvey
**[[TASI 2004 Lectures on Anomalies|papers/0509097.pdf]]
***good brief intro
A [[Clifford matrix representation]] is ''chiral'' if all [[Clifford basis vectors]] are represented by matrices non-zero only in the second and third quadrant blocks. This is the case iff the first [[Pauli matrix|Pauli matrices]] in the [[Kronecker product]] expression of each vector is either $\si^P_1$ or $\si^P_2$,
$$
\ga_\al = \si^P_{1 \, {\rm or} \, 2} \otimes \dots
$$
A representation may be said to be chiral if this holds for the first Pauli matrices in the product, or it may be chiral for the level where the [[spin connection]] lives -- to be concrete, a rep should be called ''n'th level chiral'' if the n'th Kronecker product matrices are all $\si^P_1$ or $\si^P_2$. In a (1st level) chiral representation, all odd [[Clifford grade]] elements are represented by matrices non-zero only in the second and third quadrants, while all even elements are represented by matrices only non-zero in the first and fourth quadrant,
$$
A =
\lb \begin{array}{cc}
A^e & A^o \\
A^o & A^e
\end {array} \rb
\in Cl
$$

A ''chiral spinor'' is half of a [[spinor]]. Expressed in a chiral rep, it is either ''positive chiral'',
$$
\Psi^+ = 
\lb \begin{array}{c}
\ps^+ \\
0
\end {array} \rb
$$
or ''negative chiral'',
$$
\Psi^- = 
\lb \begin{array}{c}
0 \\
\ps^-
\end {array} \rb
$$
on the level of chirality. A full spinor may be built by adding two chiral spinors, $\Psi = \Psi^+ + \Psi^-$, such as to make a [[Dirac spinor]]. A spinor may also be broken up into n'th level chiral pieces.

The ''chirality projector'', $P_\pm$, is one of a pair of Clifford elements that operate on the suitable level to project out the desired chiral degrees of freedom,
$$
\begin{array}{cc}
P_+ =
\lb \begin{array}{cc}
1 & 0 \\
0 & 0
\end {array} \rb &
P_- =
\lb \begin{array}{cc}
0 & 0 \\
0 & 1
\end {array} \rb &
\end{array}
$$
It is often built using the [[pseudoscalar]].

As a representation space of a spin algebra, the weight vectors (eigenvectors with respect to a Cartan subalgebra) have an even number of positive weights for a positive chiral spinor and an odd number of positive weights for a negative chiral spinor, or vice versa.
<<ListTagged clifford>>
A [[differential form]] [[field|cotangent bundle]], $\nf{f}(x)$, over a [[manifold]], $M$, is ''closed'' iff its [[exterior derivative]] vanishes,
$$
\f{d} \nf{f} = 0
$$
The [[vector space]] of closed $p$-forms over $M$ is labeled $C^p$.
The ''codifferential'' is the ''adjoint exterior derivative''. Operating on a [[differential form]] of grade $p$, it is defined, using the [[exterior derivative]] and [[Hodge dual]] operators, as
$$
\ve{\de} = (-1)^p *^- \f{d} \, *
$$
and decreases the grade of the form by 1. Note that this operator is different from the non-covariant $\ve{\pa} = \ve{\pa_i} \eta^{ij} \pa_j$.

The codifferential is defined to satisfy
$$
<\f{d} \nf{a},\nf{b}> = < \nf{a},\ve{\de} \nf{b}>
$$
for the scalar product of the exterior derivative of a p-form and a (p+1)-form.

Ref:
http://en.wikipedia.org/wiki/Codifferential
also see Nakahara, p253
The $p$-th (de Rham) ''cohomology'' of a [[manifold]], $M$, is the [[vector space]],
$$
H^p(M) = C^p / E^p
$$
equal to the [[coset]] of all [[closed]] $p$-forms over $M$ that are not [[exact]]. An element of the cohomology is specified by a closed coset representative, $[\nf{f^C}] \in H^p(M)$, with $\nf{f^C} \in C^p$.
The ''commutator bracket'' (or simply //''commutator''//) of any two arbitrary [[Lie algebra]] generators, [[Clifford element]]s, or operators is another generator, element, or operator equal to
$$
\lb A, B \rb = A B - B A 
$$
employing the appropriate product or operator composition. It relates to the [[antisymmetric bracket]] (and cross product) by a factor of $\ha$,
$$
A \times B = \lb A, B \rb_A = \ha \lb A, B \rb = \ha \lp A B - B A \rp 
$$

The commutator (called more precisely the ''graded commutator'') does not commute [[coordinate basis 1-forms]] -- these must be taken out of the bracket first. For example, for the commutator of two grade 1 [[Lieform]]s or [[Clifform]]s,
$$
\lb \f{A},\f{B} \rb = \f{dx^i} \f{dx^j} \lb A_i, B_j \rb
= \f{dx^i} \f{dx^j} \lp A_i B_j - B_j A_i \rp = \f{A} \f{B} + \f{B} \f{A}
$$
So, in general, for two $p$ and $k$ forms,
$$
\lb \nf{A}, \nf{B} \rb = \nf{A} \nf{B} - \lp -1 \rp^{pk} \nf{B} \nf{A} 
$$
and [[tangent vector]]s are considered $k$ forms with $k=-1$.
A ''connection'' completely encodes the local geometry of a [[fiber bundle]]. Specifically, it describes how the local trivializations change as one moves around on the base manifold. The group of these changes is the same as the structure group, $G$, of the fiber bundle. From any point, the infinitesimal change of a local trivialization when moving in any direction is described by the operation of a [[Lie algebra]] element. These changes may be described via a [[Lie algebra]] valued [[1-form]] over the base, the connection,
$$
\f{A} = \f{dx^i} A_i{}^B(x) T_B
$$
with the appropriate action on the fiber elements. Using this connection, the [[covariant derivative]] of a section, $\si(x)$, (valued in the fiber) is
$$
\f{\na} \si = \f{d} \si + \f{A} \si = \f{dx^i} \lp \pa_i \si + A_i{}^B T_B \si \rp
$$
in which the Lie algebra basis elements, $T_B$, act on the fiber. The connection changes under a [[gauge transformation]] so as to keep this derivative covariant.
The ''coordinate basis 1-forms'', $\f{dx^i}$, are [[1-form]]s dual to the [[coordinate basis vectors]],
\[ \f{dx^{i}}(\ve{\partial_j}) = \ve{\partial_j} \f{dx^{i}} = \delta_{j}^{i} \]
The ''coordinate basis forms'' (//''coordinate basis p-forms''//) are constructed by taking the [[wedge product]] of (p) [[coordinate basis 1-forms]],
$$
\nf{dx^{i \dots j}} = \f{dx^i} \dots \f{dx^j}
$$
For example, the ''coordinate basis 2-forms'' are
$$
\ff{dx^{ij}} = \f{dx^i} \f{dx^j}
$$
The wedge product between basis 1-forms is implied but never written, as the antisymmetric nature of the form product is assumed in the [[vector-form algebra]]. The coordinate basis forms are [[antisymmetric|index bracket]], changing sign under the interchange of any two adjacent indices, $\ff{dx^{ij}}=-\ff{dx^{ji}}$. On an $n$ dimensional manifold, the highest grade coordinate basis form is the ''coordinate basis $n$-form'',
\[ \nf{d^n x} = \f{dx^0} \dots \f{dx^{n-1}} \]
Technically, there is also a coordinate basis $0$-form,
$$
1
$$

For any grade, $p$, there are $\frac{n!}{\left(n-p\right)!p!}$ distinct coordinate basis $p$-forms. Adding these up over the $n+1$ possible grades, including the basis 0-form, there are $2^n$ distinct coordinate basis forms.
The coordinate basis vectors,
$$
\ve{\pa_i} = \ve{\frac{\partial}{\partial x^i}} \in T_p M
$$
with coordinate [[index|indices]], $i$, span the space, $T_p M$, of [[tangent vector]]s to possible curves passing through each point, $p$, of a [[manifold]], $M$.  The basis vectors may not be colinear, but are not otherwise inherently related &mdash; unlike the [[Clifford basis vectors]], they are not necessarily orthogonal or of unit length.

The coordinate basis vectors may be visualized as little arrows pointing along each coordinate curve.  Each ''coordinate basis vector'', $\ve{\pa_i}$, is the [[tangent vector]] to the curve formed by varying the $x^i$ coordinate while holding the others fixed.
The coordinates, $x^i$, used to describe points in a [[manifold]] patch may always be abandoned in favor of a new set of coordinates, $y^i$. Since coordinates in the old and new set describe the same manifold points, the new coordinates may be written as functions of the old, $y^i(x)$, and the old as functions of the new, $x^i(y)$. Similarly, two such sets of coordinates must be used in coordinate patch overlaps on the manifold.

The [[coordinate basis vectors]] are different in the two sets of coordinates, and are related by the partial derivatives of the old and new coordinates as functions of each other:
$$
\ve{\pa^y_i} = \ve{\fr{\pa}{\pa y^i}} = \fr{\pa x^j}{\pa y^i} \ve{\fr{\pa}{\pa x^j}} = \fr{\pa x^j}{\pa y^i} \ve{\pa^x_j}
\quad \quad \quad
\ve{\pa^x_i} = \ve{\fr{\pa}{\pa x^i}} = \fr{\pa y^j}{\pa x^i} \ve{\fr{\pa}{\pa y^j}} = \fr{\pa y^j}{\pa x^i} \ve{\pa^y_j}
$$
With the partial derivative matrices satisfying
$$
\fr{\pa x^j}{\pa y^i} \fr{\pa y^i}{\pa x^k} = \de_k^j
$$
Since $x$ and $y$ are coordinates for the same point, the partial derivative matrices may equivalently be considered functions of $x$ or $y$ as necessary. Similarly, the [[coordinate basis 1-forms]] are related by
$$
\f{dy^i} = \fr{\pa y^i}{\pa x^j} \f{dx^j}
\quad \quad \quad
\f{dx^i} = \fr{\pa x^i}{\pa y^j} \f{dy^j}
$$
and the [[partial derivative]]s, $\pa_i$, of a function (or field components), $f(x)$, in different coordinate systems are related by
$$
\pa^x_i f(x) = \fr{\pa}{\pa x^i} f(x) =  \fr{\pa y^j}{\pa x^i} \fr{\pa}{\pa y^j} f(x(y)) = \fr{\pa y^j}{\pa x^i} \pa^y_j f(y)
$$

A [[natural]] geometric object is invariant under coordinate change. For example, [[tangent vector]]s and [[1-form]]s may be expressed in terms of either set of coordinate basis vectors and forms,
\begin{eqnarray}
\ve{v} &=& v^i \ve{\fr{\pa}{\pa x^i}} = v^i \fr{\pa y^j}{\pa x^i} \ve{\fr{\pa}{\pa y^j}} = v'^j \ve{\fr{\pa}{\pa y^j}} = \ve{v'}\\
\f{f} &=& f_i \f{dx^i} = f_i \fr{\pa x^i}{\pa y^j} \f{dy^j} = f'_j \f{dy^j} = \f{f'}
\end{eqnarray}
In old terminology, tangent vectors are described by "contravariant" (upper) indexed components transforming as $v'^j =  v^i \fr{\pa y^j}{\pa x^i}$ and forms are described by "covariant" (lower) indexed components transforming as $f'_j = f_i \fr{\pa x^i}{\pa y^j}$ under coordinate change. Any indexed object transforming this way under coordinate change is called a ''tensor''. 

Another way of looking at coordinate change is as the identity map from the manifold to itself -- technically a [[diffeomorphism]]. However, a coordinate change is a ''passive diffeomorphism'' as it does not move the manifold points, but only mixes (re-labels) their coordinates.
A collection of elements called a ''coset'', $G/H$, can be formed by modding a [[group]], $G$, by a [[subgroup]], H. Specifically, a ''left coset'' element, $[g] \in G/H$, consists of all elements of $G$ related by the right action of elements of $H$,
$$
[g] = gH = \left\{ gh : \forall \; h \in H \right\}
$$
A coset is not a group unless $H$ is a [[normal subgroup]], in which case $G/H$ is called the ''quotient group''.

An example, let $G$ be the set of integers,
$$
G = \left\{ \dots, -2, -1, 0, 1, 2, \dots \right\}
$$
with addition, $+$, as the group product. Choose the subgroup, $H$, to be all elements of $G$ that are multiples of $4$,
$$
H = \left\{ \dots, -8, -4, 0, 4, 8, \dots \right\}
$$
The left coset consists of four ( $=$ the ''index'' of $H$ in $G$) elements,
\begin{eqnarray}
G/H &=& \left\{
\lb \dots, -3, 1, 5, \dots \rb,
\lb \dots, -2, 2, 6, \dots \rb,
\lb \dots, -1, 3, 7, \dots \rb,
\lb \dots, 0, 4, 8, \dots \rb \right\} \\
&=& \left\{ [1], [2], [3], [0] \right\}
\end{eqnarray}
The notation "$[g]$" means that $g \in G$ is a ''coset representitive'' -- any other representative related by $h \in H$ is equivalent, $[g]=[gh]$. Every element of $G$ is in exactly one of the coset elements, and each coset element is isomorphic to $H$ -- in fact, one of the coset elements, $[0]$, is $H$. There is a product between coset elements determined by the product of their representatives and representative equivalence. For example, $[1]+[3] = [4] = [0]$. And, in this example, the coset does form a group, since $H$ is normal in $G$,
$$
ghg^- = g + h - g = h \in H
$$

A ''right coset'' element, $[g] \in G/H$, consists of all elements of $G$ related by the left action of elements of $H$,
$$
[g] = Hg = \left\{ hg : \forall \; h \in H \right\}
$$
ref:
[[Everything You Always Wanted To Know About The Cosmological Constant Problem (But Were Afraid To Ask)|http://arxiv.org/abs/1205.3365]]
The ''cotangent bundle'' (//''1-form bundle''//), $T^* M = \Om^1 M$, with $n$ dimensional base [[manifold]], $M$, is a [[vector bundle]] with $n$ fiber basis elements identified as the [[coordinate basis 1-forms]], $\f{dx^i}$, for the manifold. It is the dual bundle to the [[tangent bundle]]. The fiber at a base manifold point, $p$, is the $n$ dimensional cotangent space, $T_p^* M$, spanned by the basis 1-forms, and the cotangent bundle is the union of all cotangent spaces, $T^* M = \bigcup_{p \in M} T_p^* M$.  The transition functions for the basis elements, $\f{dx^i_2} = \lp t^{21} \rp_j{}^i \f{dx^j_1}$, over overlapping patches, $U_1$ and $U_2$, are given by the ''Jacobian matrix'',
$$\lp t^{21} \rp_j{}^i = \fr{\pa x_2^i}{\pa x_1^j}$$
The structure group is thus the group of general linear transformations, $G = GL(n,\Re)$.  A ''covector field'' (//''1-form field''//), $\f{f} = \f{f}(x) = f_i(x) \f{dx^i}$, over the manifold is a section of the cotangent bundle, and consists of a [[1-form]] at each manifold point.

When a [[metric]] exists for the tangent bundle the [[frame]] basis forms, $\f{e^\al} = \f{dx^i} \lp e_i \rp^\al$, may alternatively be used as local fiber basis elements for the cotangent bundle. The transition functions are then [[Lorentz transformations|Lorentz rotation]], $\f{e^\al_2} = \lp L^{21} \rp_\be{}^\al \f{e^\be_1}$. This [[reduction of the structure group]] is the same as for the tangent bundle. Similarly, through equating the Lorentz transition functions and using the [[frame]], $\ve{e} \f{e^\al} = \ga^\al$, the cotangent bundle may be [[associated]] to the [[Clifford vector bundle]].

Since the cotangent bundle is dual to the tangent bundle, its geometric elements -- including the [[cotangent bundle connection]], holonomy, curvature, etc. -- are the dual constructions to those for the tangent bundle, and provide no new geometric insight.

Grade $p$ [[differential form]] fields are sections of the ''p-form bundle'', $\Omega^p M$, which has the $\frac{n!}{\left(n-p\right)!p!}$ [[coordinate basis p-forms|coordinate basis forms]], $\nf{dx^{i \dots j}}=\f{dx^i} \dots \f{dx^j}$, as basis. The combined collection of these p-form product bundles is the ''differential form bundle'', $\Omega M = \bigoplus_{p=0}^{n} \Omega^{p} M$, having dimension $2^{n}$.
The [[vector bundle connection]] for the [[cotangent bundle]] is defined through the operation of the suitable [[vector bundle covariant derivative|vector bundle connection]] on [[coordinate basis 1-forms]],
\begin{eqnarray}
\na_i \f{dx^j} &=& -\Ga^j{}_{ik} \f{dx^k} \\
\f{\na} \f{dx^j} &=& -\f{\Ga}^j{}_k \f{dx^k} 
\end{eqnarray}
The coefficients, $\Ga^j{}_{ik}$, of the ''cotangent bundle connection'', $\f{\Ga}^j{}_k = \f{dx^i} \Ga^j{}_{ik}$, are referred to as the [[Christoffel symbols]], and the index positions are arranged to agree with convention rather than having the usual order for connection coefficients. These are the same Christoffel symbols that arise in the [[tangent bundle connection]] since the basis elements are duals,
$$
0 = \na_i \de_j^k = \na_i \lp \ve{\pa_j} \f{dx^k} \rp
=  \lp \Ga^m{}_{ij} \ve{\pa_m} \rp \f{dx^k} - \ve{\pa_j} \lp \Ga^k{}_{im} \f{dx^m} \rp
= \Ga^k{}_{ij} - \Ga^k{}_{ij}
$$

An alternative expression for the cotangent bundle connection may be found by calculating the covariant derivative of the  [[vielbein 1-forms|frame]],
\begin{eqnarray}
\na_i \f{e^\al} &=& \lp \pa_i \lp e_j \rp^\al - \lp e_k \rp^\al \Ga^k{}_{ij} \rp \f{dx^j} = w_{i\be}{}^\al \f{e^\be} \\
\f{\na} \f{e^\al} &=& \f{d} \f{e^\al} - \f{dx^i} \f{dx^j} \lp e_k \rp^\al \Ga^k{}_{ij} = \f{d} \f{e^\al} - \ff{T^\al}  = \f{w}{}_\be{}^\al \f{e^\be}
\end{eqnarray}
in which the [[tangent bundle spin connection|tangent bundle connection]] coefficients, $ w_{i\be}{}^\al$, appear. This last equation, involving the [[torsion]] coefficients, $T^\al{}_{ij} = 2 \Ga^\al{}_{\lb ij \rb}$, may be solved for the spin connection coefficients by solving [[Cartan's equation]].

The cotangent bundle covariant derivative extends via the [[distributive rule|derivation]] to act on [[differential form]]s of higher order,
\begin{eqnarray}
\na_i \f{dx^j} \f{dx^k} \dots \f{dx^l} &=& \lp \na_i \f{dx^j} \rp \f{dx^k} \dots \f{dx^l} + \f{dx^j} \lp \na_i \f{dx^k} \rp \dots \f{dx^l} + \f{dx^j} \f{dx^k} \dots \lp \na_i \f{dx^l} \rp \\
&=& - \Ga^j{}_{im} \f{dx^m} \f{dx^k} \dots \f{dx^l} - \f{dx^j} \Ga^k{}_{im} \f{dx^m} \dots \f{dx^l} - \f{dx^j} \f{dx^k} \dots \Ga^l{}_{im} \f{dx^m}
\end{eqnarray}
The ''covariant derivative'' operator is a grade $1$ [[derivative|derivation]] of a [[fiber bundle]] section (field) that accounts for the local trivialization (change of basis) via the appropriate [[connection]]. It may be written as a [[1-form]] operator, or as a derivative with respect to a specific coordinate direction,
$$
\f{\na} = \f{dx^i} \na_i
$$
It is defined to have the following properties,
$$
\f{\na} \lp \nf{B} + f \nf{C} \rp = \f{\na} \,\nf{B} + f \f{\na} \, \nf{C} + \lp \f{d} f \rp \nf{C}
$$
where $f$ is any scalar [[function]] over the base manifold, $\f{d}$ is the [[exterior derivative]], and $\nf{B}$ and $\nf{C}$ are any tangent vector, differential form, Clifford element, or generally any fiber bundle section or direct product of sections. Using the [[partial derivative]] and connection, the covariant derivative of a section is
\begin{eqnarray}
\na_i B &=& \pa_i B + A_i B \\
\f{\na} B &=& \f{\pa} B + \f{A} B
\end{eqnarray}
A section is ''horizontal'' at a point iff its covariant derivative vanishes, $\f{\na} B = 0$.

In general, the covariant derivative may be defined for any fiber valued form that varies under a [[gauge transformation]] as $\nf{B'} = g \, \nf{B}$. Using the [[exterior derivative]],
$$
\f{\na} \nf{B} = \f{d} \nf{B} + \f{A} \nf{B}
$$
This operator generalizes further to a covariant derivative operating as forms that transform arbitrarily under the gauge group and aren't necessarily sections. For example, operating on the the [[curvature]], which transforms as $\ff{F'} = g \, \ff{F} \, g^-$, the covariant derivative is
$$
\f{\na} \ff{F} = \f{d} \ff{F} + \f{A} \ff{F} - \ff{F} \f{A}
$$
"Covariance" refers to the property that the covariant derivative of any field transforms under the same group action as the field. The covariant derivative thus plays an essential role in constructing gauge invariant objects, and this restriction provides the rule for the behavior of the connection under gauge transformation. When it is not obvious, the covariant derivative should be labeled with the symbol(s) of the connection(s) for the bundle for which it is covariant,
$$
\f{\na^A} C = \f{dx^i} \na^A_i C = \f{dx^i} \lp \pa_i C + A_i C \rp = \f{d} C + \f{A} C
$$
The curvature is perhaps the most important object characterizing the local geometry of a [[fiber bundle]] and [[connection]]. Its expression and action depends on the action of the structure group. Taking, for example, the structure group to act from the left, the ''curvature'' is then a local, [[Lie algebra]] (of the structure group) valued [[2-form|differential form]] defined as
\begin{eqnarray}
\ff{F}(x) &=& \ha \f{dx^i} \f{dx^j} F_{ij}{}^B T_B \\
&=& \f{d} \f{A} + \f{A} \f{A} \\
&=& \f{d} \f{A} + \f{A} \times \f{A} \\
&=& \f{d} \f{A} + \ha \lb \f{A}, \f{A} \rb
\end{eqnarray}
The curvature coefficients are
$$
F_{ij}{}^C = \pa_i A_j{}^C - \pa_j A_i{}^C + A_i{}^A A_j{}^B C_{AB}{}^C 
$$
in which $C_{AB}{}^C$ are the [[structure constants|Lie algebra]].

The curvature is most intuitively derived in terms of the [[holonomy]] of infinitesimal loops.

It may also be derived by applying the [[covariant derivative]] twice to any fiber bundle section,
$$
\f{\na} \f{\na} C = \lp \f{d} + \f{A} \rp \lp \f{d} + \f{A} \rp C =
\f{d} \f{d} C + \f{d} \f{A} C + \f{A} \f{d} C + \f{A} \f{A} C
 = \lp \f{d} \f{A} + \f{A} \f{A} \rp C = \ff{F} C
$$ 

The curvature changes under [[gauge transformation]]s, $C \mapsto C'=gC$, as $\ff{F} \mapsto \ff{F'}=g \ff{F} g^-$.

Note again that the expression of the curvature, and its action, depends on the form of the group action.
[[Contracting|vector-form algebra]] the [[coordinate basis vectors]] with the [[Ricci curvature]] and the [[tangent bundle]] [[metric]] gives the ''curvature scalar'' (//''Ricci scalar''//),
$$
R = g^{im} \ve{\pa_i} \f{R}{}_m = R^i{}_i
$$
This equals a full contraction of the [[Riemann curvature]] tensor,
$$
R = g^{mj} R_{ij}{}^i{}_m = 2 g^{mj} \lp \pa_{\lb i \rd} \Ga^i{}_{\ld j \rb m} + \Ga^i{}_{\lb i \rd l} \Ga^l{}_{\ld j \rb m} \rp
$$
In terms of the [[tangent bundle spin connection|tangent bundle connection]] and [[frame]], the curvature scalar is
$$
R = \ve{e^\al} \f{R}{}_\al = \ve{e^\al} \ve{e_\be} \ff{R}^\be{}_\al
= 2 \lp e_\al \rp^j \lp e_\be \rp^i \lp \pa_{\lb i \rd} w_{\ld j \rb}{}^\be{}_\al + w_{\lb i \rd}{}^\be{}_\ga w_{\ld j \rb}{}^\ga{}_\al \rp
$$
If the spin connection is torsionless, the curvature scalar may also be written as
\begin{eqnarray}
R = R_\al{}^\al &=& 2 \pa_\be w_\al{}^{\be \al} + w_{\be \al}{}^\ep w_\ep{}^{\be \al} - w_\be{}^{\ep \be} w_{\al \ep}{}^\al
\end{eqnarray}
''De Sitter spacetime'' is the unique geometry of a [[spacetime]], $M$, satisfying [[Einstein's equation]] with no matter and a positive cosmological constant, $\La$. It may be embedded in the five dimensional, flat, Lorentzian $(\et_{00}=1)$ spacetime, $\mathbb{R}(1,4)$ -- in which it is a [[hyperboloid|http://en.wikipedia.org/wiki/Hyperboloid]] of one sheet:
$$
x^0 x^0 - x^w x^w = - \al^2
$$
The spaces corresponding to each time, $x^0=t$, are [[3-sphere]]s -- growing larger for $t>0$ and for $t<0$. De Sitter spacetime may also be described as a [[homogeneous space]], $M = SO(1,4)/SO(1,3)$.

The geometry is succinctly expressed by the [[frame]],
$$
\f{e} = \f{dt} \ga_0 + e^{\fr{t}{\al}} \f{d x^\pi} \ga_\pi
$$
with flat spatial sections at each $x^0 = t$, and the $\pi$ [[index|indices]] ranging over $\{1,2,3\}$. The coframe is
$$
\ve{e} = \ga^0 \ve{\pa_t} + e^{-\fr{t}{\al}} \ga^\pi \ve{\pa_\pi}
$$
The [[torsion]]less [[spin connection]] for the [[Clifford vector bundle]], found by solving [[Cartan's equation]], $0=\f{d} \f{e} + \f{\om} \times \f{e}$, is
\begin{eqnarray}
\f{\om} &=& - \ve{e} \times \f{d} \f{e} + \fr{1}{4} \lp \ve{e} \times \ve{e} \rp \lp \f{e} \cdot \f{d} \f{e} \rp \\
&=& -\fr{1}{\al} e^{\fr{t}{\al}} \f{d x^\pi} \ga_{0 \pi}
\end{eqnarray}
The [[Clifford-Riemann curvature]] is
\begin{eqnarray}
\ff{R} &=& \f{d} \f{\om} + \ha \f{\om} \, \f{\om} \\
&=& - \fr{1}{\al^2} e^{\fr{t}{\al}} \f{d t} \f{d x^\pi} \ga_{0 \pi} 
 - \fr{1}{2 \al^2} e^{\fr{2 t}{\al}} \f{d x^\rh} \f{d x^\pi} \ga_{\rh \pi} \\
&=& - \fr{1}{2 \al^2} \f{e} \f{e}
\end{eqnarray}
The [[Clifford-Ricci curvature]] is
$$
\f{R} = \ve{e} \times \ff{R} = - \fr{1}{2 \al^2} \ve{e} \times  \f{e} \f{e} = - \fr{3}{\al^2} \f{e} = - \La \f{e}
$$
showing that the de Sitter spacetime satisfies the vacuum [[Einstein's equation]] with positive cosmological constant, $\La = \fr{3}{\al^2}$.

Since our universe appears to have a positive cosmological constant, at large $t$ this dominates the matter content and our universe is well approximated by a de Sitter spacetime at large $t$. In such a universe, the distant galaxies will accelerate away from us until, at the ''de Sitter horizon'', $r=\al$, they are receding from us faster than the speed of light -- so their light cannot reach us. Our galaxy appears to have a lonely future.

Ref:
*http://en.wikipedia.org/wiki/De_Sitter_space
*[[The Case for a Gravitational de Sitter Gauge Theory|papers/9610068.pdf]]
**Overview of how a Poincare gauge theory description of gravity, which has no Lagrangian formulation, needs to have terms added in order to be renormalizeable. These terms turn out to produce de Sitter gauge theory, with Lagrangian.
*[[Some Implications of the Cosmological Constant to Fundamental Physics|papers/0702065.pdf]]
Any graded operator, $\nf{D}$, that acts distributively over [[products|vector-form algebra]] of [[differential form]]s of grades $f$ and $g$ according to the ''graded Liebniz rule'',
$$
\nf{D} \nf{F} \nf{G} = \lp \nf{D} \nf{F} \rp \nf{G} + \lp -1 \rp^{df} \nf{F} \nf{D} \nf{G}
$$
is a ''graded derivation'' (or simply //''derivation''//) of grade $d$.

The most general grade $d$ derivation operator may be written as
$$
\nf{D} = {\cal L}_{\nf{\ve{K}}} + \nf{\ve{L}}
$$
in which $\nf{\ve{K}}$ is a vector valued $d$-form (a [[vector valued form]] of total grade $(d-1)$) and $\nf{\ve{L}}$ is a vector valued $(d+1)$-form (a vector valued form of grade $d$) and ${\cal L}$ is the [[FuN derivative]].
The ''determinant'' of a real $n\times n$(square) matrix, such as the [[frame]] matrix, $A_i{}^\al$, is a real number equal to
\begin{eqnarray}
\ll A \rl &=& \det ( A_i{}^\al ) = \va^{ij\dots k} A_i{}^0 A_j{}^1 \dots A_k{}^{n-1} \\
&=& \va^{ij\dots k} A_i{}^\al A_j{}^\be \dots A_k{}^\ga \fr{1}{n!} \ep_{\al \be \dots \ga} \\
&=& A_0{}^\al A_1{}^\be \dots A_{n-1}{}^\ga \ep_{\al \be \dots \ga}
\end{eqnarray}
(using [[permutation symbol]]s)

The determinant of an [[exponentiated|exponentiation]] matrix is the exponential of the [[trace]] of the matrix,
$$
\ll e^A \rl = e^{\li A \ri}
$$
*<<slider chkSlidernatF natF 'nat >' 'natural operators, vectors, forms'>>
*<<slider chkSliderfbF fbF 'fb >' 'fiber bundles'>>
*<<slider chkSliderpbF pbF 'pb >' 'principal bundles'>>
*<<slider chkSliderhamF hamF 'ham >' 'Hamiltonian dynamics, symplectic geometry'>>
*<<slider chkSliderssF ssF 'ss >' 'homogeneous spaces'>>
*<<slider chkSlidercartanF cartanF 'cartan >' 'Cartan geometry'>>
<<ListTagged dg>>
A ''diffeomorphism'' is a smooth invertible map from one [[manifold]] to another, or to itself. A diffeomorphism from a manifold to itself is also an [[automorphism]], so should probably be called an autodiffeomorphism -- but it's usually just called a diffeomorphism of a manifold.

An ''autodiffeomorphism'', $\ph : x \mapsto y$, typically maps points, $x$, of a manifold, $M$, to different points, $y$, of $M$. Typically each of these points is identified by coordinates, $x^i$ and $y^i$, on manifold patches, and a diffeomorphism is written as $y^i(x)$ or as $\phi^i(x)$. If these points are all the same, $x=y \forall x$, it is a passive diffeomorphism, doing nothing but [[changing the coordinates|coordinate change]].

Differential forms [[pull back|pullback]] and tangent vectors push forward under a diffeomorphism; both pull back and push forward under an autodiffeomorphism.
A ''differential form'', or //''p-form''//, or //''grade p form''// is a geometric object acting antisymmetrically on $p$ [[tangent vector]]s at a point to give a real number.  It generalizes [[1-form]]s, and may be visualized as a $p$ dimensional volume element sitting at a [[manifold]] point.  A ''2-form'' may be written in terms of real coefficients times the [[coordinate basis forms]] as
\[ \ff{a} = \ha a_{ij} \f{dx^i}\f{dx^j} \]
Such a 2-form may be visualized as an infinitesimal area element.  The coefficients are [[antisymmetric|index bracket]] in the indices,
\[ a_{ij} = a_{\lb ij \rb} = \ha \lp a_{ij} - a_{ji} \rp \]
A general p-form may be written as
\[ \nf{b} = \fr{1}{p!} b_{i \dots k} \f{dx^i} \dots \f{dx^k} \]
The ''vector-form decoration'' convention requires that [[tangent vector]]s have an over-arrow, while grade $p$ forms have $p$ under-arrows or an under-bar if $p$ is unspecified or greater than 2.  Multi-tangent vectors of grade $p$, which arise in [[vector-form algebra]], may be referred to as ''(-p)-form''s and are decorated with $p$ over-arrows or an over-bar. Unlike the case for [[Clifford element]]s, no use is made of differential forms of mixed grade. In an $n$ dimensional manifold, the highest grade form is an $n$-form,
\[ \nf{z} = z_v \f{dx^0} \dots \f{dx^{n-1}} = z_v \nf{d^n x} \]
in which $\nf{d^n x}$ is the coordinate basis n-form. Also, technically, a real number at a manifold point is a 0-form.

Any differential form may also be written in terms of the [[frame]] basis forms as
\[ \nf{b} = \fr{1}{p!} b_{\al \dots \be} \f{e^\al} \dots \f{e^\be} \]
<<ListTagged dirac>>
Consider an $n$ dimensional [[manifold]] and its [[tangent bundle]]. At each manifold point, $x$, the tangent bundle fiber, $V_x = T_x M$, is a vector space spanned by the $n$ [[coordinate basis vectors]], $\ve{\pa_i} \in V_x$. An $m$ dimensional subspace, $V^s_x \subset V_x$, can be spanned at each manifold point by a set of $m$ linearly independent basis vectors, $\ve{s_a}$. A collection of such subspaces, one defined at each manifold point, is a ''distrubution'', 
$$
\ve{\De} = \left\{ \ve{s_1}, \ve{s_2}, \dots, \ve{s_m} \right\}
$$
specified by $m$ linearly independent vector fields, $\ve{s_a}(x)$. A distribution is a subbundle of the tangent bundle.

A distribution is ''involutive'' (//in ''involution''//) iff the basis vector fields of the distribution close under the [[Lie bracket|Lie derivative]],
$$
\lb \ve{s_a}, \ve{s_b} \rb_L \in V^s
$$
This is sometimes written as $\lb \ve{\De}, \ve{\De} \rb_L = \ve{\De}$. An involutive distribution may be integrated to give the ''foliation'' of the manifold by the collection of $m$ dimensional [[submanifold]]s which have $V^s_x$ as their tangent vector space at each point.

The ''divergence'' of a [[vector field|tangent bundle]], $\ve{v}$ is a function describing how much the vector field is spreading away from (or converging towards) each point. The divergence operator takes a vector field as argument and returns a real valued field, and is defined implicitly by the [[Lie derivative]] of the [[volume form]] along the vector field,
$$
{\cal L}_{\ve{v}} \nf{e} = \f{d} \lp \ve{v} \nf{e} \rp = \nf{e} \, \mathrm{div}(\ve{v})
$$
In terms of the [[tangent bundle covariant derivative|tangent bundle connection]] it is
$$
\mathrm{div}(\ve{v}) = D_i v^i = \pa_i v^i + v^j \Ga^i{}_{ij} = D_\al v^\al = \pa_\al v^\al + v^\al \om_\be{}^\be{}_\al  
$$
This involves the [[trace]] of the [[Christoffel symbols]] or [[spin connection]], which can be used to produce an important formula relating the divergence to the [[partial derivative]] of the [[frame determinant|volume form]],
$$
\pa_i \ll e \rl v^i = \ll e \rl D_i v^i
$$
A ''division algebra'' over a field is an algebra (having addition and multiplication) that also has division, so that for any $a$ and non-zero $b$, there is a unique $x$ such that $a=b x$, and a unique $y$ such that $a=y b$. The possible normed division algebras are the reals, complex numbers, split-complex numbers, [[quaternion]]s, split-quaternions, [[octonion]]s, and split-octonions. 
The idea that Lorentz transformations should be modified to preserve a constant minimum length as well as the speed of light.

This makes sense, since it's basically a modification of special relativity to take place in a [[de Sitter spacetime]] instead of Minkowski space -- which is appropriate in a universe with a positive cosmological constant. But I don't see how the effect is going to be measurable in QFT, since this [[spacetime]] curvature is usually so small. Proponents of "DSR" claim the effect is increased locally by large local energy density. But it seems to me like a hack -- and what we really want to do is QFT in an arbitrarily curved spacetime.

Good introductory paper:
http://arxiv.org/abs/gr-qc/0207085

Speculation:
A momentum cutoff... maybe the momentum is a closed manifold rather than a plane, similar to how the tangent space in [[Cartan geometry]] is a curved surface rather than a plane. Ah, this is supported by these papers:
http://arxiv.org/abs/hep-th/0207279
http://arxiv.org/abs/gr-qc/0612093
and mentioned here:
http://math.ucr.edu/home/baez/week232.html
Hey, does Derek's paper on [[Cartan geometry]] mention that?

ah, here, relation to [[de Sitter gravity]]:
*[[de Sitter special relativity|paper/0606122.pdf]]
**wow, the connection between this and a minimal length is rather tenuous -- it relies on the assumption that a high energy process will change the local value of $\La$. Why would that happen? Should consider non-constant $\La$...
**basically, approximate the whole universe by de Sitter spacetime instead of by a flat Minkowski spacetime, and do particle physics in this universe the way it's normally done in Minkowski. This is less general than our reality, which is a bumpy spacetime.

Hmm, Lorentz transformations shouldn't need to be modified to preserve a minimal finite [[proper time]].
The rank $6$ exceptional [[Lie group]], [[E6]], is described by its $78$ dimensional [[Lie algebra]], ''e6''. This Lie algebra may be decomposed as a $45$ dimensional [[symmetric|symmetric space]] [[subgroup]], the [[special orthogonal group]] Lie algebra, $so(10)$, acting on the $32$ dimensional space of, real, positive, $Cl(10)$ [[spinor]]s, $S^{\lp10\rp}$, and a $u(1)_{PQ}$,
$$
e6 = so(10) + u(1) + S^{\lp10\rp}
$$
Also, using [[f4]] and its fundamental representation,
$$
e6 = f4 + 26
$$
The fundamental representation of $e6$ is $27$.
The rank $8$ exceptional [[Lie group]], [[E8]], is described by its $248$ dimensional [[Lie algebra]], ''e8''. This Lie algebra may be decomposed as a $120$ dimensional subalgebra, the [[spin Lie algebra]], $spin(16)$, acting on the $128$ dimensional representation space of, real, positive [[chiral]], [[Cl(16)]] [[spinor]]s, $S^{\lp16\rp+}$. In this way, any $e8$ element may be written in terms of basis generators as:
\begin{eqnarray}
E &=& B + \Ps = \ha b^{\al\be} \ga^{(16)+}_{\al\be} + \ps^a Q^+_a \\
&& \in spin(16)^+ + S^{(128)+} = {\rm Lie}(E8)
\end{eqnarray}
Explicitly, a $spin(16)$ element is expressed above as the first (upper left) quadrant of a Cl(16) bivector, $B = \ha b^{\al \be} \ga^{\lp16\rp+}_{\al \be}$, in a real, [[chiral]] [[Clifford matrix representation]], with $1 \le \al,\be \le 16$. This is a $128\times128$ real, antisymmetric matrix that is part of a $256\times256$ dimensional matrix of the Cl(16) rep. In terms of matrix components, with matrix indices $1 \le a,b \le 128$, this positive chiral part of the bivector is
$$
\lp B \rp^a{}_b = \ha b^{\al \be} \lp \ga^{\lp16\rp+}_{\al \be} \rp^a{}_b
$$
The $120$ unique, positive chiral, basis bivectors, $\ga^{\lp16\rp+}_{\al \be} \sim T_A$, are Lie algebra generators of $spin(16)$ and of $e8$, represented as $128\times128$ matrices. A positive chiral, real spinor, $\Ps = \ps^a Q^+_a$, is a column of $128$ real numbers on which these bivectors act. In terms of matrix components, these generators are $\lp Q^+_a \rp^b = \de_a^b$ and the spinor components are $\lp \Ps \rp^b = \ps^b$. The action of a bivector on a spinor gives a spinor, which can be written three different ways as:
\begin{eqnarray}
\ps'^{a} &=& \lp B \rp^a{}_b \psi^b \\
\ps'^{d} \lp Q^+_d \rp^a &=& \ha b^{\al \be} \lp \ga^{\lp16\rp+}_{\al \be} \rp^a{}_b \psi^{c} \lp Q^+_c \rp^b \\
\Ps' = \ps'^c Q^+_c &=& B \Ps = \ha b^{\al \be} \psi^{c} \ga^{\lp16\rp+}_{\al \be} Q^+_c
\end{eqnarray} 
The $248$ dimensional Lie algebra, e8, is spanned by these two sets of generators. The Lie brackets between bivectors, and between bivector generators, are determined by a [[Clifford basis product identity|Clifford basis product identities]],
\begin{eqnarray}
\lb B_1, B_2 \rb &=& B_1 B_2 - B_2 B_1 \\
\lb \ga^{\lp16\rp+}_{\al \be}, \ga^{\lp16\rp+}_{\ga \de} \rb &=& 2 \left\{ - \et_{\al \ga} \ga^{\lp16\rp+}_{\be \de} + \et_{\al \de} \ga^{\lp16\rp+}_{\be \ga} + \et_{\be \ga} \ga^{\lp16\rp+}_{\al \de} - \et_{\be \de} \ga^{\lp16\rp+}_{\al \ga} \right\}
\end{eqnarray}
giving the same structure constants as for the spin Lie algebra,
$$
C_{\lb\al\be\rb\lb\ga\de\rb}{}^{\lb\ep\up\rb} = 2 \left\{ - \et_{\al \ga} \de^{\lb\ep \up\rb}_{\be \de} + \et_{\al \de} \de^{\lb\ep \up\rb}_{\be \ga} + \et_{\be \ga} \de^{\lb\ep \up\rb}_{\al \de} - \et_{\be \de} \de^{\lb\ep \up\rb}_{\al \ga} \right\}
$$
in which the appropriate Clifford algebra metric for Cl(16,0) is $\et_{\al \be} = \de_{\al \be}$. The Lie brackets between bivector and spinor, and between their generators, are
\begin{eqnarray}
\lb B, \Ps \rb &=& B \Ps \\
\lb \ga^{\lp16\rp+}_{\al \be}, Q^+_a \rb &=& \ga^{\lp16\rp+}_{\al \be} Q^+_a = \lp \ga^{\lp16\rp+}_{\al \be} \rp^b{}_c \lp Q^+_a \rp^c Q^+_b
\end{eqnarray}
giving structure constants:
$$
C_{\lb\al\be\rb a}{}^{b} = \lp \ga^{\lp16\rp+}_{\al \be} \rp^b{}_a = - \lp \ga^{\lp16\rp+}_{\al \be} \rp_a{}^b
$$
Finally, the e8 Lie algebra description is completed by letting the structure constants be completely antisymmetric -- the [[Killing form]] identity,
$$
C_{ab}{}^{\lb\al\be\rb} = C^{\lb\al\be\rb}{}_{a b}  = \lp {\ga^{\lp16\rp+}}^{\al \be} \rp_{ba} = - \lp {\ga^{\lp16\rp+}}^{\al \be} \rp_{ab}
$$
giving the Lie brackets between spinor generators,
\begin{eqnarray}
\lb Q^+_a, Q^+_b \rb &=& - \lp {\ga^{\lp16\rp+}}^{\al \be} \rp_{ab} \ga^{\lp16\rp+}_{\al \be} \\
\lb \Ps_1, \Ps_2 \rb &=& - \ps_1^a \ps_2^b \lp {\ga^{\lp16\rp+}}^{\al \be} \rp_{ab} \ga^{\lp16\rp+}_{\al \be}
\end{eqnarray}

To summarize the above expressions, if elements of $e8$ are expressed as a combinations of $16\times16$ antisymmetric matrices of coefficients, $b = - b^T$, and $128$ elements columns, $\ps$, then the $e8$ Lie brackets can be defined heuristically in terms of matrix operations between these elements as:
\begin{eqnarray}
\lb b_1, b_2 \rb_{e8} &=& 2 \lp b_1 \et b_2 - b_2 \et b_1 \rp \\
\lb b, \ps \rb_{e8} &=& \big< \ha b \ga^{(16)+} \big> \ps \\
\lb \ps_1, \ps_2 \rb_{e8} &=& - 2 \big( \ps_1^T \ga^{(16)+} \ps_2 \big)_B
\end{eqnarray}

The Killing form for e8 is
\begin{eqnarray}
g_{\lb \al \be \rb \lb \ga \de \rb} &=&
C_{\lb \al \be \rb \lb \ep \up \rb}{}^{\lb \ze \et \rb} C_{\lb \ga \de \rb \lb \ze \et \rb}{}^{\lb \ep \up \rb}
+ C_{\lb \al \be \rb a}{}^b C_{\lb \ga \de \rb b}{}^a
= 240 \lp \et_{\al \de} \et_{\be \ga} - \et_{\al \ga} \et_{\be \de} \rp \\
g_{ab} &=& 2 C_{a \lb \al \be \rb}{}^c C_{b c}{}^{\lb \al \be \rb}
= 2 \lp \ga^{\lp16\rp+}_{\al \be} \rp_a{}^c \lp {\ga^{\lp16\rp+}}^{\al \be} \rp_{cb} = 2 \lp \de_\al^\be \de_\be^\al - \de_\al^\al \de_\be^\be \rp \de_{ab} = - 480 \, \de_{ab}  
\end{eqnarray}

Since we use a real representation for Cl(16,0), the above describes the compact real form of E8. If we use a complex rep for Cl(16,0), we get a complex form of e8. And if we use a Clifford algebra of different signature, like [[Cl(1,15)]], we get a (real or complex) non-compact E8. For all of these choices, the above structure constants remain symbolically the same, with the appropriate choice of $\et_{\al \be}$. Note though that it is always necessary to choose a chiral rep for Cl.

The $e8$ Lie algebra has many subalgebras other than the $spin(16)$ which has been discussed above. Two maximal subgroups of $E8$ are $(E7 \times SU(2))/(\mathbb{ Z}/2 \mathbb{ Z})$ and $(E6 \times SU(3))/(\mathbb{ Z}/3 \mathbb{ Z})$ -- involving the other exceptional groups, [[E7]] and [[E6]], and the special unitary groups, [[SU(3)]] and [[SU(2)]]. One particularly interesting way $e8$ can be broken down is:
\begin{eqnarray}
e8 &=& e6 + su(3) + 54 \times 3 \\
 &=& so(10) + u(1) + 32 + su(3) + 54 \times 3\\
 &=& so(4) + su(2) + su(2) + u(1) + 4 \times 8 + u(1) + 32 + su(3) + 54 \times 3
\end{eqnarray}
Yet another way $e8$ can be broken up is via the [[e8 triality decomposition]]:
\begin{eqnarray}
e8 &=& spin(8) + spin(8) + V^{(8)} \! \times \! V^{(8)} + S^{(8)+} \! \times \! S^{(8)+} + S^{(8)-} \! \times \! S^{(8)-} \\
 &=& so(4) + so(4) + 4 \times 4 + so(6) + so(2) + 6 \times 2  + 3 \times 8 \times 8 \\
 &=& so(4) + su(2) + su(2) + 4 \times 4 + su(4) + u(1) + 6 \times 2  + 3 \times 8 \times 8
\end{eqnarray}
I'm currently trying to use these to build a [[T.O.E.|theory of everything]]

Calculate e8 structure constants from those of two [[f4]]'s.

Ref:
*[[http://en.wikipedia.org/wiki/E8_(mathematics)|http://en.wikipedia.org/wiki/E8_(mathematics)]]
*G,S,&W, [[Superstring Theory|http://www.amazon.com/Superstring-Cambridge-Monographs-Mathematical-Physics/dp/0521357527/ref=pd_bbs_sr_3/104-9709999-3726336?ie=UTF8&s=books&qid=1179001057&sr=8-3]]
*J. F. Adams, [[Lectures on Exceptional Lie groups|http://www.amazon.com/gp/reader/0226005267/ref=sib_dp_pt/104-6593454-7361512#reader-link]]
*S. Adler
**[[Should E8 SUSY Yang-Mills be Reconsidered as a Family Unification Model?|http://arxiv.org/abs/hep-ph/0201009]]
*P. Ramond
**[[Exceptional Groups and Physics|http://arxiv.org/abs/hep-th/0301050]]
[[e8]] [[Lie algebra structure]]

@@display:block;text-align:center;[img[images/png/e8 root system.png]]@@

Ref:
*[[David Richter|http://homepages.wmich.edu/~drichter/]]
**[[Triacontagonal coordinates for the E8 root system|papers/0704.3091.pdf]]
**[[Gosset's figure in a Clifford algebra|papers/gossetfigurecliffordalgebra2004.pdf]]
*Mark W Hopkins? (sci.physics.research poster)
**[[Standard Model|papers/MH Standard Model.pdf]]
*Richard Koch
**[[HyperSolids|http://www.uoregon.edu/~koch/hypersolids/hypersolids.html]]
The [[e8]] [[Lie algebra]], $e8={\rm Lie}(E8)$, corresponding to the [[Lie group]], [[E8]], breaks into a $120$ dimensional $spin(16)$ [[spin Lie algebra]] acting on a $128$ dimensional [[chiral]] [[Cl(16)]] [[spinor]], $S^{\lp16\rp+}$. However, this $spin(16)$ can be decomposed into two $28$ dimensional $spin(8)$'s and a $64$ dimensional piece, related to two pieces of the $128=64+64$ spinor through [[triality]] -- a decomposition described by [[John Baez]] in [[TWF90|http://math.ucr.edu/home/baez/week90.html]]:
<<<
Emboldened with our success, we now look at the vector space

so(8) + so(8) + end(S+) + end(S-) + end(V)

Here end(S+) is the space of all linear transformations of the vector space S+, so if you like, it's just the space of 8x8 matrices. Similarly for end(S-) and end(V). Now the dimension of this space is

28   +  28    +   64   +   64    +   64   =  248

Hey! This is just the dimension of E8! Maybe this space is E8!

Yes indeed. Again, you can cook up a bracket operation on this space using all the stuff we've got. Here's the basic idea. end(S+), end(S-), and end(V) are already Lie algebras, where the bracket of two guys x and y is just the commutator [x,y] = xy - yx, where we multiply using matrix multiplication. Since so(8) has a representation as linear transformations of V, it has two representations on end(V), corresponding to left and right matrix multiplication; glomming these two together we get a representation of so(8) + so(8) on end(V). Similarly we have representations of so(8) + so(8) on end(S+) and end(S-). Putting all this stuff together we get a Lie algebra, if we do it right - and it's E8. At least that's what Kostant said; I haven't checked it.
<<<
We can build this by breaking up the $spin(16)^+ + S^{\lp16\rp+}$ generators and structure constants into the new ones. Letting the indices run $1 \le \al,\be \le 8$ and $1 \le a,b \le 8$, and using the [[chiral Cl(16) bivector|Cl(16)]] decomposition into [[Cl(8)]] elements using the [[Kronecker product]], we define the new set of e8 generators in terms of the old:
$$
\begin{array}{rclcccl}
H_{\al\be} &=& \ga^{\lp16\rp+}_{\al\be} &=& \Ga^+_{\al\be} \otimes 1 &\in& spin(8)^+ \otimes 1 \\
G_{\al\be} &=& \ga^{\lp16\rp+}_{\lp\al+8\rp\lp\be+8\rp} &=& P^{\lp8\rp}_+ \otimes \Ga_{\al\be} &\in& 1 \otimes spin(8) \\
\Ps^I_{\al\be} &=& \ga^{\lp16\rp+}_{\al\lp\be+8\rp} &=& -\Ga^+_\al \otimes \Ga_\be &\in& V^{\lp8\rp+} \otimes V^{\lp8\rp}\\
\Ps^{II}_{ab} &=& Q^+_{16\lp a-1\rp+b} &=& q^+_a \otimes q^+_b &\in& S^{\lp8\rp+} \otimes S^{\lp8\rp+}\\
\Ps^{III}_{ab} &=& Q^+_{16\lp a-1\rp+b+8} &=& q^+_a \otimes q^-_b &\in& S^{\lp8\rp+} \otimes S^{\lp8\rp-}
\end{array}
$$
in which $P^{\lp8\rp}_+ = \ha \lp 1 + \Ga \rp$ is the positive chirality projector for Cl(8), giving $\Ga^+_{\al\be} = P^{\lp8\rp}_+ \Ga_{\al\be}$ and $\Ga^+_{\al} = P^{\lp8\rp}_+ \Ga_{\al}$, and $q^\pm_a$ are positive and negative chiral Cl(8) spinors. Since this is just a re-labeling, the new Lie brackets (and structure constants) come from the old structure constants.

Since the triality decomposition of $e8$,
$$
e8 = spin(8) + spin(8) + V^{(8)} \! \times \! V^{(8)} + S^{(8)+} \! \times \! S^{(8)+} + S^{(8)-} \! \times \! S^{(8)-}
$$
close matches the triality decomposition of [[f4]],
$$
f4 = spin(8) + V^{(8)} + S^{(8)+} + S^{(8)-}
$$
the easiest way to obtain the $e8$ structure constants is by combining the structure constants of two $f4$'s. Explicitly,
\begin{eqnarray}
\lb \ga_{\al \be}, \ga_{\ga \de} \rb &=& 2 \left\{ - \et_{\al \ga} \ga_{\be \de} + \et_{\al \de} \ga_{\be \ga} + \et_{\be \ga} \ga_{\al \de} - \et_{\be \de} \ga_{\al \ga} \right\} \\
\lb \ga_{\al' \be'}, \ga_{\ga' \de'} \rb &=& 2 \left\{ - \et_{\al' \ga'} \ga_{\be' \de'} + \et_{\al' \de'} \ga_{\be' \ga'} + \et_{\be' \ga'} \ga_{\al' \de'} - \et_{\be' \de'} \ga_{\al' \ga'} \right\} \\
\lb \ga_{\al \be}, \ga_{\ga \de'} \rb &=& 2 \left\{ \et_{\be \ga} \ga_{\al \de'} - \et_{\al \ga} \ga_{\be \de'} \right\} \\
\lb \ga_{\al' \be'}, \ga_{\ga \de'} \rb &=& 2 \left\{ \et_{\be' \de'} \ga_{\ga \al'} - \et_{\al' \de'} \ga_{\ga \be'} \right\} \\
\lb \ga_{\al \be}, Q^+_{ab'} \rb &=& ( \ga^+_{\al \be} )^b{}_a Q^+_{bb'} \\
\lb \ga_{\al' \be'}, Q^+_{ba'} \rb &=& ( \ga^+_{\al' \be'} )^{b'}{}_{a'} Q^+_{bb'} \\
\lb \ga_{\al \be}, Q^-_{ab'} \rb &=& ( \ga^-_{\al \be} )^b{}_a Q^-_{bb'} \\
\lb \ga_{\al' \be'}, Q^-_{b'a} \rb &=& ( \ga^-_{\al' \be'} )^{b'}{}_{a'} Q^-_{bb'} \\
\lb \ga_{\al \al'}, \ga_{\be \be'} \rb &=& -2 \ga_{\al \be} \et_{\al' \be'} - 2 \et_{\al \be} \ga_{\al' \be'} \\
\lb Q^+_{aa'}, Q^+_{bb'} \rb &=& \ha ( \ga^+{}^{\al \be} )_{ab} \ga_{\al \be} g_{a'b'} + \ha g_{ab} ( \ga^+{}^{\al' \be'} )_{a'b'} \ga_{\al' \be'} \\
\lb Q^-_{aa'}, Q^-_{bb'} \rb &=& \ha ( \ga^-{}^{\al \be} )_{ab} \ga_{\al \be} g_{a'b'} + \ha g_{ab} ( \ga^-{}^{\al' \be'} )_{a'b'} \ga_{\al' \be'} \\
\lb \ga_{\al\al'}, Q^+_{aa'} \rb &=& ( \bar{\ga}_\al )^b{}_a ( \bar{\ga}_{\al'} )^{b'}{}_{a'} Q^-_{bb'} \\
\lb \ga_{\al\al'}, Q^-_{aa'} \rb &=& - ( \ga_\al )^b{}_a ( \ga_{\al'} )^{b'}{}_{a'} Q^+_{bb'} \\
\lb Q^+_{aa'}, Q^-_{bb'} \rb &=& - ( \ga^\al )_{ab} ( \ga^{\al'} )_{a'b'} \ga_{\al\al'}
\end{eqnarray}
in which the greek indices are raised and lowered by the Clifford signature metrics, $\eta_{\al \be}$ and $\eta_{\al' \be'}$, and the latin spinor indices are raised and lowered by the appropriate spinor metric, $g_{ab}$ or $g_{a'b'}$. Note that this description holds for the compact, split, or quaternionic real forms of $e8$, via appropriate choice of $spin(4,4)$ or $spin(8)$.

Ref:
*[[John Baez]]
**http://math.ucr.edu/home/baez/week90.html
**[[Octonions|papers/oct.pdf]]
*Barton and Sudbery
**[[Magic Squares and Matrix Models of Lie algebras|papers/0203010v2.pdf]]
<<ListTagged editing>>
Any real $n \times n$ (square) matrix, $A_i{}^j$, has a corresponding set of $n$ complex ''eigenvalues'', $\la_\al$, ''right eigenvectors'', $\lp r_i \rp^\al$, and ''left eigenvectors'', $\lp l_\al \rp^i$, satisfying the ''eigenequations'',
\begin{eqnarray}
A_i{}^j \lp r_j \rp^\be &=& \lp r_i \rp^\al \La_\al{}^\be\\
\lp l_\al \rp^i A_i{}^j &=& \La_\al{}^\be \lp l_\be \rp^j
\end{eqnarray}
in which $\La_\al{}^\be$ is the diagonal matrix with the eigenvalues, $\la_\al$, on the diagonal,
$$
\La = \lb
\begin{array}{cccc}
\la_1 & 0 & \dots & 0\\
0 & \la_2 & \dots & 0\\
\vdots & \vdots & \ddots & \vdots\\
0 & 0 & \dots & \la_n\\
\end{array}
\rb
$$
and the left eigenvectors, $l_\al = \lp l_\al \rp^i b_i$, are presumed to be elements of some [[vector space]]. From the the above equations, the eigenvalues are the roots of the ''characteristic polynomial'',
$$
0 = p_A(\la) = \ll \la - A \rl = \det \lp \la \de_\al^\be - A_\al{}^\be \rp
$$
using the [[determinant]]. The matrix is called ''singular'' iff one of the eigenvalues is $0$ and ''degenerate'' iff two or more of the eigenvalues are the same, with each eigenvalue having a ''multiplicity'', $m$. If $A$ is real and symmetric, $A^T = A$, ($A_i{}^j = A^j{}_i$), or complex and Hermitian, $A^\dagger = A$, ($A^*_i{}^j = A^j{}_i$), the eigenvalues are real and the left and right eigenvectors are dual and orthonormal. If $A$ is real but not symmetric, the eigenvalues and eigenvectors are complex. The eigenvectors are only determined up to a scaling factor by the eigenequations. Or, if $A$ is degenerate, the $m$ eigenvectors corresponding to an eigenvalue need only span that $m$ dimensional ''eigenspace''.

A real matrix, $A$, is not ''defective'' iff the left or right eigenvectors span the entire $n$ dimensional vector space, in which case the left and right eigenvectors can be scaled to satisfy the ''normality conditions'',
$$
\lp l_\al \rp^i \lp r_i \rp^\be = \de_\al^\be
$$
With that scaling done, the matrix may be written as
$$
A_i{}^j = \lp r_i \rp^\be \La_\be{}^\al \lp l_\al \rp^j
$$
the ''spectral decomposition''. A symmetric or Hermitian matrix is never defective, and has $l = r^\dagger$. The determinant of a matrix equals the product of its eigenvalues, and its [[trace]] is their sum,
$$
\ll A \rl = \ll \La \rl = \prod_\al \la_\al \;\;\;\;\;\;\;\;\; \li A \ri = \li \La \ri = \sum_\al \la_\al
$$

If we are dealing with a [[vector valued 1-form|vector valued form]], $\f{\ve{A}} = \f{dx^i} A_i{}^j \ve{\pa_j}$, then the [[vector-form algebra]] gives the eigenequations for the ''eigenforms'' and //''eigenvectors''//,
\begin{eqnarray}
\f{\ve{A}} \f{r^\be} &=& \f{r^\al} \La_\al{}^\be\\
\ve{l_\al} \f{\ve{A}} &=& \La_\al{}^\be \ve{l_\be}
\end{eqnarray}
and, if the eigenvectors and eigenforms can be scaled to satisfy $\ve{l_\al} \f{r^\be} = \de_\al^\be$, the spectral decomposition is
$$
\f{\ve{A}} = \f{r^\be} \La_\be{}^\al \ve{l_\al}
$$
A [[differential form]] [[field|cotangent bundle]], $\nf{f}(x)$, over a [[manifold]], $M$, is ''exact'' iff it is the [[exterior derivative]] of some other differential form field,
$$
\nf{f} = \f{d} \nf{g}
$$
The [[vector space]] of exact $p$-forms over $M$ is labeled $E^p$. Since the exterior derivative is nilpotent, all exact forms are [[closed]], $E^p \subset C^p$.
The algebraic structure of real, complex, [[quaternion]], and [[octonion]] [[division algebra]]s allows the existance of exceptional [[Lie algebra]]s:
| | $\mathbb{R}$ | $\mathbb{C}$ | $\mathbb{H}$ | $\mathbb{O}$ |
| $\mathbb{R}$ | [[su(2)]] | [[su(3)]] | [[sp(3)]] | [[f4]] |
| $\mathbb{C}$ | [[su(3)]] | 2[[su(3)]] | [[su(6)|special unitary group]] | [[e6]] |
| $\mathbb{H}$ | [[sp(3)]] | su(6) | [[so(12)|special orthogonal group]] | [[e7]] |
| $\mathbb{O}$ | [[f4]] | [[e6]] | [[e7]] | [[e8]] |
This construction relies on [[triality]] automorphisms. For example,
\begin{eqnarray}
su(3) &=& u(1) + u(1) + 2 + 2 +2 \\
sp(3) &=& su(2) + su(2) + su(2) + V^{(4)} + S^{(4)+} + S^{(4)-} \\
f4 &=& so(8) + V^{(8)} + S^{(8)+} + S^{(8)-} \\
e6 &=& u(1) + u(1) + so(8) + 2 \otimes V^{(8)} + 2 \otimes S^{(8)+} + 2 \otimes S^{(8)-} \\ 
e7 &=& su(2) + su(2) + su(2) + so(8) + V^{(4)} \otimes V^{(8)} + S^{(4)+} \otimes S^{(8)+} + S^{(4)-} \otimes S^{(8)-}
\end{eqnarray}
We can also see that $e7$ can be constructed from $sp(3)$ and $f4$. These constructions extend to noncompact versions of these Lie algebras by using the split-quaternions and [[split-octonion]]s.

Further details are available in:
*Barton and Sudbury, "Magic squares and matrix models of Lie algebras," http://arxiv.org/abs/math/0203010
Any algebraic element may be ''exponentiated'',
\[ e^A = 1 + A + \ha A A + \fr{1}{3!} A A A + \dots \]
Or, equivalently, ''exponentiation'' may be defined as
\[ e^A = \lim_{N \to \infty} \lp 1 + \fr{1}{N} A \rp^N \]

The derivative of a parameterized exponential is
\[ \fr{d}{dt} e^{tA} = A + t A^2 + \fr{1}{2!} t^2 A^3 + \dots = A e^{tA} = e^{tA} A \]
More rigorously, the solution of any set of first order ODE's,
$$
\fr{d}{dt} E = A E
$$
in which $A$ is a linear operator, is used to define the exponentiation of that operator,
$$
E(t) = e^{t A} E(0)
$$

If $A$ may be written in terms of an [[adjoint|Clifford adjoint]] operator and a diagonal matrix of [[eigen]]values, $A = U \La U^-$, then
\[ e^A = U e^\La U^- \]
is easily computed by exponentiating the eigenvalues.
The ''exterior derivative'' operator is the [[partial derivative]] operator, $\f{d}=\f{\pa}$, applied to [[differential form]]s,
$$
\f{d} \nf{A} = \f{\pa} \nf{A} = \lp \f{dx^i} \pa_i \rp \lp \f{dx^j} \dots \f{dx^k} \fr{1}{p!} A_{j \dots k} \rp = \f{dx^i} \f{dx^j} \dots \f{dx^k} \fr{1}{p!} \pa_i A_{j \dots k} 
$$
This operation is conventionally written as $dA$ but is written in this work using the under-arrow since it has a form grade of $1$. Even though it is defined using the un-natural partial derivative it is a [[natural]] (coordinate independent) operator since the non-tensor terms arising from [[coordinate change]], $x \mapsto x(y)$, vanish by the symmetry of partial derivation and antisymmetry of collections of forms,
$$
\f{d} \f{f} = \f{dx^i} \f{dx^j} \pa^x_i f_j = \f{dy^k} \f{dy^m} \fr{\pa x^i}{\pa y^k} \fr{\pa x^j}{\pa y^m} \fr{\pa y^n}{\pa x^i} \pa^y_n f_j = \f{dy^k} \f{dy^m} \lp \pa^y_k \fr{\pa x^j}{\pa y^m} f_j - \fr{\pa^2 x^j}{\pa y^k \pa y^m} f_j \rp = \f{dy^k} \f{dy^m} \pa^y_k f'_m = \f{d'} \f{f'}
$$
This doesn't work for the partial derivative applied to [[tangent vector]]s, so there is no such thing as the exterior derivative of a [[vector valued form]]. The exterior derivative of a scalar function, $\f{d} f = \f{dx^i} \pa_i f$, is called the ''gradient'' in old fashioned vector calculus, while the exterior derivative of a 1-form, $\f{d} \f{f}$, is associated to the ''curl'' in three dimensional space.

The operator is nilpotent,
$$
\f{d} \f{d} = \f{dx^i} \pa_i \f{dx^j} \pa_j = \f{dx^i} \f{dx^j} \pa_i \pa_j = 0 
$$
and, as a grade $1$ [[derivation]], distributes over the [[product|vector-form algebra]] of a $f$-form, $\nf{F}$, and $g$-form, $\nf{G}$, via the graded Liebniz rule,
$$
\f{d} \lp \nf{F} \nf{G} \rp = \lp \f{d} \nf{F} \rp \nf{G} + \lp -1 \rp^f \nf{F} \lp \f{d} \nf{G} \rp
$$
But it does not distribute over the product of vectors and forms, since instead
$$
\f{d} \lp \ve{v} \nf{G} \rp = \f{\pa} \lp \ve{v} \nf{G} \rp = \lp \f{\pa} \ve{v} \rp \nf{G} - \ve{v} \lp \f{d} \nf{G} \rp + \lp \ve{v} \f{\pa} \rp \nf{G} = {\cal L}_{\ve{v}} \nf{G} - \ve{v} \lp \f{d} \nf{G} \rp 
$$
in which the pair of unnatural terms reassemble into the natural [[Lie derivative]].
The rank $4$ exceptional [[Lie group]], [[F4]], is described by its $52$ dimensional [[Lie algebra]], ''f4''. This Lie algebra may be decomposed as a $36$ dimensional subalgebra, the [[spin Lie algebra]], $spin(9)$, acting on the $16$ dimensional space of, real, positive [[chiral]], Cl(9) [[spinor]]s, $S^{\lp9\rp}$,
$$
f4 = spin(9) + S^{(9)} = spin(8) + V^{(8)} + S^{(8)+} + S^{(8)-}
$$
which breaks up further into $28$ dimensional $spin(8)$ and three $8$ dimensional elements: the vector, $V^{(8)}$, positive [[chiral]] [[spinor]], $S^{(8)+}$, and negative chiral spinor, $S^{(8)-}$ -- all related through [[triality]]. Explicitly, the Lie algebra brackets between generators, for the compact real form of f4 or for the split real form using the $spin(4,4)$ subalgebra, are:
\begin{eqnarray}
\lb \ga_{\al \be}, \ga_{\ga \de} \rb &=& 2 \left\{ - \et_{\al \ga} \ga_{\be \de} + \et_{\al \de} \ga_{\be \ga} + \et_{\be \ga} \ga_{\al \de} - \et_{\be \de} \ga_{\al \ga} \right\} \\
\lb \ga_{\al \be}, \ga_{\ga} \rb &=& 2 \left\{ \et_{\be \ga} \ga_{\al} - \et_{\al \ga} \ga_{\be} \right\} \\
\lb \ga_{\al \be}, Q^+_a \rb &=& ( \ga^+_{\al \be} )^b{}_a Q^+_b \\
\lb \ga_{\al \be}, Q^-_a \rb &=& ( \ga^-_{\al \be} )^b{}_a Q^-_b \\
\lb \ga_{\al}, \ga_{\be} \rb &=& -2 \ga_{\al \be} \\
\lb Q^+_a, Q^+_b \rb &=& \ha ( \ga^+{}^{\al \be} )_{ab} \ga_{\al \be} \\
\lb Q^-_a, Q^-_b \rb &=& \ha ( \ga^-{}^{\al \be} )_{ab} \ga_{\al \be} \\
\lb \ga_\al, Q^+_a \rb &=& ( \bar{\ga}_\al )^b{}_a Q^-_b \\
\lb \ga_\al, Q^-_a \rb &=& - ( \ga_\al )^b{}_a Q^+_b \\
\lb Q^+_a, Q^-_b \rb &=& - ( \ga^\al )_{ab} \ga_\al
\end{eqnarray}
in which the greek indices are raised and lowered by the Clifford signature metric, $\eta_{\al \be}$, and the latin spinor indices are raised and lowered by $g_{ab} = (\ga_1 \ga_2 \ga_3 \ga_4)^+_{ab}$ for the split real form, in which those $\ga_a$ are the negative signature ones.

The smallest irreducible representation of f4 is 26 dimensional.
<<ListTagged fb>>
A ''fiber bundle'' is a [[manifold]], $E$, the ''total space'' (''//entire space//''), along with a ''defining map'', $\pi$, to a separate ''base manifold'', $M$.  Locally, a patch of the total space, $E_{U_a} \sim U_a \otimes F$, is the product of a base patch, $U_a$, and a ''typical fiber'', $F$ -- and $\pi : E_{U_a} \rightarrow U_a$ is a projection, $\pi : z \mapsto x$. A fiber bundle may be visualized as the base manifold with a copy of the fiber attached at each base manifold point -- the fiber is said to be "over" the base manifold. There are thus two equivalent ways of describing a fiber bundle: as things happening in fibers over the base space, $M$, or as things in the total space, $E$. Each fiber is a [[submanifold]] of $E$, but there is not necessarily any submanifold of $E$ associated with $M$.

The best we can usually do is specify the explicit maps from all the $E_{U_a}$ to $U_a \otimes F$ via a ''local trivialization'', $\phi_a : z \mapsto (x,f)$. By inverting these maps, each typical fiber element, $f \in F$, gives a local section, $\ph_a^-(x,f)$, in $E_{U_a}$ over each patch. A ''local section'', $\si$, is a map from base manifold patches to the total space patches, $\si:U_a \rightarrow E_{U_a}$, which projects trivially, $\pi(\si(x))=x \; \forall \; x \in U_a$. The local trivializations over the patches are glued together such that the ''transition functions'',
$$
t_{ab}(z) = \ph^-_a \circ \ph_b(z) 
$$
at each overlap point, $x=\pi(z)$, are [[autodiffeomorphism|diffeomorphism]]s of the fiber there, $\ld t_{ab} \rl_x : F \to F$, corresponding to the action of an element of the ''structure group'', $G$, of the fiber bundle on the fibers, $G:F \to F$. Another way of writing this, letting $z=\ph^-_b(x,f)$, is
$$
t_{ab} \circ \ph^-_b(x,f) = \ph^-_a(x,f) = \ph^-_b(x,g_{ab}(x) f)
$$
in which $g_{ab}(x) \in G$, acts on each fiber element at each overlap point via the [[left action|group]] of the [[Lie group]], $G$. When describing a fiber bundle it is necessary to specify the base, the fiber, the structure group, and the group action on the fiber.

With a local trivialization in hand, a local section, $\si(x) = \ph_a(x,f_\si(x))$, can be specified by choosing a typical fiber element, $f_\si(x)$, at each $x \in U_a$. The collection of fiber valued functions, $f_\si(x) \in F$ (which is sometimes just written as $\si(x) \in F$), is refered to by physicists as a ''field'' over the base manifold. A complete collection of local sections can be glued together to give a ''global section'' iff
$$
\ph^-_a(x,f^a_\si(x)) = \ph^-_b(x,g_{ab}(x) f^a_\si(x))
$$
and hence iff
$$
f^b_\si(x) = g_{ab}(x) f^a_\si(x)
$$
In this way, a ''global section'' (//''section''//) associates $M$ with a particular submanifold, $\si$, of $E$. This change of how a section is represented when the local trivialization is changed, $\si'(x) = g(x) \si(x)$, is the most basic type of [[gauge transformation]].

The [[partial derivative]] is zero when acting on a ''constant'' section, $f_\si(x) = \si(x) = \si$, that is specified locally by a constant field, $\pa_i \si=0$. This derivative doesn't properly keep track of the local trivialization or gluing between patches. To remedy this, a [[covariant derivative]] is introduced which, via a [[connection]], keeps track of how the local trivialization changes over the base when taking the derivative of a section -- it co-varies with a gauge transformation. Using the covariant derivative, any fiber element may be [[parallel transport]]ed along any path on the base to obtain a new fiber element at any point along the path. For a closed path, the parallel transport of a fiber element is represented by a [[holonomy]] -- an element of the structure group which acts on the fiber element. For a small closed path, or loop, the holonomy is given approximately by the [[curvature]] -- an important geometric descriptor of the fiber bundle and connection.

The above description, employing local trivializations, treats fiber bundle geometry as something happening over a base space. Fiber bundle geometry may be described more naturally over the total space by employing an [[Ehresmann connection]]. The defining map, $\pi$, of a fiber bundle gives an involutive [[distribution]], $\ve{\De_\pi}$, and foliation of the total space, $E$, by fibers. This distribution is the kernel of the [[pushforward|pullback]] of the map, $\pi_* \ve{\De_\pi}=0$, and is tangent to the foliating fibers of $E$.

Refs:
*A more thorough description is available at http://en.wikipedia.org/wiki/Fiber_bundle
*[[A Route Towards Gauge Theory|papers/A Route Towards Gauge Theory.pdf]]
*[[Geometrical aspects of local gauge symmetry|papers/Geometrical aspects of local gauge symmetry.pdf]]
*[[Preparation for Gauge Theory|papers/9902027.pdf]]
**excellent mathematical review of the basics
Consider a [[vector field|tangent bundle]], $\ve{v}(x)$ over a manifold. There are unique [[path]]s, $x(t)$, called ''integral curves'' of the vector field, such that the [[tangent vector]] at each point along the path is equal to the vector of the vector field at that point,
$$
\fr{d x^i(t)}{d t} = v^i(x(t))
$$
This relation may be thought of, and solved, as a set of ODE's. Consider the integral curves, with $\ve{v}$ as tangent vectors, starting from each manifold point, $x$, with these paths parameterized such that $t=0$ at these points. From these initial conditions, the point of each path, $y(t,x)$, is determined as a function of parameter and starting point, $x$. This is a ''flow'' &mdash; a parameterized [[autodiffeomorphism|diffeomorphism]], $\ph_t(x)=y(t,x)$, satisfying the "time symmetry" rule:
$$
\ph_t(\ph_{t'}(x))  = y(t,y(t',x)) = y(t+t',x) = \ph_{t+t'}(x) 
$$
A flow may be visualized as a movement of the manifold points beneath any overlying geometric elements. Any chosen initial point, $x$, is carried along by the flow along the path, $\ph_t(x)$, defined by the ''flow equation'',
$$
\lb \fr{\pa}{\pa t} \ph_t^i(x) \rb \ve{\pa_i} = \ve{v}(x)
$$
for all $t$. The flow is completely determined by the vector field, $\ve{v}(x)$ &mdash; the ''vector field generator'' of the flow. An observer attached to such a point carried by the flow may either consider herself to be moving through a field of geometric elements or, alternatively, to be having the geometric elements change over her. For short times, the flow is (in terms of coordinates)
$$
\ph_t^i(x) \simeq x^i + t v^i(x)
$$
The solution to the flow equation may be written heuristically as the [[exponentiation]] of the flow,
$$
\ph_t(x) = e^{t\ve{v}\f{d}} x = e^{t {\cal L}_{\ve{v}}} x
$$
and any geometric object may be [[pushed forward|pullback]] along the flow by exponentiating the [[Lie derivative]], with
$$
\ph^*_t X = e^{t {\cal L}_{\ve{v}}} X
$$

It is possible to have a parameterized autodiffeomorphism, $\ph_t(x)$, that satisfies $\ph_0(x)=x$ but does not satisfy the time symmetry rule. This is a ''time dependent flow'', and produces two distinct, time dependent velocity fields. The first, the ''Lagrangian flow field'', is the velocity of each initial manifold point, $x_0$, wherever it might be carried on the manifold: 
$$
\ve{v_t}(x_0) = \lb \fr{\pa}{\pa t} \ph_t^i(x_0) \rl_t \ve{\pa_i}
$$
The second, the ''Euler flow field'', is the velocity at each manifold point at time $t$:
$$
\ld \ve{v_t}(x) \rl_{x=\ph_t(x_0)} = \lb \fr{\pa}{\pa t} \ph_t^i(x_0) \rl_t \ve{\pa_i}
$$
If the Euler flow field is constant in time, it is the vector field generator for the corresponding flow.
<<ListTagged folder>>
The ''equivalence principle'' of General Relativity states that physics in a sufficiently small region around each point in an $n$ dimensional curved [[spacetime]] is locally indistinguishable from physics in a flat spacetime around that point.  The mathematical implication is that at each point there is a set of $n$ ''orthonormal basis vectors'' (a.k.a. //''frame vectors''//), which can be written in terms of the [[coordinate basis vectors]] as
\[ \ve{e_\al} = \lp e_\al \rp^i \ve{\pa_i} \]
They are orthonormal, $\lp \ve{e_\al},\ve{e_\be} \rp = \et_{\al \be}$, under use of a [[metric]], in which $\et_{\al \be}$ is a [[Minkowski metric]]. The set of their [[1-form]] duals constitute the ''coframe 1-forms'' (a.k.a. //''coframe''//, //''vielbein''//, //''tetrad''//, //''frame 1-forms''//, or sometimes also just called the //''frame''//),
\begin{eqnarray}
\f{e^\al} &=& \f{dx^i} \lp e_i \rp^\al\\
\ve{e_\al} \f{e^\be} &=& \lp e_\al \rp^i \lp e_j \rp^\be \ve{\pa_i} \f{dx^i} = \lp e_\al \rp^i \lp e_i \rp^\be = \de_\al^\be
\end{eqnarray}
As the set of "rulers" on the manifold, the coframe matrix components have [[units]] of time, $T$. The ''orthonormal basis vector matrix'', or //''frame matrix''//, $\lp e_\al \rp^i$, is the inverse of the ''coframe matrix'', $\lp e_i \rp^\al$ &mdash; they satisfy $\lp e_\al \rp^i \lp e_i \rp^\be = \de_\al^\be$ and $\lp e_i \rp^\al \lp e_\al \rp^j = \de_i^j$. (The ${}^-$ in $\lp e^-_\al \rp^i$ is not written but is implied from the position of the [[indices]].)

Physically, at every point the coframe encodes a map from [[tangent vector]]s to vectors in a [[rest frame]].  It is very useful to employ the [[Clifford basis vectors]] as the fundamental geometric basis vector elements of this rest frame, $\ve{e_\al} \leftrightarrow \ga_\al$. The ''coframe'', as a Clifford vector valued 1-form, is a map from the [[tangent bundle]] to the [[Clifford vector bundle]] &mdash; a map from tangent vectors to Clifford vectors &mdash; and is written as
\[ \f{e} = \f{e^\al} \ga_\al = \f{dx^i} \lp e_i \rp^\al \ga_\al \]
It is a [[Clifform]].  Using the coframe, any tangent vector, $\ve{v}$, on the manifold may be mapped to its corresponding Clifford vector, $v$, via [[vector-form algebra]],
$$
\ve{v} = v^\al \ve{e_\al} \leftrightarrow v = \ve{v} \f{e} = v^i \ve{\pa_i} \f{dx^j} \lp e_j \rp^\al \ga_\al = v^i \lp e_i \rp^\al \ga_\al = v^\al \ga_\al
$$
Taking the [[Clifford algebra]] dot product of this Clifford vector with the frame gives the ''1-form dual'' of the vector,
$$
\f{v} = v \cdot \f{e} = v^\al \f{e^\be} \ga_\al \cdot \ga_\be = \lp v^\al \et_{\al \be} \rp \f{e^\be} = v_\be \f{e^\be} = \lp \ve{v} \f{e} \rp \cdot \f{e}
$$
satisfying $\ve{v} \f{v} = v^\al v_\al = \lp \ve{v}, \ve{v} \rp$. The frame or coframe matrices multiply indexed objects and change their indices between coordinate indices and orthonormal basis labels, $T_{\al i} \lp e_j \rp^\al \lp e_\ga \rp^i = T_{j \ga}$. The Clifford vectors corresponding to the orthonormal basis vectors are the Clifford basis vectors,
$$
\ve{e_\al} \leftrightarrow \ve{e_\al} \f{e} = \ga_\al
$$
The ''frame'', as a Clifford vector valued vector, is defined as
$$\ve{e}=\ga^\al \ve{e_\al}=\ga^\al \lp e_\al \rp^i \ve{\pa_i}$$
and satisfies
$$\ve{e} \f{e} = \ga^\al \lp e_\al \rp^i \ve{\pa_i} \f{dx^j} \lp e_j \rp^\be \ga_\be
= \ga^\al \lp e_\al \rp^i \lp e_i \rp^\be \ga_\be
= \ga^\al \ga_\al
= n$$
Using the frame, any [[differential form]], $\f{f}$, on the manifold may be mapped to its corresponding Clifford vector,
$$
\f{f} = \f{e^\al} f_\al \leftrightarrow f = \ve{e} \f{f} = \ga^\al \lp e_\al \rp^i \ve{\pa_i} \f{dx^j} f_j = \ga^\al \lp e_\al \rp^i f_i = f_\al \ga^\al
$$
(Since the frame and coframe are used similarly and, as one is the inverse of the other, carry the same information, they are both often collectively referred to as the //''frame''//.)

The frame (or coframe), along with the Minkowski metric encoding the signature, completely determines a metric on the manifold. But the converse is not true &mdash; the metric only determines a frame up to a [[Lorentz transformation|Lorentz rotation]].  So the frame (or coframe) should be considered the fundamental object encoding the geometry of the manifold. It is possible to make the correspondence between frame and metric one-to-one by imposing a coordinate dependent restriction on the form of the frame matrix &mdash; such as restricting to the use of a [[UT frame]].
[[geodesic]]


''momentum'' conserved along [[Killing vector]] directions.

Rovelli p122 for free particle in Minkowski space.
A ''function'' over a [[manifold]], $M$, is usually written as $f(x)$ &mdash; a function of the corresponding coordinates, $f \circ x_a^{-} : \Re^n \rightarrow M \rightarrow \Re$.  In this case it is a map from the points of the manifold, in the various charts, into the real numbers. The $x$ in $f(x)$ is shorthand for the set of coordinates, $x^i$, over each manifold patch.

A function is more thoroughly described as a section of a [[fiber bundle]].
The rank $2$ exceptional [[Lie group]], [[G2]], is described by its $14$ dimensional [[Lie algebra]], ''g2''. This Lie algebra may be decomposed as a $8$ dimensional [[symmetric|symmetric space]] [[subgroup]], the [[special unitary group]] Lie algebra, [[su(3)]], acting on the standard $3$ representation and its dual, $\bar{3}$,
$$
g2 = su(3) + 3 + \bar{3}
$$
This also relates to
$$
g2 \subset so(7) = so(6) + 6
$$
$$
so(7) = g2 + 7
$$
which gives the fundamental $7$ rep.

An explicit construction of the Lie algebra and the group can be found in:

Ref:
*[[Cerchiai - Euler angles for G2|papers/Cerchiai - Euler angles for G2.pdf]]
A ''gauge transformation'' (//''passive gauge transformation''//) is a transformation of the section (//''gauge''//), $\si(x)$, of a [[fiber bundle]] by an element, $g \in G$, of the structure [[group|Lie group]] to another gauge, $\si'(x) = g \, \si(x)$. A gauge transformation is ''local'' if it has a position dependent $g(x)$ and ''global'' if it isn't. Alternatively, a ''passive coordinate gauge transformation'' may be considered -- and treated equivalently --  that is nothing but a description of how the representation of the section changes under a change of local trivialization. 

The [[covariant derivative]], written using a [[connection]], varies covariantly,
\begin{eqnarray}
\f{\na'} \si' &=& g(x) \f{\na} \si\\
\lp \f{d} g \rp \si + g \f{d} \si + \f{A'} g \si &=& g \f{d} \si + g \f{A} \si\\
\end{eqnarray}
giving the transformation law for the connection under a gauge transformation,
$$
\f{A'} = g \f{A} g^- - \lp \f{d} g \rp g^- = g \f{A} g^- + g \lp \f{d} g^- \rp
$$
An infinitesimal transformation, $g \simeq 1 + G^A T_A = 1 + G$, changes the connection to
\begin{eqnarray}
\f{A'} &\simeq& \f{A} - \f{d} G - \f{A} G + G \f{A} = \f{A} - \f{\na} G \\
\de_G \f{A} &=& - \f{\na} G
\end{eqnarray}
Under a gauge transformation, the [[curvature]] changes to
\begin{eqnarray}
\ff{F'} &=& \f{d} \f{A'} + \f{A'} \f{A'} \\
&=& \f{d} \lb g \f{A} g^- - \lp \f{d} g \rp g^- \rb + \lb g \f{A} g^- - \lp \f{d} g \rp g^- \rb \lb g \f{A} g^- - \lp \f{d} g \rp g^- \rb \\
&=& g \lp \f{d} \f{A} + \f{A} \f{A} \rp g^- \\ 
&=& g \ff{F} g^- = A_g \ff{F} \\
&\simeq& \ff{F} + G \ff{F} - \ff{F} G = \ff{F} + \lb G , \ff{F} \rb \\
\de_G \ff{F} &=& \lb G , \ff{F} \rb
\end{eqnarray}

All physical, measurable quantities in physics are invariant under gauge transformations.

The above description of gauge transformations presumes the connection to be a field over the base manifold. A gauge transformation may also be described from the viewpoint of structures over the total space of the fiber bundle, as an [[Ehresmann gauge transformation]]. In this space, an ''active gauge transformation'' is a vertical [[autodiffeomorphism|diffeomorphism]] -- this gauge transformation, which transforms the connection over the total space while leaving the local sections fixed, is equivalent to the passive gauge transformation, which transforms the local sections while leaving the connection fixed.
A ''geodesic'' on a [[manifold]] with a [[metric]] or [[frame]] is a [[path]], $x(t)$, of extremal length (or time).  This [[proper time]], in temporal [[units]] such as seconds, is the integral of the speed between two points,
$$S = \Delta \ta = \int \f{dt} \ll v \rl = \int \f{dt} \sqrt{\ll v \cdot v \rl} = \int \f{dt} \sqrt{\ll \lp \ve{v} \f{e} \rp \cdot \lp \ve{v} \f{e} \rp \rl} = \int \f{dt} \sqrt{\ll \fr{d x^i}{d t} \fr{d x^j}{d t} g_{ij}(x) \rl} $$
with respect to parameter time, $t$.  Extremizing this length with respect to variation of $x^i(t)$ gives a geodesic equation invariant with respect to reparameterization.  Alternatively, a geodesic also extremizes the action,   
$$S = \int \f{d \ta} \ha v^2 = \int \f{d \ta} \ha \fr{d x^i}{d \ta} \fr{d x^j}{d \ta} g_{ij}(x) = \int \f{d \ta} \ha \fr{d x^i}{d \ta} \fr{d x^j}{d \ta} g_{ij}(x) $$
which is a simpler variation and produces the affine geodesic equation, with the parameter set equal to the proper time along the curve, $t=\ta$.  The variation is
\begin{eqnarray}
\de S &=& \int \f{d \ta} \lc \ha \fr{d x^i}{d \ta} \fr{d x^j}{d \ta} \de x^k \pa_k g_{ij} + \fr{d \de x^i}{d \ta} \fr{d x^j}{d \ta} g_{ij} \rc \\
&=&  \int \f{d \ta} \de x^k  \lc \ha \fr{d x^i}{d \ta} \fr{d x^j}{d \ta} \pa_k g_{ij} - \fr{d}{d \ta} \lp \fr{d x^j}{d \ta} g_{kj} \rp \rc
\end{eqnarray}
and gives the geodesic equation,
$$\fr{d^2 x^k}{d \ta^2} = \fr{d x^i}{d \ta} \fr{d x^j}{d \ta} \lp \ha g^{mk} \pa_m g_{ij} - g^{mk} \pa_i g_{mj} \rp 
= - \fr{d x^i}{d \ta} \fr{d x^j}{d \ta} \Ga^k{}_{ij}$$
This determines the path on a manifold followed by any freely falling body, given its initial position and velocity.  This motion of a free particle also has a [[Hamiltonian formulation|free particle Hamiltonian]]. The geodesic dependends on derivatives of the metric via the [[Christoffel symbols|tangent bundle connection]], and may be expressed by derivatives of the frame via the spin connection.  (This relationship also gives a quick way of calculating connection coefficients by varying the metric or frame.) Note that geodesics aren't influenced by [[torsion]] since only the symmetric part of the Christoffel symbols, $\Ga^k{}_{\lp ij \rp}$, enter the geodesic equation.

The [[tangent vector]] to a geodesic is [[parallel transport|tangent bundle parallel transport]]ed along the path,
$$0 = \lp \ve{v} \f{\nabla} \rp \ve{v} = v^i \nabla_i v^k \ve{\pa_k} = \lp \fr{d v^k}{d \ta} + v^i v^j \Ga^k{}_{ij} \rp \ve{\pa_k}$$
by virtue of the geodesic equation.  If the velocity is expressed as a [[Clifford element]] using the frame, $v = \ve{v} \f{e}$, then
$$0 = \ve{v} \f{\nabla} v = v^i \nabla_i v^\al \ga_\al = \lp \fr{d v^\al}{d \ta} - v^i v^\be \om_i{}_\be{}^\al \rp \ga_\al$$
in terms of the [[spin connection]].  The speed along a geodesic with respect to proper time is constant, and its square, $v^2$, is either $1$, $0$, or $-1$ depending on whether the velocity is timelike, null, or spacelike.

The component of a geodesic's velocity along any [[Killing vector]] is constant along the path,
$$
...
$$
//add equation//
*<<slider chkSliderhamF hamF 'ham >' 'Hamiltonian dynamics, symplectic geometry'>>
*<<slider chkSliderkkF kkF 'kk >' 'Kaluza-Klein theory, Killing vector fields'>>
*<<slider chkSlidercosmoF cosmoF 'cosmo >' 'cosmology'>>
*<<slider chkSlidergrscalF grscalF 'grscal >' 'gr plus a scalar field, Brans-Dicke theories, conformal transformations'>>
*<<slider chkSliderlqgF lqgF 'lqg >' 'loop quantum gravity, loops, spin foams, spin networks'>>
*<<slider chkSlidertorsF torsF 'tors >' 'torsion, teleparallel gravity'>>
<<ListTagged gr>>
Ref:
*Fabrizio Nesti
**[[Standard Model and Gravity from Spinors|http://arxiv.org/abs/0706.3304]]
***Uses $so(3,1)$ spin connection. Split this into self-dual part for gravity and anti-self-dual part for either electroweak $su(2)_L$ or $su(2)_R$. Nesti breaks both up out of $SO(3,1,\mathbb{C})$. He also goes on to talk about Pati-Salam, but his frame and Higgs are messed up and he doesn't use MacDowell-Mansouri. Ooh, he is getting close to what I'm doing though -- mentions embedding in non-orthogonal groups in a foottiddler. Mirror fermion problem. [[Coleman-Mandula]] doesn't apply because the [[frame]] doesn't have Poincare symmetry.
**[[Gravi-Weak Unification|http://arxiv.org/abs/0706.3307]]
***Uses $Cl(4,\mathbb{C})$ for a Pati-Salam model.
*Stephon Alexander
**[[Isogravity: Toward an Electroweak and Gravitational Unification|http://arxiv.org/abs/0706.4481]]
A ''group'' is a collection of elements, $g \in G$, along with an ordered group product by which one element times a second equals a third, $g_1 g_2 = g_3$. A group has the following properties:
*it includes the ''identity element'', $g 1 = 1 g = g$
*every element has an [[inverse]], $g g^- = g^- g = 1$
*its product is ''associative'', $a(bc)=(ab)c$
*its product is ''closed'', $ab \in G$

One group element, $h$, may act on another, $g$, via three different ''group action''s:
*''left action'': $L_h g = h g$
*''right action'': $R_h g = g h$
*''conjugation'', also known as the inner [[automorphism]] or ''adjoint action'': $A_h g = L_h R_{h^-} g = h g h^-$

A group is ''abelian'' iff $gh = hg$ for all $h, g \in G$.

An example of a group is the set of integers, $G = \left\{ \dots, -3, -2, -1, 0, 1, 2, 3, \dots \right\}$, with addition, $+$, as the group product. For this abelian group, $0$ is the identity element.
*James Ryan
**[[A new proposal for group field theory I: the 3d case|http://arxiv.org/abs/gr-qc/0611080]]
<<ListTagged grscal>>
<<ListTagged gut>>
<<ListTagged ham>>
<<ListTagged higgs>>
The ''holonomy'' is the [[path holonomy]], $U$, for an arbitrary closed path on the base manifold of a [[fiber bundle]]. It may be written heuristically as
$$
U = Pe^{-\oint \f{A}}
$$
in which the [[connection]] is integrated all the way around the path. (The only real meaning of this expression is that it is a solution for the path holonomy at the end point (which equals the initial point) of the path.) 

It is enlightening to calculate the approximate holonomy for a small, square-ish path. Such a path may be specified by choosing two orthonormal vectors, $\ve{u}$ and $\ve{v}$, at a point $x_{0}$ and making a closed path by going $\va$ in the $\ve{u}$ direction, then $\va$ along $\ve{v}$, $\varepsilon$ along $-\ve{u}$, then $\va$ along $-\ve{v}$ back to $x_{0}$. These four path segments, each parameterized by $0 \leq t \leq \va$, are given by
$$
\va_{1}^{i}=tu^{i}, \quad
\va_{2}^{i}=\va u^{i}+tv^{i}, \quad
\va_{3}^{i}=\va u^{i}+\va v^{i}-tu^{i},\quad
\va_{4}^{i}=\va v^{i}-tv^{i}
$$
and produce an anti-symmetric [[second order path dependence|path holonomy]],
$$
\va^{ij} = \int_{0}^{\va}\f{dt}\,\fr{d\va_{1}^{i}}{dt}\va_{1}^{j}
+\int_{0}^{\va}\f{dt}\,\fr{d\va_{2}^{i}}{dt}\va_{2}^{j}
+\int_{0}^{\va}\f{dt}\,\fr{d\va_{3}^{i}}{dt}\va_{3}^{j}
+\int_{0}^{\va}\f{dt}\,\fr{d\va_{4}^{i}}{dt}\va_{4}^{j}
=\va^{2} \lp v^{i}u^{j}-v^{j}u^{i} \rp
$$
implying a [[loop|vector-form algebra]] described by a tangent 2-vector,
$$
\vv{L}=\ha L^{ij}\ve{\pa_i}\ve{\pa_j}
=\ha \va^{ij}\ve{\pa_i}\ve{\pa_j}
=\va^{2}v^{i}u^{j}\ve{\pa_i}\ve{\pa_j}
=\va^{2}\ve{v}\,\ve{u}
$$
The holonomy around this small loop is approximately the [[path holonomy]] to second order,
$$
U \simeq 1 + \va^{ij} \lb - \pa_{j} A_{i} + A_{i} A_{j}\rb
=1 + \ha \va^{ij} \lb \pa_i A_j - \pa_j A_i + 2 A_i \times A_j \rb
=1 + \ha \va^{ij} F_{ij}
=1 - \vv{L} \ff{F}
$$
with the (defining) appearance of the [[curvature]],
$$
\ff{F} = \f{d} \f{A} + \f{A} \f{A} = \f{dx^i} \f{dx^j} \lp \pa_i A_j + A_i \times A_j \rp = \ha \f{dx^i} \f{dx^j} F_{ij} 
$$
The contraction of the loop with the curvature, $\vv{L} \ff{F}$, is a nice example of [[vector-form algebra]]. Any fiber element, $C$, parallel transported around a small loop, $\vv{L}$, is transformed to
$$
C \mapsto C' = UC \simeq C - \vv{L} \ff{F} C
$$
to first order in loop area, $\va^2$. This provides a nice alternative definition of curvature in terms of parallel transport around small closed paths.
A ''homogeneous space'' (a.k.a. //''coset space''//, //''quotient space''//, or //''Klein geometry''//), $S=G/H$, is both a left [[coset]] and [[manifold]] formed by modding a [[Lie group geometry]], $G$, by a [[subgroup]], $H$. The points, $x \in S$, of a homogeneous space are specified by their coset representatives, $r(x) \in G$, up to right action by $H$,
$$
x \sim \lb r(x) \rb = \lb r(x) \, h(y) \rb = r(x) \, H = \left\{ r(x) \, h(y) : \forall \, h(y) \in H \right\}
$$
The homogeneous space is the base space, $M = S = G/H$, of a [[principal bundle]] with total space $E=G$ and $F=H$ as the structure group and fiber. The defining map is $\pi : g \mapsto [g]$ and the choice of ''coset representative section'' (//''homogeneous reference section''//), $r : S \rightarrow G$, (a function of points on the base space, valued in the total space) serves as a [[reference section|Ehresmann principal bundle connection]] and provides the [[local trivialization|fiber bundle]], $\ph : (x,h) \mapsto r(x) \, h \in G$. A homogeneous space has a natural ''zero point'' corresponding to the equivalence class of the identity in $G$, so the coset representative section and coordinates are chosen so $r(0) = 1$. The [[Maurer-Cartan form]], $\f{\cal I} = g^- \f{d} g$, over $G$ [[pulls back|pullback]] along the reference section, $g=r(x)$, to give the ''Maurer-Cartan frame'' over the homogeneous space,
$$
\f{I}(x) = r^* \f{\cal I} = r^- \f{d} r \in \f{\rm Lie}(G)
$$
which leads to a description of the [[homogeneous space geometry]] over $S$ or to the [[Ehresmann homogeneous space geometry]] over $G$. Note that a homogeneous space, $S=G/H$, is itself a Lie group if $H$ is a [[normal subgroup]] of $G$.
A [[principal bundle]] gauge transformation corresponds to a transformation of a reference section, $r(x) \in G$ of a [[homogeneous space]],
$$
r'(x) = r(x) \, h(x)
$$
by $h(x) \in H$. This transformation doesn't move the points of the homogeneous space, since $[r'(x)] = [r(x) \, h(x)] = [r(x)]$, but just moves the section up or down the fibers at those points by a [[diffeomorphism]], $\ph(x,y)=(x,y_\ph(x,y))$, in accordance with the point of view of the [[Ehresmann principal bundle gauge transformation]]. The [[Maurer-Cartan frame|homogeneous space]] over $S=G/H$ transforms to
$$
\f{I'}(x) = \lp r h \rp^* \f{\cal I} = h^- r^- \f{d} \lp r h\rp = h^- \f{I} h + h^- \f{d} h = \f{e'_S} + \f{A'_S}
$$
corresponding to the [[homogeneous space frame|homogeneous space geometry]], $\f{e_S} = \f{e_S^A} K_A$, and [[homogeneous H-connection|homogeneous space geometry]], $\f{A_S} = \f{A_S^P} H_P$, changing to
\begin{eqnarray}
\f{e'_S} &=& h^- \f{e_S} h \\
\f{A'_S} &=& h^- \f{A_S} h + h^- \f{d} h
\end{eqnarray}
since $H$ is taken to be [[reductive]] in $G$, which implies $A_{h^-} {\rm Lie}(G/H) \in {\rm Lie}(G/H)$, $A_{h^-} {\rm Lie}(H) \in {\rm Lie}(H)$, and $h^- \pa_a h \in {\rm Lie}(H)$. The gauge transformation of the H-connection is familiar as the [[gauge transformation]] of a principal bundle connection, and the transformation of the homogeneous space frame, $\f{e'_S^A} = \lp K^A , h^- K_B h \rp \f{e_S^B} = L^A{}_B \f{e_S^B}$, is a [[Lorentz rotation]], familiar as the co[[tangent bundle gauge transformation]]. 

For an infinitesimal gauge transformation, $h \simeq 1 + h^P H_P = 1 + H$, these transformations are
\begin{eqnarray}
\f{e'_S} &\simeq& (1 - H) \f{e_S^B} K_B (1 + H) \simeq \f{e_S} + \lb \f{e_S} , H \rb \\
\f{A'_S} &\simeq& (1 - H) \f{A_S^R} H_R (1 + H) + (1 - H) \f{d} (1 + H) \simeq \f{A_S} + \f{d} H + \lb \f{A_S} , H \rb 
\end{eqnarray}
to first order in the gauge parameters, $h^P$.

A more general possible gauge transformation is
$$
r'(x) = r(x) \, g(x)
$$
by $g(x) \in G$. This transformation may move the points of the homogeneous space, with diffeomorphism $\ph(x,y)=(x_\ph(x,y),y_\ph(x,y))$. The Maurer-Cartan frame over $S$ transforms to
$$
\f{I'}(x) = g^- \f{I} g + g^- \f{d} g = \f{e'_S} + \f{A'_S}
$$
which possibly mixes the homogeneous space frame and H-connection. For an infinitesimal gauge transformation,
$$
g \simeq 1 + g^A K_A + g^P H_P = 1 + K + H
$$
this transformation is
\begin{eqnarray}
\f{I'} &\simeq& \lp 1 - K - H \rp \lp \f{e_S^B} K_B + \f{A_S^Q} H_Q \rp \lp 1 + K + H \rp + \f{d} K + \f{d} H \\
&\simeq& \f{I} + \lb \f{e_S} , K \rb + \lb \f{A_S} , K \rb + \lb \f{e_S} , H \rb + \lb \f{A_S} , H \rb + \f{d} K + \f{d} H \\
&=& \f{I} - g^A \f{e_S^B} \lp C_{AB}{}^C K_C + C_{AB}{}^P H_P \rp - g^A \f{A_S^Q} C_{AQ}{}^C K_C - g^P \f{e_S^B} C_{PB}{}^C K_C - g^P \f{A_S^Q} C_{PQ}{}^R H_R + \f{d} g^A K_A + \f{d} g^P H_P
\end{eqnarray}
giving
\begin{eqnarray}
\f{e'_S} &\simeq& \f{e_S} + \lp - g^A \f{e_S^B} C_{AB}{}^C - g^A \f{A_S^Q} C_{AQ}{}^C - g^P \f{e_S^B} C_{PB}{}^C + \f{d} g^C \rp K_C \\
&=& \f{e_S} + \f{d} K + \lb \f{A_S} , K \rb + \lb \f{e_S} , H \rb + \lb \f{e_S} , K \rb_K \\
\f{A'_S} &\simeq& \f{A_S} + \lp - g^A \f{e_S^B} C_{AB}{}^R - g^P \f{A_S^Q} C_{PQ}{}^R + \f{d} g^R \rp H_R \\
&=&  \f{A_S} + \f{d} H + \lb \f{A_S} , H \rb + \lb \f{e_S} , K \rb_H
\end{eqnarray}
\begin{eqnarray}
to first order in the gauge parameters, $g^I$. There is, though, a potential problem with this type of gauge transformation: If we choose $g(x)=r^-(x)$ then the section transforms to $r'(x) = r(x) \, g(x) = r \, r^- = 1$, which is no longer a section since $\pi \circ r'$ is not the identity map on $S$. It is still interesting to consider though, as this choice results in $\f{I'} = r'^- \f{d} r' = 0$, but such a gauge transformation may not be allowed as it may not come from a diffeomorphism, unless the space is contractable.
A [[homogeneous space]], $S=G/H$, built from a [[Lie group]], $G$, and a [[subgroup]], $H$, inherits a geometry from the [[Lie group tangent bundle geometry]] of $G$ and how the $H$ subgroup -- a [[submanifold]] of $G$ -- sits in $G$. The flows on the [[Lie group manifold|Lie group geometry]] are described by the [[Lie algebra]] generators, $T_I \in {\rm Lie}(G)$, with [[index|indices]] $I$ running from $1$ to $n_G$ -- the dimension of $G$. These generators are presumed to be rotated so the [[Killing form]],
$$
\lp T_I, T_J \rp = g_{IJ} = C_{IK}{}^L C_{JL}{}^K 
$$
is diagonal and the structure constants satisfy $C_{IKL} = - C_{ILK}$. The subgroup, $H$, is taken to be [[reductive]] in $G$, with generators $H_P = T_P \in {\rm Lie}(H)$, (with $P$-series indices running from $1$ to $n_H$). The $n_S = (n_G - n_H)$ remaining generators are the ''coset generators'', $K_A = T_A$, spanning the [[vector space]], ${\rm Lie}(G/H)$. We take $z^i$ to be coordinates for $G$, $y^p$ to be coordinates for $H$, and $x^a$ to be coordinates for $S$. A good choice for [[coset representative|homogeneous space]]s is the [[exponentiation]] of the coset generators,
$$
r(x) = e^{x^a K_a} \in G
$$
A coset representative is not necessarily a subgroup of $G$, but it is a section and a [[submanifold]]. If $G$ has a matrix representation, the exponentiation above gives the explicit form of $r(x)$ as a matrix -- a good way to think of it. Whatever the choice of coset representative, the [[Maurer-Cartan form]] over $G$ [[pulls back|pullback]] to give the [[Maurer-Cartan frame|homogeneous space]] over $S$,
$$
\f{I}(x) = \f{e_S} + \f{A_S} = r^- \f{d} r
$$
which splits into the ''homogeneous space frame'', $\f{e_S} = \f{e_S^A} K_A$, and the ''homogeneous H-connection'', $\f{A_S} = \f{A_S^P} H_P$. The coefficients are determined by the Lie group geometry and may be computed explicitly using the Killing form,
\begin{eqnarray}
\f{e_S^A} &=& \lp K^A, \f{I} \rp \\
\f{A_S^P} &=& \lp H^P, \f{I} \rp
\end{eqnarray}
These homogeneous space frame 1-forms may be used as the [[frame]] 1-forms for the [[tangent bundle]] over $S$, with resulting ''homogeneous space [[metric]]'',
$$
\lp \ve{u}, \ve{v} \rp = \lp \ve{u} \f{e_S}, \ve{v} \f{e_S} \rp = u^A v^B \lp K_A, K_B \rp = u^A v^B g_{AB} = u^a v^b \lp e^S_a \rp^A \lp e^S_b \rp^B g_{AB} = u^a v^b g_{ab}
$$
Note that this metric, $g_{ab} = \lp e^S_a \rp^A \lp e^S_b \rp^B g_{AB}$, over $S$ does NOT correspond to the natural [[submanifold geometry]] of the coset representative in $G$ -- that metric would be:
$$
g'_{ab} = \lp e^S_a \rp^A \lp e^S_b \rp^B g_{AB} + \lp A^S_a \rp^P \lp A^S_b \rp^Q g_{PQ}
$$
Rather, the homogeneous space metric is independent of the choice of coset representative, and thus necessarily independent of the homogeneous H-connection. The homogeneous H-connection, $\f{A_S}(x)$, is a particular (constrained to be part of the Maurer-Cartan frame) [[principal bundle]] connection when the homogeneous space is viewed as the base of a principal $H$-bundle.

With the homogeneous space frame in hand it is straightforward to calculate the [[homogeneous space tangent bundle geometry]] based on a choice of [[torsion]]; and to calculate the [[homogeneous space geometry symmetries]] -- the [[Killing vector]]s of the homogeneous space frame.

Refs:
*Roberto Camporesi
**http://calvino.polito.it/~camporesi/
*Leonardo Castellani
**http://www.mfn.unipmn.it/%7ecastella/
**[[On G/H geometry and its use in M-theory compactifications|papers/9912277.pdf]]
**[[Symmetries of Coset Spaces and Kaluza-Klein Supergravity|papers/Symmetries of Coset Spaces and Kaluza-Klein Supergravity.pdf]]
*[[Super coset space geometry|papers/0610039.pdf]]
*Excellent new paper:
**[[Heat Kernel on Homogeneous Bundles over Symmetric Spaces|papers/0701489.pdf]]
Not surprisingly, a [[homogeneous space geometry]] has a large symmetry group, described by [[Killing vector]] fields, $\ve{\xi}(x)$, over $S=G/H$. Most of the symmetries, $\ve{\xi_I}$, correspond to the [[left action|group]] of the [[Lie group]], $G$, through its [[Lie algebra]] generators, $T_I$. However, more symmetries, $\ve{\xi'_X}$, come from the right action of a different group, though some of these may correspond to some of the $\ve{\xi_I}$.

An element $g \in G$, acts from the left on the coset element, $[r(x)] \in S$, to give another coset element, $[g \, r(x)] = [r(x')]$. This implies that the left action of $g$ on the coset representative, $r(x)$, is
$$
g \, r(x) = r(x') h
$$
for some $h \in H$. If the group element is approximated near the identity by $g \simeq (1 + \ep^I T_I)$, with a corresponding [[flow]] on the coset manifold of $x' \simeq x + \ep^I \xi_I$, and $h \simeq (1 + \ep^I h_I^P(x) H_P)$, the above equation,
$$
(1 + \ep^I T_I) \, r \simeq (r + \ep^I \ve{\xi_I} \f{d} r) (1 + \ep^I h_I^P H_P)
$$
gives, to first order in $\ep^I$,
$$
T_I r = \ve{\xi_I} \f{d} r + h_I^P r H_P
$$
Multiplying on the left by $r^-$ and using the Maurer-Cartan frame over $S$,
$$
\f{I} = r^- \f{d} r = \f{e_S^A} K_A + \f{A_S^P} H_P
$$
gives
$$
r^- T_I r = \lp \xi_I \rp^a \lp e^S_a \rp^A K_A + \lp \xi_I \rp^a \lp A^S_a \rp^P H_P + h_I^P H_P
$$
plugging this into the ${\rm Lie}(G)$ [[Killing form]] with the generator duals of the $K_A \in {\rm Lie}(G/H)$ and $H_P \in {\rm Lie}(H)$ gives explicit expressions for the coefficients of the ''left Killing vector fields on the homogeneous space geometry'',
$$
\lp \xi_I \rp^a(x) = \lp e^S_A \rp^a \lp K^A , r^- T_I r \rp
$$
and the ''H-compensator'',
$$
h_I{}^P(x) = -\lp \xi_I \rp^a \lp A^S_a \rp^P + \lp H^P , r^- T_I r \rp
$$
These $n$ Killing vector fields comprise the left action flows of $G$ on $S$. They are demonstrably Killing. Using the definition of the [[Lie derivative]],
$$
{\cal L}_{\ve{\xi_I}} \f{e_S^A} = \ve{\xi_I} \lp \f{d} \f{e_S^A} \rp +\f{d} \lp \ve{\xi_I} \f{e_S^A} \rp
$$
and an expression from the [[homogeneous space tangent bundle geometry]],
\begin{eqnarray}
\ve{\xi_I} \lp \f{d} \f{e_S^A} \rp &=& \ve{\xi_I} \lp - \ha \f{e_S^C} \f{e_S^B} C_{CB}{}^A - \f{A_S^P} \f{e_S^B} C_{PB}{}^A \rp \\
&=& - \lp \xi_I \rp^C \f{e_S^B} C_{CB}{}^A - \lp \ve{\xi_I} \f{A_S^P} \rp \f{e_S^B} C_{PB}{}^A + \f{A_S^P} \lp \xi_I \rp^B C_{PB}{}^A
\end{eqnarray}
combined with the coset representative relation, a Killing form identity, and the commutation relations for the [[reductive]] coset,
\begin{eqnarray}
\f{d} \lp \ve{\xi_I} \f{e_S^A} \rp &=& \f{d} \lp K^A , r^- T_I r \rp = \lp K^A , \lb r^- T_I r, \f{I} \rb \rp = - \lp \lb K^A , \f{I} \rb, r^- T_I r \rp \\
&=& - \f{e_S^B} C^A{}_B{}^P \lp H_P, r^- T_I r \rp - \lp \f{e_S^B} C^A{}_B{}^C + \f{A_S^P} C^A{}_P{}^C \rp \lp K_C, r^- T_I r \rp \\
&=& - \f{e_S^B} C^A{}_B{}^P \lp h_{IP} + \lp \xi_I \rp^a \lp A^S_a \rp_P \rp - \lp \f{e_S^B} C^A{}_B{}^C + \f{A_S^P} C^A{}_P{}^C \rp \lp \xi_I \rp_C
\end{eqnarray}
gives, after happy cancellation,
$$
{\cal L}_{\ve{\xi_I}} \f{e_S^A} = - \f{e_S^B} h_I{}^P C_P{}^A{}_B
$$
with nice Killing rotation coefficients, $\lp B_I \rp_B{}^A = h_I{}^P C_{PB}{}^A$.

The right action of $g \in G$ on a coset element, $[r] = r H$, only makes sense if $g$ "gets past" the $H$ so that $R_g \, [r] = r H g = r g H = [r g]$, which is true iff $g$ is in the [[normalizer]], $g \in N_G(H) = N(H)$. Of course, if $g \in H$ its right action has no effect on $[r]$, so the ''right action'' group of symmetries is the group $N(H)/H$. The right Killing vectors, $\ve{\xi'_X}$, and H-compensators, $h'$, corresponding to a $g \in N(H)/H$ come from
\begin{eqnarray}
r \, g &=& r(x') h' \\
r \, (1 + \ep^X K_X) &\simeq& (r + \ep^X \ve{\xi'_X} \f{d} r) (1 + \ep^X h'_X^P H_P) \\
K_X &=& \lp \xi'_X \rp^a \lp e^S_a \rp^A K_A + \lp \xi'_X \rp^a \lp A^S_a \rp^P H_P + h'_X^P H_P
\end{eqnarray}
(in which $K_X \in {\rm Lie}(N(H)/H) \subset {\rm Lie}(G/H)$ are a reduced linear combination of $K_A$) and are
\begin{eqnarray}
\lp \xi'_X \rp^a(x) &=& \lp e^S_X \rp^a \\
h'_X{}^P(x) &=& -\lp \xi'_X \rp^a \lp A^S_a \rp^P + \de_I^P
\end{eqnarray}
Since the right Killing vectors are a reduced linear combination of the orthonormal basis vectors, the Killing equation is
\begin{eqnarray}
{\cal L}_{\ve{\xi'_X}} \f{e_S^A} &=& {\cal L}_{\ve{e_X}} \f{e_S^A} = \ve{e_X} \lp \f{d} \f{e_S^A} \rp \\
&=& \ve{e^S_X} \lp - \ha \f{e_S^C} \f{e_S^B} C_{CB}{}^A - \f{A_S^P} \f{e_S^B} C_{PB}{}^A \rp \\
&=& \f{e_S^B} \lp -C_{XB}{}^A - \lp A^S_X \rp^P C_{PB}{}^A + \lp A^S_B \rp^P C_{PX}{}^A \rp
\end{eqnarray}
showing that $\ve{\xi'_X}$ is Killing, with Killing rotation coefficients $\lp B'_X \rp_B{}^A = \lp -C_{XB}{}^A - \lp A_X \rp^P C_{PB}{}^A \rp$, if the last term above vanishes -- which it does since the reductivity of $H$ in $G$ implies $\lb {\rm Lie}(H), {\rm Lie}(G/H) \rb \subset {\rm Lie}(G/H)$, and that $H$ is normal in $N(H)/H$ implies $\lb {\rm Lie}(H), {\rm Lie}(N(H)/H) \rb \subset {\rm Lie}(H)$, and together these imply $\lb {\rm Lie}(H), {\rm Lie}(N(H)/H) \rb = 0$, which gives $ C_{PX}{}^A = 0$.

After this analysis one might think the full symmetry group of the homogeneous space geometry is $G \times N(H)/H$, and that is the biggest it could be, but it might be smaller since some of the right Killing vectors, $\ve{\xi'_X}$, may not be independent of the left Killing vectors, $\ve{\xi_I}$.
An analysis of [[reductive]] [[homogeneous space geometry]] resulted in the Maurer-Cartan frame,
$$
\f{I} = \f{e_S} + \f{A_S} = r^- \f{d} r
$$
over the [[homogeneous space]], $S=G/H$, which split into the homogeneous space frame, $\f{e_S} = \f{dx^a} \lp e^S_a \rp^A K_A$, and homogeneous H-connection, $\f{A_S} = \f{dx^a} \lp A^S_a \rp^P H_P$. As the [[pullback]] of the [[Maurer-Cartan form]] [[curvature]], the ''homogeneous space curvature'' vanishes,
$$
0 = \ff{F}(x) = \f{d} \f{I} + \ha \lb \f{I}, \f{I} \rb = \f{d} \f{e_S} + \f{d} \f{A_S} + \ha \lb \f{e_S}, \f{e_S} \rb + \lb \f{A_S}, \f{e_S} \rb + \ha \lb \f{A_S}, \f{A_S} \rb 
$$
which, via the [[reductive]] homogeneous space commutation relations, gives
\begin{eqnarray}
0 &=& \f{d} \f{e_S^A} + \f{A_S^P} \f{e_S^B} C_{PB}{}^A + \ha \f{e_S^C} \f{e_S^B} C_{CB}{}^A \\  
0 &=& \f{d} \f{A_S^P} + \ha \f{A_S^Q} \f{A_S^R} C_{QR}{}^P + \ha \f{e_S^C} \f{e_S^D} C_{CD}{}^P
\end{eqnarray}
This implies the ''[[principal bundle]] curvature of the homogeneous H-connection'' is
$$
\ff{F_H} = \ff{F_H^P} H_P  = \f{d} \f{A_S} + \ha \lb \f{A_S}, \f{A_S} \rb  = - \ha \f{e_S^F} \f{e_S^E} C_{FE}{}^P H_P
$$
But what [[tangent bundle connection]], $\f{w}^A{}_B$, do we choose to build over $S$ when treating $\f{e_S^A}$ as the [[frame]] 1-forms? I.e., what do we choose for the [[torsion]],
$$
\ff{T^A} = \f{d} \f{e_S^A} + \f{\om}^A{}_B \f{e_S^B} = ?
$$
There are two decent looking choices:

If we choose a torsion of $\ff{T^A} = - \ha \f{e_S^C} \f{e_S^B} C_{CB}{}^A$, the ''torsionful homogeneous space connection'' is $\f{w}^A{}_B = \f{A_S^P} C_{PB}{}^A$, which produces a nice [[Riemann curvature]] of
\begin{eqnarray}
\ff{R}^A{}_B &=& \f{d} \f{w}^A{}_B + \f{w}^A{}_C \f{w}^C{}_B \\
&=& \f{d} \f{A_S^P} C_{PB}{}^A + \f{A_S^Q} C_{QC}{}^A \f{A_S^R} C_{RB}{}^C \\
&=& - \ha \f{e_S^F} \f{e_S^E} C_{FE}{}^P C_{PB}{}^A + \f{A_S^Q} \f{A_S^R} \lp - \ha C_{QR}{}^P C_{PB}{}^A + C_{QC}{}^A C_{RB}{}^C \rp \\
&=& - \ha \f{e_S^F} \f{e_S^E} C_{FE}{}^P C_{PB}{}^A
\end{eqnarray}
by the [[Jacobi identity|Lie algebra]]. This is nice because it matches the H-connection curvature,
$$
\ff{R}^A{}_B = \ff{F_H^P} C_{PB}{}^A
$$

But we often want torsion to vanish, $\ff{T^A} = 0$. This choice results in the ''torsionless homogeneous space connection'',
$$
\f{w}^A{}_B = \ha \f{e_S^C} C_{CB}{}^A + \f{A_S^P} C_{PB}{}^A
$$
which produces the Riemann curvature,
\begin{eqnarray}
\ff{R}^A{}_B &=& \f{d} \f{w}^A{}_B + \f{w}^A{}_C \f{w}^C{}_B \\
&=& \ha \f{d} \f{e_S^C} C_{CB}{}^A + \f{d} \f{A_S^P} C_{PB}{}^A + \lp \ha \f{e_S^F} C_{FC}{}^A + \f{A_S^Q} C_{QC}{}^A \rp \lp \ha \f{e_S^E} C_{EB}{}^C + \f{A_S^R} C_{RB}{}^C \rp \\
&=& \ha \f{e_S^F} \f{e_S^E} \lp - \fr{1}{4} C_{FE}{}^C C_{CB}{}^A - C_{FE}{}^P C_{PB}{}^A \rp \\
&=& \ha \f{e_S^F} \f{e_S^E} R_{FE}{}^A{}_B
\end{eqnarray}
and a [[Ricci curvature]] of
\begin{eqnarray}
\f{R}{}_B &=& \ve{e^S_A} \ff{R}^A{}_B = \f{e_S^E} R_{AE}{}^A{}_B \\
&=& \f{e_S^E} \lp - \fr{1}{4} C_{AE}{}^C C_{CB}{}^A - C_{AE}{}^P C_{PB}{}^A \rp \\
&=& - \ha \f{e_S^E} \lp g_{EB} - \ha C_{EA}{}^C C_{BC}{}^A \rp
\end{eqnarray}
which gives a [[curvature scalar]] of
$$
R = \ve{e^S_B} \f{R}{}^B = - \ha \lp n_S - \ha C_{BA}{}^C C^B{}_C{}^A \rp
$$

For a [[symmetric space]], $C_{AB}{}^C=0$, there is no choice -- the torsion vanishes and the ''symmetric space connection'' is $\f{w}^A{}_B = \f{A_S^P} C_{PB}{}^A$.
The analogue of [[spherical coordinates]] in $n$ dimensions are given in terms of $r$ and the $(n-1)$ ''angular coordinates'', $a^w$, by
\begin{eqnarray}
x^1 &=& r \cos(a^1) \\
x^2 &=& r \sin(a^1) \cos(a^2) \\
x^3 &=& r \sin(a^1) \sin(a^2) \cos(a^3) \\
& \vdots \\
x^{n-1} &=& r \sin(a^1) \dots \sin(a^{n-2}) \cos(a^{n-1}) \\
x^n &=& r \sin(a^1) \dots \sin(a^{n-2}) \sin(a^{n-1})
\end{eqnarray}

Ref:
*http://en.wikipedia.org/wiki/Hypersphere#Hyperspherical_coordinates
<<ListTagged illus>>
An ''antisymmetric index bracket'' is used to produce an indexed quantity (tensor) that is antisymmetric in its indices. For example, for a tensor with two [[indices]], $a_{ij}$, the antisymmetric part of this tensor is 
\[ a_{\lb ij \rb} = \ha \lp a_{ij} - a_{ji} \rp \]
A tensor is antisymmetric in its indices iff it equals its corresponding antisymmetric part, $a_{ij} = a_{\lb ij \rb}$. Such an antisymmetric tensor changes sign under the interchange of any two neighboring indices, $a_{ij}=-a_{ji}$.

For a tensor with three indices, $b_{ijk}$, the antisymmeterized part is
\[ b_{\lb ijk \rb} = \fr{1}{3!} \lp b_{ijk}+b_{jki}+b_{kij}-b_{jik}-b_{ikj}-b_{kji}  \rp \]

The ''antisymmeterized index bracket'' is similar in operation to the [[antisymmetric bracket]].

A ''symmetric index bracket'' is used to produce an indexed quantity (tensor) that is symmetric in its indices. For example, for a tensor with two indices, $a_{ij}$, the symmetric part of this tensor is 
\[ a_{\lp ij \rp} = \ha \lp a_{ij} + a_{ji} \rp \]
A tensor is symmetric in its indices iff it equals its corresponding symmetric part, $a_{ij} = a_{\lp ij \rp}$. Such a symmetric tensor is invariant under the interchange of any two neighboring indices, $a_{ij}=a_{ji}$.

For a tensor with three indices, $b_{ijk}$, the symmeterized part is
\[ b_{\lp ijk \rp} = \fr{1}{3!} \lp b_{ijk}+b_{jki}+b_{kij}+b_{jik}+b_{ikj}+b_{kji}  \rp \]
Unless stated otherwise, repeated indices in expressions are summed &mdash; for example,
\[ \ve{v} \f{f} = v^i f_i = \sum_{i=0}^{n-1} v^i f_i \]
This, Einstein's summation convention, loses the information on the range of the sum.  To remedy this deficiency, indices from different parts of the alphabets are taken to range over different integers corresponding to the spaces they coordinatize or label:
| !Latin index | !Greek index | !Range over |!For |
| $i,j,k,l,m,n$ | $\al,\be,\ga,\de,\ep,\up$ | $0 \dots (n-1)  \, {\rm or} \, 1 \dots n$ |all $n$, or any appropriate subset |
| $a,b,c,d$ | $\mu,\nu,\ka,\la$ | $1,2,3, (4 \, {\rm or} \, 0)$ |[[spacetime]] |
| $e,f,g,h$ | $\va,\ze,\ta,\io$ | $1,2,3$ |space |
| $w,x,y,z$ | $\pi,\rh,\si,\xi$ | $1 \dots (n-1)$ |spatial |
| $p,q,r,u$ | $\th,\ph,\ch,\ps$ | $4 \dots (n-1) \, {\rm or} \, 5 \dots n$ |Kaluza-Klein or fiber coordinates &mdash; i.e. other than spacetime |
| $A,B,C$ | | $1 \dots$ ? |Lie algebra elements |
Lower case Latin indices are for [[coordinates|manifold]], Greek indices are [[Clifford algebra]] basis element labels, and upper case Latin indices are [[Lie algebra]] generator element labels.
The integral over a volume, $V$, of the [[exterior derivative]] of a [[differential form]] equals the integral of that form over the boundary, $\pa V$, of that volume,
$$
\int_V \f{d} \nf{F} = \int_{\pa V} \nf{F}
$$

This, ''Stoke's theorem'', may be used to evaluate integrals by finding an ''antiderivative'' of the integrand. For example,
$$
\int_{\lb 0,1 \rb} \f{dx} \, x = \lb \ha x^2 \rl_0^1 = \ha
$$
in which $F = \ha x^2$ is the antiderivative of $\f{d} F = \f{dx} \, x$, and $\pa V$ consists of the boundary points $0$ and $1$. The "integral" over two points is simply the ordered sum of the integrand evaluated at those points. 

Stoke's theorem is a generalization of the ''fundamental theorem of calculus''.

//need to patch together simply connected regions for this definition to work. Betti number? DeRham chains?//
[[Donald Knuth|http://en.wikipedia.org/wiki/Knuth]] suggested the use of a minus sign for group inverses,
\begin{eqnarray}
g^- &=& g^{-1}\\
gg^- &=& g^- g = 1
\end{eqnarray}
during a talk on notation, http://scpd.stanford.edu/scpd/students/Dam_ui/pages/ArchivedVideoList56K.asp?Include=musings.  It's more compact and makes sense, since raising a group element to a power is not always natural, but the inverse is.
Using the definition for the [[determinant]] of a matrix, such as the [[frame]] matrix, $\lp e_i\rp^\al$, and its [[matrix inverse]], $\lp e^-_\al \rp^i = \lp e_\al \rp^i$, with the [[permutation symbol]] gives
\begin{eqnarray}
\ep^{\al \be \dots \ga} &=& \ep^{ij \dots k} \lp e_i\rp^\al \lp e_j\rp^\be \dots \lp e_k\rp^\ga\\
\ep^{\al \be \dots \de \ga} \lp e_\ga \rp^k &=& \ep^{ij \dots m k} \lp e_i\rp^\al \lp e_j\rp^\be \dots \lp e_m\rp^\de\\
\ep^{\al \be \dots \ep \de \ga} \lp e_\de \rp^m \lp e_\ga \rp^k &=& \ep^{ij \dots n m k} \lp e_i\rp^\al \lp e_j\rp^\be \dots \lp e_n\rp^\ep
\end{eqnarray}
Combining these identities with the [[permutation identities]] allows the [[matrix inverse]] to be written explicitly, as well as giving other expressions, such as
\begin{eqnarray}
\lp e_{\lb \ga \rd}\rp^k \lp e_{\ld \de \rb} \rp^m &=& \fr{\ll \et \rl}{2 \lp n-2 \rp!} \ep^{ij\dots nmk} \lp e_i \rp^\al \lp e_j \rp^\be \dots \lp e_n \rp^\ep \ep_{\al \be \dots \ep \ga \de} \\
&=& \fr{1}{2 \ll e \rl \lp n-2 \rp!} \va^{ij\dots nmk} \lp e_i \rp^\al \lp e_j \rp^\be \dots \lp e_n \rp^\ep \ep_{\al \be \dots \ep \ga \de}
\end{eqnarray}
Jet spaces are spaces of derivatives of sections.

Refs:
*Gennadi Sardanashvily
**[[Ten Lectures on Jet Manifolds in Classical and Quantum Field Theory|papers/0203040.pdf]]
<<ListTagged kk>>
The [[left action|group]] of one [[Lie group]] element on all others induces a [[diffeomorphism]], $\ph_h(x)$ on the [[Lie group manifold|Lie group geometry]],
$$
L_h g(x) = h g(x) = g(\ph_h(x))
$$
A vector field on the Lie group manifold is ''left invariant'' iff it is invariant under the [[pushforward|pullback]] of this diffeomorphism for arbitrary $h$,
$$
L_{h*} \ve{v}(x) = \ve{v} \f{\pa} \ph_h(x) = \ve{v}(\ph_h(x))
$$
The partial derivative of the diffeomorphism in the above expression is computed explicitly by using the [[chain rule|partial derivative]],
$$
\f{\pa} g(\ph_h(x)) = \lp \f{\pa} \ve{\ph_h}(x) \rp \f{\pa} g(\ph_h) = h \f{\pa} g(x)
$$
and the defining equation for the [[right action vector fields and 1-forms|Lie group geometry]],
$$
\f{\pa} g = g T_B \f{\xi_R^B}
$$
to write
$$
\lp \f{\pa} \ve{\ph_h}(x) \rp h g T_B \f{\xi_R^B}(\ph_h) = h g T_B \f{\xi_R^B}(x)
$$
and get
$$
\f{\pa} \ve{\ph_h}(x) = \f{\xi_R^B}(x) \ve{\xi^R_B}(\ph_h(x))
$$
This implies the right action vector fields are left invariant,
$$
L_{h*} \ve{\xi^R_C}(x) =  \ve{\xi^R_C}(x) \f{\pa} \ph_h(x) = \ve{\xi^R_C}(x) \f{\xi_R^B}(x) \ve{\xi^R_B}(\ph_h) = \ve{\xi^R_C}(\ph_h(x))
$$

A [[differential form]] is left invariant iff it is invariant under the [[pullback]], $L_h^* \nf{F}(\phi_h(x)) = \nf{F}(x)$. The 1-form duals to left invariant vector fields, such as the duals to the right action vector fields, are left invariant. Since autodiffeomorphisms are invertible, these statements may be summarized by defining any form or [[vector valued form]] to be left invariant iff it is invariant under the pushforward, $L_{h*} \nf{\ve{K}}(x) = \nf{\ve{K}}(\phi_h(x))$, or pullback.
The defining equations for the left and right action Killing vector fields, $\ve{\xi^L_B}$ and $\ve{\xi^R_B}$, over a [[Lie group geometry]],
\begin{eqnarray}
\ve{\xi^L_B} \f{d} g &=& T_B g \\
\ve{\xi^R_B} \f{d} g &=& g T_B
\end{eqnarray}
and for their [[1-form]] duals,
\begin{eqnarray}
\f{\xi_L^B} T_B &=& \lp \f{d} g \rp g^- \\
\f{\xi_R^B} T_B &=& g^- \f{d} g
\end{eqnarray}
satisfying $\ve{\xi^L_B} \f{\xi_L^C} = \de_B^C$ and $\ve{\xi^R_B} \f{\xi_R^C} = \de_B^C$, combine with the [[Killing form]] to give the ''left-right rotator'',
$$
L^C{}_B = \ve{\xi^L_B} \f{\xi_R^C} = \ve{\xi^L_B} \lp T^C ,  g^- \f{d} g \rp = \lp T^C, g^- T_B g \rp 
$$
This is a [[Lorentz rotation]],
\begin{eqnarray}
L^A{}_B L^C{}_D g_{AC} &=& \lp T^A, g^- T_B g \rp \lp T^C, g^- T_D g \rp g_{AC} \\
&=& \lp g T^A g^-, T_B \rp \lp g T_A g^-, T_D \rp \\
&=& \lp T'^A, T_B \rp \lp T'{}_A, T_D \rp \\
&=& \lp T_B, T_D \rp
= g_{BD}
\end{eqnarray}
giving one set of Killing vector fields in terms of the other,
\begin{eqnarray}
\ve{\xi^L_B} &=& L^C{}_B \ve{\xi^R_C} \\
\ve{\xi^R_B} &=& L_B{}^C \ve{\xi^L_C}
\end{eqnarray}

A Lorentz rotation of the structure constants by the left-right rotator leaves them invariant,
\begin{eqnarray}
L^C{}_D L^B{}_E C_{CB}{}^A L_A{}^F &=& \lp T^C, g^- T_D g \rp \lp T^B, g^- T_E g \rp \lp \lb T_C,T_B \rb, T^A \rp \lp T_A, g^- T^F g \rp \\
&=& \lp \lb g^- T_D g,g^- T_E g \rb, g^- T^F g \rp = \lp g^- \lb T_D , T_E \rb g, g^- T^F g \rp \\
&=& C_{DE}{}^F
\end{eqnarray}

The [[exterior derivative]] of the left-right rotator is
\begin{eqnarray}
\f{d} L^C{}_B &=& \f{d} \lp T^C, g^- T_B g \rp \\
&=&  \lp T^C, \lb g^- T_B g \, , \, g^- \f{d} g \rb \rp \\
&=&  \lp \lb \f{\xi_R^A} T_A , T^C \rb , g^- T_B g \rp \\
&=&  \f{\xi_R^A} C_A{}^C{}_D L^D{}_B
\end{eqnarray}
The ''left/right [[chiral]]ity projector''s,
$$
P_{L/R} = \ha \lp 1 \mp i \ga \rp
$$
are built using the spacetime Clifford algebra, [[Cl(1,3)]], [[pseudoscalar]], $\ga = \ga_0 \ga_1 \ga_2 \ga_3$. In the [[Weyl representation|Dirac matrices]], they are
\begin{eqnarray}
P_L &=&
\lb \begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array} \rb \\
P_R &=&
\lb \begin{array}{cccc}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array} \rb
\end{eqnarray}
<<ListTagged lie>>
ref:
[[Little Higgs Review|papers/0502182.pdf]]
nice review
<<ListTagged lqg>>
[<img[images/png/manifold.png]]An oriented $n$ dimensional differentiable ''manifold'', $M$, may be visualized as a curved $n$ dimensional surface embedded in a higher dimensional, pseudo-Euclidean space. A manifold is described mathematically by a collection of coordinate charts (patches), $\left\{ \left( U_a, \: x_a \right) \right\}$, with the open sets, $U_a$, labeled by $a$, covering $M$, and the coordinates, $x_a : U_a \rightarrow \mathbb{R}^n$, homeomorphic maps into open subsets of $\mathbb{R}^n$ such that overlap maps, $x_a \circ x_b^{-} : \mathbb{R}^n \rightarrow M \rightarrow \mathbb{R}^n$, defined on $x_b( U_a \cap U_b)$, are infinitely differentiable. So, every point, $x$, on the manifold is labeled by a set of $n$ real ''coordinates'', $x_a^i(x)$, in some chart, $U_a$, with coordinate [[indices]], $i$, typically running from $1$ to $n$ or from $0$ to $(n-1)$. In most practical cases the chart label, $a$, is not written and the coordinates are simply written as $x^i$ with some chart implied.

For more on manifolds, see http://en.wikipedia.org/wiki/Manifold
The [[inverse]] of a square matrix, such as the [[frame]] matrix, $\lp e_i \rp^\al$, may be written explicitly as
\begin{eqnarray}
\lp e^-_\ga \rp^k &=& \fr{\ll \et \rl}{\lp n-1 \rp!} \ep^{ij\dots mk} \lp e_i \rp^\al \lp e_j \rp^\be \dots \lp e_m \rp^\de \ep_{\al \be \dots \de \ga} \\
&=& \fr{1}{\ll e \rl \lp n-1 \rp!} \va^{ij\dots mk} \lp e_i \rp^\al \lp e_j \rp^\be \dots \lp e_m \rp^\de \ep_{\al \be \dots \de \ga}
\end{eqnarray}
by using the [[inverse matrix identities]] and [[permutation identities]].  It satisfies $\lp e^-_\al \rp^i \lp e_i \rp^\be = \de_\al^\be$ and $\lp e_i \rp^\al \lp e^-_\al \rp^j = \de_i^j$.
*<<slider chkSlidereditingF editingF 'editing >' 'tips on editing and authoring tiddlers, including all sorts of tools'>>
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<<ListTagged meta>>
A ''metric'', $g$, for a [[vector space]] is a an object that takes two vectors and spits out a number -- it determines the symmetric ''scalar product'',
$$
\lp \ve{u}, \ve{v} \rp = u^i v^j \lp \ve{\pa_i}, \ve{\pa_j} \rp = v^i u^j g_{ij} \in \mathbb{R}
$$
A metric on a manifold gives the scalar product between any two [[coordinate basis vectors]] at any point, $g_{ij} = \lp \ve{\pa_i}, \ve{\pa_j} \rp$.  When a [[frame]] exists on the manifold it determines this scalar product and metric, using the [[Clifford algebra]] dot product and the [[vector-form algebra]], as
\[ \lp \ve{u}, \ve{v} \rp = \lp \ve{u} \f{e} \rp \cdot \lp \ve{v} \f{e} \rp = u^\al \ga_\al \cdot v^\be \ga_\be = u^\al v^\be \et_{\al \be} = u^i \lp e_i \rp^\al v^j \lp e_j \rp^\be \et_{\al \be} = u^i v^j g_{ij} \]
with the use of frame coefficients and the [[Minkowski metric]] replacing the use of a metric if desired.  Using component [[indices]], the ''metric matrix'' (often just abbreviated as "metric") in terms of the frame matrix is
\[ g_{ij} = \lp e_i \rp^\al \lp e_j \rp^\be \et_{\al \be} \]
The metric is invariant under [[Lorentz transformations|Lorentz rotation]] of the frame,
\[ g'_{ij} = \lp e'_i \rp^\al \lp e'_j \rp^\be \et_{\al \be} = \lp e_i \rp^\ga L^\al{}_\ga \lp e_j \rp^\de L^\be{}_\de \et_{\al \be} = \lp e_i \rp^\ga \lp e_j \rp^\de \et_{\ga \de} = g_{ij} \]
Another way of seeing this is that the scalar product of two tangent vectors is invariant under [[Clifford adjoint]] transformations of the frame,
$$\f{e} \mapsto \f{e'} = U \f{e} U^-$$
$$\lp \ve{u} \f{e'} \rp \cdot \lp \ve{v} \f{e'} \rp = \lp \ve{u} U \f{e} U^- \rp \cdot \lp \ve{v} U \f{e} U^- \rp =
\lp \ve{u} \f{e} \rp \cdot \lp \ve{v} \f{e} \rp$$
with Lorentz transformations forming a subset of these.
The combined spacetime curvature is:
$$
\ff{F_s} = \ha \lp \ff{R} + \fr{\La}{6} \f{e} \f{e} \rp
$$
in which $\ff{R}$ is the [[Clifford vector bundle]] curvature, $\La$ is the [[cosmological constant|Einstein's equation]], and $\f{e}$ is the [[frame]]. The ''modified BF action for gravity'' over a four dimensional base [[spacetime]] is:
$$
S_s = \int \li \ff{B_s} \ff{F_s} - \fr{g \La}{48} \ff{B_s} \ff{B_s}) \ga \ri
$$
in which $\ff{B_s}$ is the ''dual bivector valued 2-form'', $g$ is some small coupling constant, and $\ga$ is the spacetime [[pseudoscalar]]. Insisting that $\de S_s =0$ under $\de \ff{B_s}$ gives:
$$
\ff{B_s} = \fr{12}{g \La} \lp \ff{R} + \fr{\La}{6} \f{e} \f{e} \rp \ga^-
$$
Plugging this back into the action gives:
$$
S_s = \fr{3}{g \La} \int \li \lp \ff{R} + \fr{\La}{6} \f{e} \f{e} \rp \lp \ff{R} + \fr{\La}{6} \f{e} \f{e} \rp \ga^- \ri
$$
Multiplying this out gives a Chern-Simons boundary term,
$$
\li \ff{R} \ff{R} \ga^- \ri = \f{d} \li \lp \f{\om} \f{d} \f{\om} + \fr{1}{3} \f{\om} \f{\om} \f{\om} \rp \ga^- \ri
$$
as well as the [[Clifford curvature scalar]],
$$
\li \f{e}\f{e} \ff{R} \, \ga^- \ri = \nf{e} R
$$
and a [[volume form]] term,
$$
\li \f{e}\f{e} \f{e} \f{e} \ga^- \ri = 4! \, \nf{e}
$$
Dropping the boundary term, the action is the Einstein-Hilbert action,
$$
S_s = \fr{1}{g} \int \nf{e} \lp R + 2 \La \rp
$$

Varying the frame in the modified BF action for gravity gives the equation of motion:
$$
0 = \f{e} \cdot \ff{B_s} = \ha \lp \f{e} \cdot \ff{R} + \fr{\La}{6} \f{e} \f{e} \f{e} \rp
$$
Taking the [[Hodge dual]] and [[Clifford dual]] of these trivector valued 3-forms gives
\begin{eqnarray}
* \lp \f{e} \cdot \ff{R} \rp \ga^- &=& \lp * \f{e^\mu} \f{e^\nu} \f{e^\rh} \rp \fr{1}{4} R_{\nu\rh}{}^{\ka\la} \lp \ga_{\mu\ka\la} \ga^- \rp \\
&=& \lp \fr{1}{3!} \ep^{\mu\nu\rh\de} \f{e_\de} \rp \fr{1}{4} R_{\nu\rh}{}^{\ka\la} \lp \fr{1}{3!} \ep_{\mu\ka\la\ga} \ga^\ga \rp \\
&=& \f{e_\de} \fr{\ll \et \rl}{3! \,4} \de^{\nu\rh\de}_{\lb \ka\la\ga \rb} R_{\nu\rh}{}^{\ka\la} \ga^\ga \\
&=& - \fr{\ll \et \rl}{3! \, 3!} \lp \f{R} - \fr{1}{2}\f{e} R \rp
\end{eqnarray}
and
$$
* \lp \f{e} \f{e} \f{e} \rp \ga^- = \lp \fr{1}{3!} \ep^{\mu\nu\rh\de} \f{e_\de} \rp \lp \fr{1}{3!} \ep_{\mu\ka\la\ga} \ga^\ga \rp
= \fr{\ll \et \rl}{3!} \f{e}
$$
and we see that this equation of motion is [[Einstein's equation]],
$$
\f{R} - \fr{1}{2} \f{e} R = \La \f{e}
$$

Ref:
*K. Krasnov
**[[Non-metric gravity: A status report|http://arxiv.org/abs/0711.0697]]
***This looks interesting, have only read some and need to finish.
<<ListTagged nat>>
When we consider a [[manifold]] there are ''natural'' geometric objects that arise "for free" -- without the addition of any further algebraic structure. [[Path|path]]s on the manifold lead to the definition of [[tangent vector]]s and the [[tangent bundle]], their dual [[1-form]]s and the [[cotangent bundle]] lead to [[differential form]]s and [[vector valued form]]s. Such objects are invariant under [[coordinate change]]. Explicitly, a natural object has coordinate indexed components that all transform as a [[tensor|coordinate change]]. Since coordinates and algebraic objects are a computational artifice, only natural objects may be expected to be physically meaningful.

A ''natural operator'', such as the [[exterior derivative]], [[Lie derivative]], [[FuN derivative]], [[covariant derivative]], or [[vector-form algebra]] product acts on natural objects and produces natural objects as a result. Another way of understanding this is that natural operators commute with [[diffeomorphism]]s.

To be more explicit, a tangent vector is natural since between two coordinate systems
$$
\ve{v} = v^j \ve{\pa^x_j} = v^j \fr{\pa y^p}{\pa x^j} \ve{\pa^y_p} = v'^p \ve{\pa^y_p} = \ve{v'}
$$
An example of an operation that is not natural (an ''unatural'' operator) is the [[partial derivative]] acting on tangent vectors, since between two coordinate systems
\begin{eqnarray}
\f{\pa} \ve{v} &=& \f{dx^i} \pa^x_i v^j \ve{\pa_j} = \f{dy^m} \fr{\pa x^i}{\pa y^m} \lp \fr{\pa y^k}{\pa x^i} \pa^y_k v^j \rp \fr{\pa y^p}{\pa x^j} \ve{\pa^y_p}
= \f{dy^m} \lp \pa^y_m v^j \rp \fr{\pa y^p}{\pa x^j} \ve{\pa^y_p} \\
\neq \f{\pa'} \ve{v'} &=& \f{dy^m} \pa^y_m \lp v^j \fr{\pa y^p}{\pa x^j} \rp \ve{\pa^y_p} = \f{dy^m} \lp \pa^y_m v^j \rp \fr{\pa y^p}{\pa x^j} \ve{\pa^y_p} + \f{dy^m} v^j \lp \pa^y_m \fr{\pa y^p}{\pa x^j} \rp \ve{\pa^y_p}
\end{eqnarray}
The resulting object, $\f{\pa} \ve{v}$, is not natural because that last term does not vanish. However, unnatural objects can sometimes be assembled into natural objects if such terms are made to cancel. For example, subtracting
$$
\ve{u'} \f{\pa'} \ve{v'} = u^q \fr{\pa y^m}{\pa x^q} \lp \pa^y_m v^j \rp \fr{\pa y^p}{\pa x^j} \ve{\pa^y_p} + u^q v^j \lp \fr{\pa y^m}{\pa x^q} \pa^y_m \fr{\pa y^p}{\pa x^j} \rp \ve{\pa^y_p}
$$
from
$$
\ve{v'} \f{\pa'} \ve{u'} = v^q \fr{\pa y^m}{\pa x^q} \lp \pa^y_m u^j \rp \fr{\pa y^p}{\pa x^j} \ve{\pa^y_p} + v^q u^j \lp \fr{\pa y^m}{\pa x^q} \pa^y_m \fr{\pa y^p}{\pa x^j} \rp \ve{\pa^y_p}
$$
the last terms of each cancel to give a natural object, the Lie bracket,
$$
\ve{v'} \f{\pa'} \ve{u'} - \ve{u'} \f{\pa'} \ve{v'} = \ve{v} \f{\pa} \ve{u} - \ve{u} \f{\pa} \ve{v} = \lb \ve{v} , \ve{u} \rb_L 
$$
Geometry over a disconnected manifold?

This seems to give an elementary picture:
http://arxiv.org/abs/hep-th/9401145
A [[subgroup]], $N \subset G$, of a [[group]], $G$, is called a ''normal subgroup'', $N \triangleleft G$, iff it is invariant under [[conjugation|group]]; that is, for each $n \in N$ and all $g \in G$ the element $A_g n = g n g^- \in N$. Another way of saying this is that for each $n \in N$  and all $g \in G$ there is an $n' \in N$ such that $gn = n'g$. Or, equivalently, $gN=Ng$ for all $g \in G$.

If $N$ and $G$ are [[Lie group]]s, their elements near the identity may be approximated by [[Lie algebra]] elements,
\begin{eqnarray}
g &\simeq& 1 + x^I T_I \\ 
n &\simeq& 1 + n^P N_P
\end{eqnarray}
in which $T_I$ and $N_P$ are the ${\rm Lie}(G)$ and ${\rm Lie}(N)$ generators. Iff $N$ is a normal subgroup, $N \triangleleft G$, then, collecting orders of $x^I$ gives
\begin{eqnarray}
g n g^- &=& n' \\
\lp 1 + x^I T_I \rp \lp 1 + n^P N_P \rp \lp 1 - x^J T_J \rp &\simeq& \lp 1 + n^P N_P + x^I n_I^P N_P \rp \\
\lb T_I, n^P N_P \rb &=& n_I^P N_P
\end{eqnarray}
and so the Lie algebras satisfy
$$
\lb {\rm Lie}(G), {\rm Lie}(N) \rb \subset {\rm Lie}(N)
$$
The ''normalizer'' of a subset, $S$, in $G$ is the [[subgroup]] consisting of all elements of $G$ that leave $S$ invariant under conjugation,
$$
N_G(S) = \lc n \in G \; | \; nSn^- = S \rc
$$
Another way of thinking of this is that $N_G(S)$ consists of all elements, $n \in G$, satisfying $ns=sn'$ for each $s \in S$ and some $n' \in N_G(S)$. The normalizer of a single element, $N_G(a) = C_G(a)$, is the [[centralizer]], and in general the centralizer is a [[normal subgroup]] of the normalizer, $C_G(S) \triangleleft N_G(S)$. If $H$ is a subgroup of $G$, the normalizer, $N_G(H)$, is the largest subgroup of $G$ having $H$ as a normal subgroup, $H \triangleleft N_G(H)$.

If $H$ and $N = N_G(H)$ are [[Lie group]]s, their [[Lie algebra]] generators satisfy
$$
\lb {\rm Lie}(N), {\rm Lie}(H) \rb \subset {\rm Lie}(H)
$$
The ''octonions'' are an eight-dimensional [[division algebra]], spanned by eight basis elements, $e_0,...,e_7$. The octonion identity element is $e_0=1$, and the other seven can be thought of as different imaginary directions, squaring to $-1$. Octonionic multiplication is non-commutative and, unusually, non-associative. A multiplication table for the basis elements is
| | $e_0$ | $e_1$ | $e_2$ | $e_3$ | $e_4$ | $e_5$ | $e_6$ | $e_7$ |
| $e_0$ | $e_0$ | $e_1$ | $e_2$ | $e_3$ | $e_4$ | $e_5$ | $e_6$ | $e_7$ |
| $e_1$ | $e_1$ | $-e_0$ | $e_4$ | $e_7$ | $-e_2$ | $e_6$ | $-e_5$ | $-e_3$ |
| $e_2$ | $e_2$ | $-e_4$ | $-e_0$ | $e_5$ | $e_1$ | $-e_3$ | $e_7$ | $-e_6$ |
| $e_3$ | $e_3$ | $-e_7$ | $-e_5$ | $-e_0$ | $e_6$ | $e_2$ | $-e_4$ | $e_1$ |
| $e_4$ | $e_4$ | $e_2$ | $-e_1$ | $-e_6$ | $-e_0$ | $e_7$ | $e_3$ | $-e_5$ |
| $e_5$ | $e_5$ | $-e_6$ | $e_3$ | $-e_2$ | $-e_7$ | $-e_0$ | $e_1$ | $e_4$ |
| $e_6$ | $e_6$ | $e_5$ | $-e_7$ | $e_4$ | $-e_3$ | $-e_1$ | $-e_0$ | $e_2$ |
| $e_7$ | $e_7$ | $e_3$ | $e_6$ | $-e_1$ | $e_5$ | $-e_4$ | $-e_2$ | $-e_0$ |
which can be written using an ''octonion multiplication coefficient matrix'' as
$$
e_a e_b = M_{ab}{}^c e_c
$$
so, for example, $e_1 e_2 = e_4$ and $M_{12}{}^4 = 1$. ''Octonionic conjugation'' is given by
$$
e_{\os{0}}=e_0 \;\;\;\;\; e_{\os{1}}=-e_1 \;\;\;\; ... \;\;\;\; e_{\os{7}}=-e_7 
$$
and satisfies $\widetilde{(e_a e_b)} = e_{\os{b}} e_{\os{a}}$, so we have $M_{ab}{}^c = M_{\os{b} \os{a}}{}^{\os{c}}$. 

The metric is defined as
$$
n_{ab}=(e_a, e_b) = \ha ( e_a e_{\os{b}} + e_b e_{\os{a}} )
$$
and for the octonions is $n_{ab}=\de_{ab}$. This metric can be used to raise or lower octonion indices (which has no effect for the octonions, but will matter for the [[split-octonion]]s).
If, in eight dimensions of signature $(8,0)$ (or $(4,4)$), we have a [[Clifford vector|Clifford basis vectors]], $v=v^c \ga_c \in V^{(8)}$, a positive real [[chiral]] [[spinor]], $\ps=\ps^a Q^+_a \in S^{(8)+}$, and a negative real chiral spinor, $\ch=\ch^b Q^-_b \in S^{(8)-}$, satisfying
$$
\ch = v \ps
$$
using the usual matrix representative product of a vector times a spinor, then we can use an [[octonionic representation of Clifford algebra]] to write this as
$$
\ch^b = v^c \Ga_c{}^b{}_a \ps^a = v^c \ps^a M_{\os{a} \os{c}}{}^b
$$
where we have replaced the Clifford matrix representative coefficients with the equivalent [[octonion]] (or [[split-octonion]]) multiplication coefficients, and used the cyclic identity. We can see that this relates directly to the [[octonion]]ic product
$$
v \ps = v^c \ps^a e_c e_a = v^c \ps^a M_{c a}{}^{\os{b}} e_{\os{b}}
= v^c \ps^a M_{\os{a} \os{c}}{}^b e_{\os{b}}
= \tilde{\ch}
$$
and octonionic conjugation. Thus we have a useful //confusion// of octonions with vectors and spinors,
\begin{eqnarray}
v = v^c e_c & \leftrightarrow & v = v^c \ga_c \\
\ps = \ps^a e_a & \leftrightarrow & \ps=\ps^a Q^+_a \\
\ch = \ch^b e_b & \leftrightarrow & \ch=\ch^b Q^-_b \\
\tilde{\ch} = v \ps & \leftrightarrow & \ch = v \ps
\end{eqnarray}
The [[octonion]] or [[split-octonion]] multiplication table can be used to define
$$
\Ga_{cab} = M_{\os{a} \os{b} c} = M_{\os{a}\os{b}}{}^d n_{dc}  
$$
which satisfies a ''cyclic identity'',
$$
\Ga_{cab} = \Ga_{abc} = \Ga_{bca} 
$$
Further defining $\overline{\Ga}_{cab} = \Ga^c{}_{ba}$, these can be used to construct a $16 \times 16$ real chiral [[Clifford matrix representation]] of [[Cl(8)]] or [[Cl(4,4)]], with the Clifford basis vectors represented as
$$
\ga_c =
\lb \begin{array}{cc}
0 & \overline{\Ga}_c{}^b{}_a \\
\Ga_c{}^b{}_a & 0
\end{array} \rb
$$
This remarkable fact holds because the octonionic product leads to satisfaction of the fundamental [[Clifford algebra]] identity,
\[ \ga_a \cdot \ga_b = \ha \lp \ga_a \ga_b + \ga_b \ga_a  \rp = n_{ab} \]
One generation of [[standard model]] fermions consists of [[left chiral|Dirac spinor]] electron neutrinos, $\nu_{e}$, electrons, $e$, up quarks, $u$, and down quarks, $d$, as well as their anti-particles, $\bar{e}, \bar{u}, \bar{d}$  (and possibly the anti-neutrino, $\bar{\nu}_e$, which might not exist). These particles have the following ''hypercharge'', $Y$, and ''weak charge'', $W$, which combine as their ''electric charge'', $Q = (Y+W)/2$:
|                 | $Y$ | $W$ | $Q$ |
| $\nu_e$ | $-1$ | $+1$ | $0$ |
| $\bar{\nu}_e$ | $0$ | $0$ | $0$ |
| $e$          | $-1$ | $-1$ | $-1$ |
| $\bar{e}$ | $+2$ | $0$ | $+1$ |
| $u$         | $+\fr{1}{3}$ | $+1$ | $+\fr{2}{3}$ |
| $\bar{u}$ | $-\fr{4}{3}$ | $0$ | $-\fr{2}{3}$ |
| $d$         | $+\fr{1}{3}$ | $-1$ | $-\fr{1}{3}$ |
| $\bar{d}$ | $+\fr{2}{3}$ | $0$ | $+\fr{1}{3}$ |
The up and down quarks also have pairs of nonzero [[su(3)]] ''color'' charges. This full set of charges is replicated for each of the other two generations of fermions in the [[Elementary particle zoo]].
A matrix, $L$, is ''orthogonal'', iff its [[inverse]], $L^- = L^T$, is its transpose. 
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<<ListTagged paper>>
A geometric object is parallel transported if it is perceived to be unmoving by an observer traveling along with it. Equivalently, a [[fiber bundle]] section, $C(x)$, having values along a [[path]], $x(t)$, is ''parallel transport''ed iff it is [[horizontal|covariant derivative]] along the path,
$$
0 = \ve{v} \f{\na} C = v^i \pa_i C + v^i A_i{}^B T_B C = \fr{d}{d t} C(x(t)) + \ve{v} \f{A} C
$$
In this equation $\ve{v} = \fr{dx^i}{dt} \ve{\pa_i}$ is the path velocity and $\f{A}C$ represents the [[connection]] acting via the left action on the fiber section -- in practice the connection will act appropriately to the specific case.

The solution to the parallel transport equation may be written via the [[path holonomy]].
The ''partial derivative'', $\partial_i$, of a [[function]], $f(x)$, over a [[manifold]] is a [[derivative|derivation]] taken with respect to one manifold coordinate, while holding the other coordinates constant,
$$
\pa_i f = \fr{\pa f}{\pa x^i} = \pa^x_i f
$$
This derivative is explicitly dependent on the choice of coordinates, and must be glued together over different manifold patches.

Partial derivatives may also be taken of geometric objects (sections of [[fiber bundle]]s), with care taken to keep track of which elements are coordinate dependent and which are constant. As an example,
$$
\pa_i \f{A} = \pa_i \f{dx^j} A_j{}^B T_B = \f{dx^j} \lp \pa_i A_j{}^B \rp T_B
$$
The partial derivatives of [[fiber basis elements|vector bundle]], including [[coordinate basis vectors]], [[coordinate basis 1-forms]], and [[Lie algebra]] basis elements, vanish,
$$
\pa_i \ve{\pa_j}=0 \qquad \pa_i \f{dx^j} = 0 \qquad \pa_i T_A=0
$$

The partial derivative may be combined with [[coordinate basis 1-forms]] to produce the ''partial derivative operator'',
$$
\f{\pa}=\f{dx^i} \pa_i = \f{dx^i} \fr{\pa}{\pa x^i}
$$
(usually also just referred to as the //''partial derivative''//). This is NOT a [[natural]] operator on [[vector valued form]]s (or [[vectors|tangent bundle]]) but it is a natural operator on [[differential form]]s -- for which it is the [[exterior derivative]],
$$
\f{\pa} \f{f} = \f{dx^i} \f{dx^j} \pa_i f_j = \f{d} \f{f}
$$
Even though it is not a natural operator on VVF's, and therefore does not by itself produce geometrically meaningful objects, it may still be used on them,
$$
\f{\pa} \f{\ve{A}} = \f{dx^i} \f{dx^j} \lp \pa_i A_j{}^k \rp \ve{\pa_k}
$$
and provides a useful, coordinate dependent but index free calculational device when combined with other terms using [[vector-form algebra]] to build coordinate invariant geometric objects.

As a useful example, the partial derivative operator satisfies the ''chain rule'',
$$
\f{\pa} f(y(x)) = \f{dx^i} \lp \pa_i y^j(x) \rp \pa_j f(y) = \lp \f{\pa} \ve{y}(x) \rp \f{\pa} f(y)
$$
with the partial derivatives taken with respect to the functional dependencies as written.
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A ''path'' or //''curve''//, $c$, on a [[manifold]] is a connected set of manifold points, with these path points usually labeled by a continuous, monotonically increasing real parameter, $t$.
$$
c:\mathbb{R} \to M
$$
$$
c(t)=x(t) \in M
$$
Paths are often written in terms of the $n$ coordinates (in some manifold patch) corresponding to the path points, $x^i(t)=x_c^i(t)=c^i(t)$, for any parameter value. Physics is invariant under arbitrary reparameterizations of the path. Each path point, labeled by $t$, has a [[tangent vector]], $\ve{v}(t)$, with amplitude dependent on the parameter. Any "initial" path point, $x(0)$, corresponding to parameter $t=0$, has nearby path points given approximately by
$$
x^i(t) \simeq x^i(0) + t v^i(0)
$$
to first order in $t$. A path, or technically a path segment, can be defined to exist for a subset of the reals, such as $t \in \lb 0,1 \rb$. Such a path is called a ''closed path'' or //''loop''// iff its ends meet, $x(0) = x(1)$, and otherwise is called an ''open path''. Paths, other than loops which intersect in one place, are usually restricted to be non-self-intersecting.
Any [[typical fiber|fiber bundle]] element at an initial point, $C_0 = C(x(0))$, may be [[parallel transport]]ed along a path, $x(t)$, by solving the (set of) first order ODE's,
$$
\fr{d}{dt} C = - \ve{v} \f{A} C
$$
Since the [[connection]] is in the Lie algebra of the structure group the solution may be expressed as $C(x(t)) = U(t) C_0$, in which $U(t) \in G$ is the ''path holonomy''. (We use the left action throughout this example, which may be adapted for the appropriate structure group action.) $U(t)$ is an element of the structure group that acts on any initial fiber element to give the solution to the parallel transport equation along a path. Plugging this form for the solution into the parallel transport equation, the path holonomy is the solution to the resulting ''holonomy equation'',
$$
\fr{d}{d t} U(t) = - \ve{v} \f{A} U
$$
from an initial condition of $U(0)=1$. (once again, the actual action of the connection on the holonomy will depend on the specific group action -- here taken to be the left action.) This equation may be readily converted to an integral equation,
$$
U(t) - 1 = - \int_0^t \f{dt} \fr{dx^i}{dt} A_i U(t) 
$$
For small displacements along the path, $x^i = x^i_0 + \va^i(t)$, the solution may be found to any order. To first order,
$$
U(t) \simeq 1 - \int_0^t \f{dt} \fr{d \va^i}{dt} A_i(x_0) U(0) = 1 - \va^i A_i
$$
and to second order,
\begin{eqnarray}
U(t) &\simeq& 1 - \int_0^t \f{dt} \fr{d \va^i}{dt} \lb A_i + \va^j \pa_j A_i \rb \lb 1 - \va^k A_k \rb \\
&\simeq& 1 - \va^i A_i + \va^{ij} \lb - \pa_j A_i + A_i A_j \rb
\end{eqnarray}
with the ''second order path dependence'' above defined as
$$
\va^{ij} = \lb \int_0^t \f{dt} \fr{d \va^i}{dt} \va^j \rb
$$

The solution to the holonomy equation may be written heuristically as
$$
U(t) = Pe^{-\int \f{A}} = Pe^{-\int_0^t \f{dt} v^i A_i}
$$
in which $P$ stands for "''path ordered''", and is there to make sure it's understood that this isn't a proper [[exponentiation]], but rather a way of heuristically writing the solution to the holonomy equation.

The path holonomy is the [[holonomy]] for a path that isn't necessarily closed.
<<ListTagged pb>>
Contracting components of $n$ dimensional [[permutation symbol]]s gives
\begin{eqnarray}
\ep^{\al \be \dots \ga} \ep_{\al \be \dots \ga} &=& \ll \et \rl n! \\
\ep^{\al \be \dots \ga} \ep_{\de \be \dots \ga} &=& \ll \et \rl \lp n-1 \rp! \de^\al_\de \\
\ep^{\al \be \ga \dots \de} \ep_{\ep \up \ga \dots \de} &=& \ll \et \rl 2! \lp n-2 \rp! \de^{\al \be}_{\lb \ep \up \rb} \\
\ep^{\al \dots \be \ga \dots \de} \ep_{\ep \dots \up \ga \dots \de} &=& \ll \et \rl p! \lp n-p \rp! \de^{\al \dots \be}_{\lb \ep \dots \up \rb}
\end{eqnarray}
The ''permutation symbol for label [[indices]]'' ranging over $n$ dimension is
$$
\ep_{\al \dots \de} = n! \de^0_{\lb \al \rd} \dots \de^{(n-1)}_{\ld \de \rb} = n! \de^{0 \dots (n-1)}_{\lb \al \dots \de \rb}
$$
using the antisymmetric [[index bracket]]. Alternatively, the indices may range from $1$ to $n$, or over any collection of $n$ numbers. It is antisymmetric in all indices &mdash; returning $1$ for positive permutations, $-1$ for negative permutations, and $0$ if any indices are repeated. The indices may be raised with the [[Minkowski metric]] to get
$$
\ep^{\al \be \dots \ga} = \et^{\al \de} \et^{\be \ep} \dots \et^{\ga \up} \ep_{\de \ep \dots \up} = \ll \et \rl \ep_{\al \be \dots \ga}
$$
Since [[Clifford basis vectors]] anti-commute, the permutation symbol arises geometrically as
\[ \ep_{\al \be \dots \ga} = \li \ga_\al \ga_\be \dots \ga_\ga \ga^- \ri \]
using the inverse of the [[pseudoscalar]], $\ga^- = \ga^{n-1} \dots \ga^1 \ga^0$, and the [[scalar part|Clifford grade]] operator.

The ''permutation symbol for coordinate [[indices]]'' ranging over $n$ dimension is
$$
\va^{i \dots j} = n! \de_{0 \dots (n-1)}^{\lb i \dots j \rb}
$$
These indices may be lowered with a [[metric]] to get
$$
\va_{i \dots j} = g_{ik} \dots g_{jl} \va^{k \dots l}
$$

Label and coordinate indices may be changed using the [[frame]] and its inverse,
$$
\begin{array}{rclcrcl}
\ep^{i \dots j} &=& \ep^{\al \dots \be} \lp e_\al \rp^i \dots \lp e_\be \rp^j & \;\;\;\;\;\; &
\va^{\al \dots \be} &=& \va^{i \dots j} \lp e_i \rp^\al \dots \lp e_j \rp^\be \\
\ep_{i \dots j} &=& \lp e_i \rp^\al \dots \lp e_j \rp^\be \ep_{\al \dots \be} & \;\;\;\;\;\; &
\va_{\al \dots \be} &=& \lp e_\al \rp^i \dots \lp e_\be \rp^j \va_{i \dots j}
\end{array}
$$
The two different permutation tensors contract to give [[determinant]]s,
\begin{eqnarray}
\ep^{\al \dots \be} \ep_{\al \dots \be} &=& \ll \et \rl n! \\
\va^{i \dots j} \va_{i \dots j} &=& \ll g \rl n! = \ll e \rl^2 \ll \et \rl n! \\
\ep^{\al \dots \be} \va_{\al \dots \be} &=& \ep^{i \dots j} \va_{i \dots j}
= \ep_{\al \dots \be} \va^{\al \dots \be} = \ep_{i \dots j} \va^{i \dots j}
= \ll e \rl n!
\end{eqnarray}
and they are related by:
$$
\va_{i \dots j} = \ll e \rl \ll \et \rl \ep_{i \dots j}
$$
//Yes, it's non-standard to have two permutation symbols -- but I didn't like all the factors of $\ll e \rl$ floating around. With the way I've defined them, the permutation tensors, $\ep_{\al \dots \be}$ and $\va^{i \dots j}$, actually ARE permutation symbols in these indices, which may be raised, lowered, or converted at will.//
<<ListTagged person>>
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*     <<slider chkSliderdgF dgF 'dg >' 'basics of differential geometry'>>
*     <<slider chkSliderkkF lieF 'lie >' 'Lie algebras, Lie groups'>>
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      <<ListTagged physics>>
<<ListTagged plugin>>
A ''principal bundle'' or //''principal $G$-bundle''// is a [[fiber bundle]] with arbitrary base, $M$, and structure group, $G$, acting via [[left action|Lie group]] on typical fibers -- [[Lie group geometries|Lie group geometry]] homeomorphic to $G$. Unlike the case for other kinds of bundles, the [[Lie group]], $G$, may also act [[transitive]]ly on the fibers via the [[right action|Lie group]]. For a section, $C(x)$, transforming under the left action [[gauge transformation]], $C \mapsto C' = g(x) C$, the [[covariant derivative]] is
$$
\f{\na} C = \f{d} C + \f{A} C
$$
with the ''principal bundle [[connection]]'' (//''gauge field''//), $\f{A} = \f{dx^i} A_i{}^B T_B$, a 1-form over $M$ valued in the [[Lie algebra]] of $G$.

Any fiber element, $C$, at $t=0$ may be [[parallel transport]]ed to $C(t) = U(t) C$ along a [[path]] on the base by a parameter dependent element $U \in G$, the [[path holonomy]], $U=Pe^{-\int_0^t \f{A}}$, satisfying the path holonomy equation,
$$
\fr{d}{dt} U(t) = - \ve{v} \f{A}  U
$$

Applying the covariant derivative twice (being careful with the signs of 1-forms hopping sides),
$$
\f{\na} \f{\na} C = \f{d} \lp \f{d} C + \f{A} C \rp + \f{A} \lp \f{d} C + \f{A} C \rp 
= \lp \f{d} \f{A} + \f{A} \f{A} \rp C
= \ff{F} C
$$
gives the ''principal bundle [[curvature]]'',
$$
\ff{F} = \f{d} \f{A} + \f{A} \f{A} = \f{d} \f{A} + \ha \lb \f{A}, \f{A} \rb 
$$
a Lie algebra valued 2-form with components,
$$
\ff{F^B} = \f{d} \f{A^B} + \ha \f{A^C} \f{A^D} C_{CD}{}^B
$$
This expression for the curvature may alternatively be obtained from the [[holonomy]].

Under a gauge transformation, $C(x) \mapsto C'(x) = g(x) C(x)$, the covariant derivative changes to
\begin{eqnarray}
\f{\na'} C' &=& g \lp \f{\na} C \rp\\
\lp \f{d} g \rp C + g \f{d} C + \f{A'} g C &=& g \f{d} C + g \f{A} C\\
\end{eqnarray}
giving the transformation law for the connection,
$$
\f{A'} = g \f{A} g^- + g \f{d} g^-
$$
An infinitesimal transformation, $g \simeq 1 + G^A T_A = 1 + G$, changes the connection to
$$
\f{A'} \simeq \f{A} - \f{d} G - \f{A} G + G \f{A} = \f{A} - \f{\na} G
$$
The curvature consequently transforms under a gauge transformation to
$$
\ff{F'} = \f{d} \f{A'} + \f{A'} \f{A'} = g \ff{F} g^- \simeq \ff{F} + \lb G, \ff{F} \rb
$$

The covariant derivative acting on a Lie algebra valued field (rather than a section) such as the curvature, transforming under the adjoint action, $\ff{F'} = g \ff{F} g^-$, is 
$$
\f{\na} \ff{F} = \f{d} \ff{F} + \f{A} \ff{F} - \ff{F} \f{A} = \f{d} \ff{F} + \lb \f{A}, \ff{F} \rb 
$$
A ''principle bundle'' is a [[principal bundle]] with strong moral fiber.
For a [[path]] designated by coordinates, $x^i(t)$, and parametrized by $t$ the [[velocity|tangent vector]] along the path with respect to this parameter is
\[ \ve{v} = v^i \ve{\pa_i} = \fr{dx^i}{dt} \ve{\pa_i} \]
and the magnitude of the velocity is
\[ \ll v \rl = \sqrt{\ll v \cdot v \rl} = \sqrt{\ll \lp \ve{v} \f{e} \rp \cdot \lp \ve{v} \f{e} \rp \rl} = \sqrt{\ll v^i v^j g_{ij} \rl} \]
using the [[frame]], $\f{e}$, to map the velocity into a [[rest frame]].  The change in ''proper time'', in seconds or other time [[units]], $T$, along the path is the integral along the path,
\[ \De \ta = \int \f{dt} \ll v \rl \]
The proper time describes how much time passes for a particle or observer moving along that path &mdash; in contrast, parameter time is not necessarily physically meaningful.  In terms of parameter time, the proper time changes as $\fr{d\tau}{dt} = \ll v \rl$.  If the path is parameterized (or reparameterized) by proper time so that $t(\tau)=\tau$, then the magnitude of the velocity with respect to proper time is one, $|v|=1$, and the parameter value, in $T$ units, marks out how a clock would read, traveling that path.

It is only possible to parameterize paths by proper time if their velocity is never null.  For positive [[signature|Minkowski metric]], $\et_{00} = +1$, a ''timelike'' path satisfies $v \cdot v > 0$, a ''null'' path satisfies $v \cdot v = 0$, and a ''spacelike'' path satisfies $v \cdot v < 0$.  This is reversed for negative signature.  A null path, $|v|=0$, is lightlike and time does not pass for a particle or observer on that path.  For a spacelike path, the proper time gives the ''spatial distance'' along the path, in "light seconds" or other spatial distance units, such as meters if the proper time is multiplied by the speed of light, $c$.   Massive particles and observers only travel timelike paths, and massless particles only travel null paths.

A path between two points that extremizes proper time is a [[geodesic]].
The Clifford ''pseudoscalar'', or ''//Clifford volume element//'', is the grade $n$ [[Clifford basis element|Clifford basis elements]], formed by the (antisymmetric) product of the $n$ [[Clifford basis vectors]],
$$
\ga = \ga_0 \ga_1 \dots \ga_{n-1} = \fr{\ll \et \rl}{n!} \ep^{\al \be \dots \ga} \ga_{\al \be \dots \ga}
$$
(giving the relation to the [[permutation symbol]]). For a [[Clifford algebra]] of [[signature|Minkowski metric]] $(p,q)$ the pseudoscalar squares to
\[ \ga \ga = \lp -1 \rp^q \lp -1 \rp^{\fr{n \lp n+1 \rp}{2}} \]
and so has the inverse
\[ \ga^- = \lp -1 \rp^q \lp -1 \rp^{\fr{n \lp n+1 \rp}{2}} \ga\]
The pseudoscalar commutes with all even [[Clifford grade]]d elements, $A^e \ga = \ga A^e$, and commutes or anticommutes with odd graded elements, dependent on overall dimension, $A^o \ga = (-1)^{n+1} \ga A^o$.

For [[Cl(1,3)]], $\ga \ga = -1$ and so $\ga^- = -\ga$.  And, since $n=1+3=4$, this [[spacetime]] pseudoscalar anticommutes with odd Clifford grade elements.
A map, $\phi:x \mapsto y$, takes a point $x$ on a [[manifold]] $M$ to a point $y$ on the manifold $N$ (which may be $M$ itself).These points have coordinates $x^i$ and $y^j = \ph^j(x)$ in some local patches. When the map is smooth (continuously differentiable) the [[partial derivative]], $\fr{\pa y^j}{\pa x^i} = \pa_i \ph^j(x)$, is well defined. In this way $\phi$ induces a map, $\phi^*$, from any [[differential form]], $\f{a}$, at $y$ to a form at $x$ -- the ''pullback'' of $\f{a}$ along $\phi$,
$$
\f{\phi^*a} = \lb \f{\phi^*a} \rl_x = \phi^* \lb \f{a} \rl_y =  \phi^* \f{a}= \phi^* \f{dy^j} a_j(y) = \f{dx^i} \lb \fr{\pa y^j}{\pa x^i} \rl_x a_j(y) = \lp \f{\pa} \ve{\ph} \rp \f{a}
$$
using the [[vector-form algebra]]. This generalizes to forms of any grade. $\phi$ also induces a map, $\phi_*$, from any [[tangent vector]], $\ve{v}$, at $x$ to a vector at $y$ -- the ''pushforward'' of $\ve{v}$ along $\phi$,
$$
\ve{\phi_*v} = \lb \ve{\phi_*v} \rl_y = \phi_* \lb \ve{v} \rl_x =  \phi_* \ve{v}= \phi_* \ve{\fr{\pa}{\pa x^i}} v_i(x) = \ve{\fr{\pa}{\pa y^j}} \lb \fr{\pa y^j}{\pa x^i} \rl_x v_i(x) = \lp \ve{v} \f{\pa} \rp \ve{\ph} 
$$
It is easy to confirm that $\ve{v} \lp \phi^* \f{a} \rp = \lp \phi_* \ve{v} \rp \f{a}$. Note that forms pull back and vectors push forward even if the map is not invertible or bijective. It is more natural for forms to pull back and for vectors to push forward under a map.

However, if the map is smooth and invertible it is a [[diffeomorphism]] and its inverse partial derivative, $\fr{\pa x^i}{\pa y^j} = \pa_j \ph^-i(y)$, is also well defined. For this kind of map, forms also push forward and vectors also pull back,
\begin{eqnarray}
\f{\phi_*a} &=& \lb \f{\phi_*a} \rl_y = \phi_* \lb \f{a} \rl_x =  \phi_* \f{a} = \phi_* \f{dx^i} a_i(x) = \f{dy^j} \lb \fr{\pa x^i}{\pa y^j} \rl_y a_i(x) = \lp \f{\pa} \ve{\ph^-} \rp \f{a} \\
\ve{\phi^*v} &=& \lb \ve{\phi^*v} \rl_x = \phi^* \lb \ve{v} \rl_y =  \phi^* \ve{v} = \phi^* \ve{\fr{\pa}{\pa y^j}} v_j(y) = \ve{\fr{\pa}{\pa x^i}} \lb \fr{\pa x^i}{\pa y^j} \rl_y v_j(y) = \lp \ve{v} \f{\pa} \rp \ve{\ph^-} 
\end{eqnarray}
This also allows the pushforward and pullback to be defined for [[vector valued form]]s,
\begin{eqnarray}
\f{\ve{\phi_*A}} &=& \lb \f{\ve{\phi_*A}} \rl_y = \phi_* \lb \f{\ve{A}} \rl_x =  \phi_* \f{\ve{A}} = \phi_* \lp \f{dx^i} A_i{}^j(x) \ve{\fr{\pa}{\pa x^j}} \rp =
 \f{dy^j} \lb \fr{\pa x^i}{\pa y^j} \rl_y A_i{}^k(x) \lb \fr{\pa y^m}{\pa x^k} \rl_x \ve{\fr{\pa}{\pa y^m}}
= \lp \f{\pa} \ve{\ph^-} \rp \lp \f{\ve{A}} \f{\pa} \rp \ve{\ph} \\
\f{\ve{\phi^*A}} &=& \lb \f{\ve{\phi^*A}} \rl_x = \phi^* \lb \f{\ve{A}} \rl_y =  \phi^* \f{\ve{A}} = \phi^* \lp \f{dy^i} A_i{}^j(y) \ve{\fr{\pa}{\pa y^j}} \rp =
 \f{dx^i} \lb \fr{\pa y^j}{\pa x^i} \rl_x A_j{}^k(y) \lb \fr{\pa x^m}{\pa y^k} \rl_y \ve{\fr{\pa}{\pa x^m}}
= \lp \f{\pa} \ve{\ph} \rp \lp \f{\ve{A}} \f{\pa} \rp \ve{\ph^-}
\end{eqnarray}
Pullbacks and/or pushforwards can also be done for vector and form fields over manifolds by extending the operation over every manifold point.
<<ListTagged qft>>
<<ListTagged qm>>
The ''quaternions'' are a four-dimensional [[division algebra]], spanned by four basis elements, $e_0,e_1,e_2,e_3$. The quaternion identity element is $e_0=1$, and the other three can be thought of as different imaginary directions, squaring to $-1$. Quaternionic multiplication is non-commutative and associative. The multiplication table for the basis elements is
|            | $e_0$ | $e_1$ | $e_2$ | $e_3$ |
| $e_0$ | $e_0$ | $e_1$ | $e_2$ | $e_3$ |
| $e_1$ | $e_1$ | $-e_0$ | $e_3$ | $-e_2$ |
| $e_2$ | $e_2$ | $-e_3$ | $-e_0$ | $e_1$ |
| $e_3$ | $e_3$ | $e_2$ | $-e_1$ | $-e_0$ |
which can be written using an ''quaternion multiplication coefficient matrix'' as
$$
e_a e_b = M_{ab}{}^c e_c
$$
so, for example, $e_1 e_2 = e_3$ and $M_{12}{}^3 = 1$. ''Quaternionic conjugation'' is given by
$$
e_{\os{0}}=e_0 \;\;\;\;\; e_{\os{1}}=-e_1 \;\;\;\; e_{\os{2}}=-e_2 \;\;\;\; e_{\os{3}}=-e_3 
$$
and satisfies $\widetilde{(e_a e_b)} = e_{\os{b}} e_{\os{a}}$, so we have $M_{ab}{}^c = M_{\os{b} \os{a}}{}^{\os{c}}$. 

The quaternion metric is defined as
$$
n_{ab}=(e_a, e_b) = \ha ( e_a e_{\os{b}} + e_b e_{\os{a}} )
$$
and for the quaternions is $n_{ab}=\de_{ab}$. This metric can be used to raise or lower quaternion indices (which has no effect for the quaternions, but matters for the ''split-quaternions'').
If, in four dimensions of signature $(4,0)$ (or $(2,2)$), we have a [[Clifford vector|Clifford basis vectors]], $v=v^c \ga_c \in V^{(4)}$, a positive real [[chiral]] [[spinor]], $\ps=\ps^a Q^+_a \in S^{(4)+}$, and a negative real chiral spinor, $\ch=\ch^b Q^-_b \in S^{(4)-}$, satisfying
$$
\ch = v \ps
$$
using the usual matrix representative product of a vector times a spinor, then we can use a [[quaternionic representation of Clifford algebra]] to write this as
$$
\ch^b = v^c \Ga_c{}^b{}_a \ps^a = v^c \ps^a M_{\os{a} \os{c}}{}^b
$$
where we have replaced the Clifford matrix representative coefficients with the equivalent [[quaternion]] (or split-quaternion) multiplication coefficients, and used the cyclic identity. We can see that this relates directly to the [[quaternion]]ic product
$$
v \ps = v^c \ps^a e_c e_a = v^c \ps^a M_{c a}{}^{\os{b}} e_{\os{b}}
= v^c \ps^a M_{\os{a} \os{c}}{}^b e_{\os{b}}
= \tilde{\ch} 
$$
and quaternionic conjugation. Thus we have a useful //confusion// of quaternions with vectors and spinors,
\begin{eqnarray}
v = v^c e_c & \leftrightarrow & v = v^c \ga_c \\
\ps = \ps^a e_a & \leftrightarrow & \ps=\ps^a Q^+_a \\
\ch = \ch^b e_b & \leftrightarrow & \ch=\ch^b Q^-_b \\
\tilde{\ch} = v \ps & \leftrightarrow & \ch = v \ps
\end{eqnarray}
The [[quaternion]] (or split-quaternion) multiplication table can be used to define
$$
\Ga_{cab} = M_{\os{a} \os{b}}{}_c = M_{\os{a}\os{b}}{}^d n_{dc}  
$$
which satisfies a ''cyclic identity'',
$$
\Ga_{cab} = \Ga_{abc} = \Ga_{bca} 
$$
Further defining $\overline{\Ga}_{cab} = \Ga^c{}_{ba}$, these can be used to construct a $8 \times 8$ real chiral [[Clifford matrix representation]] of Cl(4) (or Cl(2,2)), with the Clifford basis vectors represented as
$$
\ga_c =
\lb \begin{array}{cc}
0 & \overline{\Ga}_c{}^b{}_a \\
\Ga_c{}^b{}_a & 0
\end{array} \rb
$$
This remarkable fact holds because the quaternionic product leads to satisfaction of the fundamental [[Clifford algebra]] identity,
\[ \ga_a \cdot \ga_b = \ha \lp \ga_a \ga_b + \ga_b \ga_a  \rp = n_{ab} \]
The main idea of triality is based on an isomorphism between [[Clifford vectors|Clifford basis vectors]], positive [[chiral]] [[spinor]]s, and negative chiral spinors, in eight dimensions -- but this can also be done in four dimensions. Via [[quaternionic confusion]], this can be related to an automorphism of a set of three quaternions.

The relevant ''triality'' is a scalar function of three [[quaternion]]s (or split-quaternions), or of the equivalent vector and spinors
$$
T(v,\ps,\ch) = (\tilde{\ch},v\ps) = (\tilde{v},\ps\ch) = (\tilde{\ps},\ch v) = v^c \ch^b \ps^a \Ga_{cba}
$$
This triality can be //dualized//, producing dualities -- maps from $\{v,\ps,\ch\}$ to $\{v',\ps',\ch'\}$ -- specified by chosen unit quaternions, such as $v_0$ or $\ch_0$ (satisfying $\tilde{v}_0 v_0 = 1$ and $\tilde{\ch}_0 \ch_0 = 1$). Explicitly, such dualities are "reflections," mapping $\ps \leftrightarrow \ch$,
\begin{eqnarray}
v' &=& -v_0 \tilde{v} v_0 =  -v_0^a v^c v_0^b \Ga^d{}_{\os{a} \os{f}} \Ga^f{}_{c \os{b}} e_d \\
\ps' &=& \widetilde{(\ch v_0)} = v_0^c \ch^b \Ga^a{}_{cb} e_a \\
\ch' &=& \widetilde{(v_0 \ps)} = \ps^a v_0^c \Ga^b{}_{ac} e_b
\end{eqnarray}
and $v \leftrightarrow \ps$,
\begin{eqnarray}
v'' &=& \widetilde{(\ps \ch_0)} = \ch_0^b \ps^a \Ga^c{}_{ba} e_c \\
\ps'' &=& \widetilde{(\ch_0 v)} = v^c \ch_0^b \Ga^a{}_{cb} e_a \\
\ch'' &=& -\ch_0 \tilde{\ch} \ch_0 =  - \ch_0^a \ch^b \ch_0^c \Ga^d{}_{\os{a} \os{f}} \Ga^f{}_{b \os{c}} e_d
\end{eqnarray}
Triality is invariant under these dualities, $T(v',\ps',\ch') = T(v,\ps,\ch)$, and they can be combined to produce a ''triality automorphism'',
\begin{eqnarray}
v''' &=& \widetilde{(\ps' \ch_0)} = \tilde{\ch}_0 (\ch v_0) \\
\ps''' &=& \widetilde{(\ch_0 v')} = -  (\tilde{v}_0 v \tilde{v}_0) \tilde{\ch}_0 \\
\ch''' &=& -\ch_0 \tilde{\ch}' \ch_0 = - \ch_0 (v_0 \ps) \ch_0 
\end{eqnarray}
mapping $v \to \ps$, $\ps \to \ch$, and $\ch \to v$. Applying either duality automorphism twice is the identity, as is applying a triality automorphism thrice.
[[fiber bundle]]

[[tangent bundle]]

$$
GL \mapsto O
$$
metric

$$
GL \mapsto U
$$
Hermitian metric

Ref:
http://en.wikipedia.org/wiki/Principal_bundle#Reduction_of_the_structure_group
A [[Lie group]], $G$, has a [[Lie algebra]], ${\rm Lie}(G)$, spanned by $n$ generators, $T_I$. A [[subgroup]], $H \subset G$, has its own set of $n_H$ generators, $H_P \in {\rm Lie}(H)$, which can be chosen from some of $G$'s, $H_P = T_P$ (with $P$-series indices running from $1$ to $n_H$). The $n_S = n_G - n_H$ remaining generators, $K_A = T_A$, are the ''coset generators'' -- so $\{T_I\} = \{H_P\} \oplus \{K_A\}$. Let the [[vector space]] spanned by the coset generators be labeled ${\rm Lie}(G/H)$ (even though the [[homogeneous space]], $S=G/H$, isn't necessarily a group). The subgroup $H$ is ''reductive'' in $G$ iff the Lie algebra decomposition,
$$
{\rm Lie}(G) = {\rm Lie}(H) \oplus {\rm Lie}(G/H)
$$
is invariant under the adjoint action of ${\rm Lie}(H)$,
\begin{eqnarray}
{\rm Ad}_{{\rm Lie}(H)} {\rm Lie}(H) &=& \lb {\rm Lie}(H), {\rm Lie}(H) \rb = {\rm Lie}(H) \\
{\rm Ad}_{{\rm Lie}(H)} {\rm Lie}(G/H) &=& \lb {\rm Lie}(H), {\rm Lie}(G/H) \rb = {\rm Lie}(G/H)
\end{eqnarray}
Specifically, iff $H$ is reductive in $G$ the commutation relations are:
\begin{eqnarray}
\lb H_P, H_Q \rb &=& C_{PQ}{}^R H_R \\
\lb H_P, K_A \rb &=& C_{PA}{}^B K_B \\
\lb K_A, K_B \rb &=& C_{AB}{}^C K_C + C_{AB}{}^R H_R
\end{eqnarray}

Here's another way of looking at what this means: By choosing a particular set of generators, $T_I$, for $G$, the [[Killing form]] for ${\rm Lie}(G)$,
$$
g_{IJ} = C_{IK}{}^L C_{JL}{}^K
$$
can be diagonalized, with the structure constants then satisfying $C_{IJK} = -C_{IKJ}$ after lowering indices with $g$. A subgroup, $H$, is ''reductive'' in $G$ iff its generators can be chosen from these, $H_P = T_P$, that diagonalized the Killing form.
When a [[subgroup]], $H \subset G$, is [[reductive]] in $G$ the [[Lie group tangent bundle geometry]], represented by the [[frame]] 1-forms,
$$
\f{E^J} = \f{\xi_R^J} = \f{{\cal I}^J} = \lp T^J , \f{\cal I} \rp = \lp T^J , \f{\cal I} \rp = \lp T^J , g^-(z) \f{d} g(z) \rp
$$
splits into two parts. The coordinates, $z^i$, over the [[Lie group manifold|Lie group geometry]] split into two sets of coordinates: the coordinates, $y^p$, over $H$, and the "leftover" coordinates, $x^a$, over a base manifold, $M$, of dimension $n_S = (n_G - n_H)$. So a point (element) of $G$ is specified by
$$
g(z) = g(x,y) = r(x) \, h(y)
$$
with $h(y) \in H$ acting on $r(x) \in G$ via the [[right action|group]]. The arbitrarily chosen ''reference section'', $r : M \to G$, corresponds to the [[submanifold]] in $G$ corresponding to $y=0$. The base manifold may be thought of as the [[homogeneous space]], $M=S=G/H$. With this choice of coordinates, and reductivity assumed, the frame 1-forms over $G$ split as:
$$
\begin{eqnarray}
\f{E^A}(z) &=& \lp K^A , h^- \f{e_S} h(y) \rp = \f{e_S^B}(x) \, \lp L^h\rp^A{}_B(y) \\
\f{E^P}(z) &=& \lp H^P , h^- \f{A_S} h(y) + h^- \f{d} h(y) \rp = \f{A_S^Q}(x) \, \lp L^h \rp^P{}_Q(y) + \f{e_H^P}(y)
\end{eqnarray}
$$
in which $\f{e_S}(x) = \f{e_S^B} K_B$ is the [[homogeneous space frame|homogeneous space geometry]], $\f{A_S}(x) = \f{A_S^Q} H_Q$, is the [[homogeneous H-connection|homogeneous space geometry]], $\f{{e'}_H^P}(y)$ are the frame 1-forms over $H$. In this way, a reductive Lie group geometry is equivalent to an [[Ehresmann homogeneous space geometry]]. The above [[left-right rotator]] for $H$ in $G$ is,
$$
\lp L^h \rp^I{}_J = \lp T^I , h^- T_J h(y) \rp
$$
and the left-right rotator for $G$ splits as
$$
L^I{}_J = \lp T^I , g^- T_J g(z) \rp = \lp T^I , h^- r^- T_J r(x) h(y) \rp = \lp L^h \rp^I{}_K \, \lp L^r \rp^K{}_J
$$
in which the left-right rotator for $r$ is
$$
\lp L^r \rp^K{}_J = \lp T^I , r^- T_J r(x) \rp
$$
Rotating the frame 1-forms gives the frame of a particular [[Kaluza-Klein]] spacetime,
$$
\begin{eqnarray}
\f{E'^A}(z) &=& \lp L^h \rp_B{}^A \, \f{e^B} = \f{e_S^B}(x) \\
\f{E'^P}(z) &=& \lp L^h \rp_Q{}^P \, \f{e^Q} = \f{A_S^P}(x) + \f{e_H^Q} \lp L^h \rp_Q{}^P
\end{eqnarray}
$$
with [[spacetime]] frame 1-forms, $\f{e_S^B}(x)$, over $M$, and $\f{e'_H}(y) = \f{e_H^Q} \lp L^h \rp_Q{}^P$ identified as the frame 1-forms over the small compact Kaluza-Klein manifold, $H$.

It is straightforward to calculate the inverse to the matrix of frame 1-form components, and get the [[orthornormal basis vector fields|frame]] over $G$,
$$
\begin{eqnarray}
\ve{E_A}(z) &=& \lp L^h \rp_A{}^B \, \ve{e^S_B}(x) - \lp L^h \rp_A{}^C \lp \ve{e^S_C} \f{A_S^Q} \rp \lp L^h \rp^P{}_Q \, \ve{e^H_P}  \\
\ve{E_P}(z) &=& \ve{e^H_P}(y)
\end{eqnarray}
$$
corresponding to the [[left invariant Killing vector fields|Lie group geometry]], $\ve{\xi^R_J} = \ve{E_J}$, and satisfying $\ve{E_I} \f{E^J} = \de_I^J$. Note that, in the coordinates we have chosen, the Killing vector fields over $G$ corresponding to the [[Lie algebra]] generators of $H$ equal the Killing vectors over $H$,
$$
\ve{\xi^R_P}(z) = \ve{\xi^{HR}_P}(y) 
$$
a fact that is true iff $H$ is reductive in $G$.

The [[reductive Lie group tangent bundle geometry]], including the connection and curvature, also splits in an interesting way.
A ''reductive Lie group tangent bundle geometry'' is a [[Lie group tangent bundle geometry]] for a [[reductive Lie group geometry]]. The [[frame]] 1-forms, $\f{E^J}=\f{{\cal I}^J}$, over the Lie group manifold, $G$, split in adapted coordinates as
\begin{eqnarray}
\f{E^A}(x,y) &=& \f{e_S^B}(x) \, \lp L^h\rp^A{}_B(y) \\
\f{E^P}(x,y) &=& \f{A_S^Q}(x) \, \lp L^h \rp^P{}_Q(y) + \f{e_H^P}(y)
\end{eqnarray}
in which $\f{e_S^B}$ and $\f{A_S^Q}$ are the [[homogeneous space frame|homogeneous space geometry]] forms and [[homogeneous H-connection|homogeneous space geometry]] forms, $\lp L^h \rp^A{}_B = \lp H^A, h^- H_B h(y) \rp$ is the [[left-right rotator]] over $H$, and $\f{e_H^P}$ are the frame 1-forms over $H$. Using the [[Maurer-Cartan equation|Maurer-Cartan form]] over $G$,
$$
0 = \f{d} \f{E^J} + \ha \f{E^I} \f{E^K} C_{IK}{}^J
$$
and insisting that the [[torsion]] vanish over $G$,
$$
\ff{T^J} = 0 = \f{d} \f{E^J} + \f{W}^J{}_K \f{E^K}
$$
gives the same [[tangent bundle spin connection|tangent bundle connection]],
$$
\f{W}^J{}_K = \ha \f{E^I} C_{IK}{}^J
$$
over $G$, as for a Lie group tangent bundle geometry. These split to:
\begin{eqnarray}
\f{W}^B{}_C &=& \ha \f{e_S^D} \, \lp L^h\rp^A{}_D C_{AC}{}^B + \ha \lp \f{A_S^Q} \, \lp L^h \rp^P{}_Q + \f{e_H^P} \rp C_{PC}{}^B \\
\f{W}^B{}_R &=& \ha \f{e_S^D} \, \lp L^h\rp^A{}_D C_{AR}{}^B \\
\f{W}^Q{}_R &=& \ha \lp \f{A_S^Q} \, \lp L^h \rp^P{}_Q + \f{e_H^P} \rp C_{PR}{}^Q
\end{eqnarray}
Note that these are the same values taken by the [[Cartan tangent bundle spin connection]] when $\f{e^A}=\f{e_S^A}$ and $\f{A^P}=\f{A_S^P}$, with
\begin{eqnarray}
F^{HS}_{DEQ} &=& \ve{e^S_E} \ve{e^S_D} \lp \f{d} \f{A^S_Q} + \ha \f{A_S^P} \f{A_S^R} C_{PRQ} \rp = -C_{DEQ} \\
\f{\nu^S}_{EF} &=& - \ha \f{e_S^D} C_{DEF} - \f{A_S^Q} C_{QEF}
\end{eqnarray}
The [[Riemann curvature]] is also the same as for the Lie group tangent bundle geometry, $\ff{R}{}^J{}_K = - \fr{1}{4} \f{E^I} \f{E^M} C_{KLI} C_M{}^{JL}$, which splits as:
\begin{eqnarray}
\ff{R}{}^B{}_C &=& - \fr{1}{4} \f{e_S^E} \, \lp L^h\rp^A{}_E \, \f{e_S^F} \, \lp L^h\rp^D{}_F \, C_{CLA} C_D{}^{BL} \\
\ff{R}{}^B{}_R &=& - \fr{1}{4} \lp \f{A_S^Q} \, \lp L^h \rp^P{}_Q + \f{e_H^P} \rp \f{e_S^F} \, \lp L^h\rp^D{}_F \, C_{RLP} C_D{}^{BL} \\
\ff{R}{}^Q{}_R &=& - \fr{1}{4} \lp \f{A_S^U} \, \lp L^h \rp^P{}_U + \f{e_H^P} \rp \lp \f{A_S^V} \, \lp L^h \rp^T{}_V + \f{e_H^T} \rp C_{RLP} C_T{}^{QL}
\end{eqnarray}
The [[Ricci curvature]] is $\f{R}{}_J = - \fr{1}{4} \f{E}{}_J$ and the [[curvature scalar]] is $R = -\fr{1}{4} n_G$.
[[Carlo Rovelli]] has a nice paper out on a local interpretation of EPR setup:
[[Relational EPR|http://arxiv.org/abs/quant-ph/0604064]]

Conventionally, an observer, $A$, at $\al$ measures a state,
$$\ll \ps \ri = \fr{1}{\sqrt{2}} \lp \ll + \ri^z_\al \ll - \ri^z_\be - \ll - \ri^z_\al \ll + \ri^z_\be \rp$$
and thus collapses the wave function for the spin partner at spatially distant $\be$.  Rovelli says $A$ only measures and determines the new information locally, with an observation $S^z_{A, \al} = \pm$, adding to the known square of total spin, $S^2_{A, \al + \be} = 0$, and thus allowing $A$ to infer the state that will be measured at $\be$ once it is back in causal contact.

Suggests the wave function, $\psi$, should be static and represent state of information, while observables (operators) should evolve in time -- i.e. the Heisenberg picture.

Perhaps the wavefunction, and its collapse, is a poor descriptor?  Since what's really going on with "collapse" is just new information being acquired by an observer.

Hmm, this seems similar to Von Neuman's treatment of QM using a density matrix.
[[Lie algebra]]


Ref:
*Clara Loeh
**[[Representation Theory of Lie Algebras|papers/Loeh - Representation Theory of Lie Algebras.pdf]]
A ''rest frame'', or //''Minkowski space''// is an inertial reference frame.  It is flat, having no curvature, and has cartesian coordinates, $x^\al$, which multiply orthonormal vectors, $\ga_\al$, to designate points, $x=x^\al \ga_\al$, in the space.  Since the space is flat and cartesian, the basis vectors, $\ga_\al$, [[Clifford basis vectors]], serve as both unit rulers on the space and as unit vectors at every space point.  The coordinates carry [[units]] of time, $[x^\al]=T$, which may be in seconds or other time units for $x^0$, and in ''light seconds'' (with the same unit, $T$) for [[spatial|indices]] coordinates, $x^\pi$.  (spatial distances in meters, or other spatial length units, are obtained by multiplication with the speed of light, $c$.)

For a [[path]] in this space, $x(t)$, parameterized by $t$, the [[velocity|tangent vector]] along the path, with respect to this parameter, is
\[ v = \fr{dx}{dt} = \fr{dx^\al}{dt} \ga_\al = v^\al \ga_\al \]
Along a path segment, the [[proper time]], $\tau$, in seconds or other time units, steps forward as
\[ \f{d\tau} = \sqrt{\ll \f{dx^\al} \f{dx^\be} \et_{\al \be} \rl} \]
in which $\et_{\al \be}$ is the [[Minkowski metric]].  In terms of parameter time, the proper time changes as
\[ \fr{d\tau}{dt} = \sqrt{\fr{dx^\al}{dt} \fr{dx^\be}{dt} \et_{\al \be}} = \sqrt{v^\al v^\be \et_{\al \be}} = \sqrt{v \cdot v} = \ll v \rl \]
If the path is parameterized (or reparameterized) by proper time, $t(\tau)=\tau$, then the magnitude of the velocity with respect to proper time is one, $|v|=1$.  The proper time describes how time is experienced by a particle or observer moving along that path.  The time coordinate, $x^0$, in the rest frame is the proper time for a path along that coordinate line &mdash; a line having constant velocity $v=\ga_0$.  ''Null lines'', paths followed by massless particles such as light, are ''null paths'', $|v|=0$, of constant velocity, $\fr{dv}{dt}=0$.

A rest frame is only a local, flat approximation to a curved [[manifold]] at a point.  The frame coordinates, $x^\al$, do not have a meaningful mapping to manifold coordinates, $x^i$, which have no units.  However, the tangent vectors and [[differential form]]s on a manifold do map back and forth to vectors and multivectors in a rest frame via the [[frame]],
\[ v = \ve{v} \f{e} \]
which has temporal units, $[\f{e}]=T$.  Since all physics can be described by local interactions, physics described locally in a rest frame can be mapped back and forth to physics on a curved manifold via the frame.  In a heuristic sense, the frame coefficients are $\lp e_i \rp^\al = \fr{\pa x^\al}{\pa x^i}$ even though the frame coordinates, $x^\al$, are not well defined functions of manifold coordinates, $x^i$, for an arbitrarily curved manifold.
The [[right action|group]] of one [[Lie group]] element on all others induces a [[diffeomorphism]], $\ph_h(x)$ on the [[Lie group manifold|Lie group geometry]],
$$
R_h g(x) = g(x) h = g(\ph_h(x))
$$
A vector field on the Lie group manifold is ''right invariant'' iff it is invariant under the [[pushforward|pullback]] of this diffeomorphism for arbitrary $h$,
$$
R_{h*} \ve{v}(x) = \ve{v} \f{\pa} \ph_h(x) = \ve{v}(\ph_h(x))
$$
The partial derivative of the diffeomorphism in the above expression is computed explicitly by using the [[chain rule|partial derivative]],
$$
\f{\pa} g(\ph_h(x)) = \lp \f{\pa} \ve{\ph_h}(x) \rp \f{\pa} g(\ph_h) = \f{\pa} g(x) h
$$
and the defining equation for the [[left action vector fields and 1-forms|Lie group geometry]],
$$
\f{\pa} g = \f{\xi_L^B} T_B g
$$
to write
$$
\lp \f{\pa} \ve{\ph_h}(x) \rp \f{\xi_L^B}(\ph_h) T_B g h = \f{\xi_L^B}(x) T_B g h
$$
and get
$$
\f{\pa} \ve{\ph_h}(x) = \f{\xi_L^B}(x) \ve{\xi^L_B}(\ph_h(x))
$$
This implies the left action vector fields are right invariant,
$$
R_{h*} \ve{\xi^L_C}(x) =  \ve{\xi^L_C}(x) \f{\pa} \ph_h(x) = \ve{\xi^L_C}(x) \f{\xi_L^B}(x) \ve{\xi^L_B}(\ph_h) = \ve{\xi^L_C}(\ph_h(x))
$$

A [[differential form]] is right invariant iff it is invariant under the [[pullback]], $R_h^* \nf{F}(\phi_h(x)) = \nf{F}(x)$. The 1-form duals to right invariant vector fields, such as the duals to the left action vector fields, are right invariant. Since autodiffeomorphisms are invertible, these statements may be summarized by defining any form or [[vector valued form]] to be right invariant iff it is invariant under the pushforward, $R_{h*} \nf{\ve{K}}(x) = \nf{\ve{K}}(\phi_h(x))$, or pullback.
Ref:
*Jeffrey D. Olson
**[[Instantons and Self-Dual Gauge Fields|papers/selfdual.ps]]
A connected [[Lie group]], $G$, is ''simple'' iff it has no [[normal subgroup]]s. In this sense, other Lie groups can have simple Lie groups as "prime factors." A Lie algebra, ${\rm Lie}(G)$, is ''simple'' iff it's only [[ideal|spinor]] is itself -- i.e. there is no other ${\rm Lie}(H) \in {\rm Lie}(G)$ such that ${\rm Lie}(G) {\rm Lie}(H) = {\rm Lie}(H)$. A Lie group is simple iff its Lie algebra is simple.

A Lie algebra is ''semi-simple'' iff it is the direct sum of simple Lie algebras. A connected Lie group is ''semi-simple'' iff its Lie algebra is semi-simple.
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The [[Lie algebra]] of the symplectic group of rank $3$ and dimension $3(2\times3+1)=21$, also labeled $C_3$. In general, the ''symplectic group'', $Sp(n)$, having dimension $n(2n+1)$, is the compact real form of the complex Lie group $Sp(2n,\mathbb{C})$, which has as split real form, $Sp(2n,\mathbb{R})$. (Yes, this naming convention is pretty messed up!)

The noncompact real symplectic Lie algebra of rank $n$, $Sp(2n,\mathbb{R})$, corresponds to $2n \times 2n$ complex matrices, $A$, satisfying
$$
\Om A^T \Om = A  
$$
in which
$$
\Om=\left[\begin{array}{cc}
0 & I_n\\
-I_n & 0\end{array}\right]
$$
which may be rearranged by row and column.

For $Sp(6,\mathbb{R})$, $A$ may be written as a $3 \times 3$ matrix of [[quaternion]]s.
''Spacetime'' (//''Lorentzian spacetime''//) is a four dimensional [[manifold]], $M$, with a [[metric]], $g_{ab}$. This metric may be derived from four ''spacetime [[orthonormal basis vectors|frame]]'', $\ve{e_\mu} = \lp e_\mu \rp^a \ve{\pa_a}$ (spanning the ''spacetime [[tangent bundle]]''), with appropriate [[indices]], along with the [[Minkowski metric]] (chosen to have positive time signature, unless stated otherwise). The ''spacetime [[torsion]]'', usually taken to be zero, determines the ''spacetime [[tangent bundle spin connection|tangent bundle connection]]'', $\f{w}^\mu{}_\nu = \f{dx^a} w_a{}^\mu{}_\nu$, which in turn determines the ''spacetime [[Riemann curvature]]''. The spacetime orthonormal basis vectors have an inverse, the ''spacetime [[frame]] 1-forms'', $\f{e^\mu} = \f{dx^a} \lp e_a \rp^\mu$.

This structure matches that of a [[Clifford vector bundle]] with the spacetime manifold as its base. The Clifford algebra fiber of this bundle is the ''spacetime [[Clifford algebra]]'', [[Cl(1,3)]], generated by four [[Clifford basis vectors]], or with the other choice of signature, Cl(3,1). For this bundle, the [[spacetime frame]] is $\f{e} = \f{dx^a} \lp e_a \rp^\mu \ga_\mu$ and the [[spacetime spin connection]] is $\f{\om} = \f{dx^a} \ha w_a{}^{\mu \nu} \ga_{\mu \nu}$, which is determined by the spacetime torsion,
$$
\ff{T} = \f{d} \f{e} + \f{\om} \times \f{e}
$$
The spacetime spin connection determines the ''spacetime [[Clifford-Riemann curvature]]'',
$$
\ff{R} = \f{d} \f{\om} + \ha \f{\om} \f{\om}
$$
which has coefficients equal to the spacetime Riemann curvature tensor, $R_{ab}{}^{\mu \nu}$. This, along with the spacetime frame, determines the ''spacetime [[Clifford-Ricci curvature]]'' and ''spacetime [[Clifford curvature scalar]]''.

Sometimes //''spacetime''// is used to refer to manifolds of dimension higher than four, along with a metric. In these cases the word should be used with qualifiers such as "any" or "generalized". A ''Riemannian spacetime'' is like a Lorentzian spacetime, but with a positive definite metric.
A [[frame]] over [[spacetime]] is a [[Cl(1,3)]] vector valued [[1-form]] field (a [[Clifform]]),
$$
\f{e} = \f{dx^a} \lp e_a \rp^\mu \ga_\mu
$$
which may be written as a matrix valued 1-form, using the [[Weyl representation|Dirac matrices]], as
\begin{eqnarray}
\f{e} &=& \f{e^\mu} \ga_\mu =
\lb \begin{array}{cc}
0 & \f{e_L} \\
\f{e_R} & 0
\end{array} \rb
=
\lb \begin{array}{cc}
0 & \f{e^0} - \f{e^\va} \si^P_\va \\
\f{e^0} + \f{e^\va} \si^P_\va & 0
\end{array} \rb
\\
&=& 
\lb \begin{array}{cccc}
0 & 0 & \f{e^0}-\f{e^3} & -\f{e^1}+i\f{e^2} \\
0 & 0 & -\f{e^1}-i\f{e^2} & \f{e^0}+\f{e^3} \\
\f{e^0}+\f{e^3} & \f{e^1}-i\f{e^2} & 0 & 0 \\
\f{e^1}+i\f{e^2} & \f{e^0}-\f{e^3} & 0 & 0
\end{array} \rb
\end{eqnarray}
with left and right [[chiral]] parts, $\f{e_{L/R}}$, represented by $2\times2$ complex matrix (or [[quaternion]]) valued 1-forms. The inverse of the frame is Cl(1,3) vector valued [[tangent vector]] field,
$$
\ve{e} = \ga^\mu \lp e_\mu \rp^a \ve{\pa_a}
$$
which may be written as a matrix valued tangent vector as
\begin{eqnarray}
\ve{e} &=& \ga^\mu \ve{e_\mu} =
\lb \begin{array}{cc}
0 & \ve{e_R} \\
\ve{e_L} & 0
\end{array} \rb
=
\lb \begin{array}{cc}
0 & \ve{e_0} + \si_P^\va \ve{e_\va} \\
\ve{e_0} - \si_P^\va \ve{e_\va} & 0
\end{array} \rb
\\
&=& 
\lb \begin{array}{cccc}
0 & 0 & \ve{e_0}+\ve{e_3} & \ve{e_1}-i\ve{e_2} \\
0 & 0 & \ve{e_1}+i\ve{e_2} & \ve{e_0}-\ve{e_3} \\
\ve{e_0}-\ve{e_3} & -\ve{e_1}+i\ve{e_2} & 0 & 0 \\
-\ve{e_1}-i\ve{e_2} & \ve{e_0}+\ve{e_3} & 0 & 0
\end{array} \rb
\end{eqnarray}
A [[spin connection]] over [[spacetime]] is a [[Cl(1,3) bivector]] valued 1-form (a [[Clifform]]),
$$
\f{\om} = \f{dx^a} \ha \om_a{}^{\mu \nu} \ga_{\mu \nu}
$$
which may be written as a matrix valued 1-form, using the [[Weyl representation|Dirac matrices]], as
\begin{eqnarray}
\f{\om} &=& \ha \f{\om^{\mu \nu}} \ga_{\mu \nu} =
\lb \begin{array}{cc}
\f{\om_L} & 0 \\
0 & \f{\om_R}
\end{array} \rb
=
\lb \begin{array}{cc}
\f{\om^{0 \va}} \si^P_\va - i \ha \f{\om^{\va \ze}} \ep_{\va \ze \ta} \si^P_\ta & 0 \\
0 & -\f{\om^{0 \va}} \si^P_\va - i \ha \f{\om^{\va \ze}} \ep_{\va \ze \ta} \si^P_\ta
\end{array} \rb \\
&=&
\lb \begin{array}{cccc}
\f{\om^{03}}- i \f{\om^{12}} & \f{\om^{01}}+\f{\om^{13}}-i \f{\om^{02}}- i \f{\om^{23}} & 0 & 0 \\
\f{\om^{01}}- \f{\om^{13}}+i \f{\om^{02}}-i \f{\om^{23}} & -\f{\om^{03}}+i \f{\om^{12}} & 0 & 0 \\
0 & 0 & -\f{\om^{03}}- i \f{\om^{12}} & -\f{\om^{01}}+\f{\om^{13}}+i \f{\om^{02}}- i \f{\om^{23}} \\
0 & 0 & -\f{\om^{01}}- \f{\om^{13}}-i \f{\om^{02}}-i \f{\om^{23}} & -\f{\om^{03}}+i \f{\om^{12}}
\end{array} \rb
\end{eqnarray}
with left and right [[chiral]] parts, $\f{\om_{L/R}}$, projected out by the [[left/right chirality projector]]. These $2\times2$ complex matrix (or [[quaternion]]) valued 1-forms satisfy $\f{\om_L}^\dagger = - \f{\om_R}$, using Hermitian conjugation.
A rotation is particularly easy to express as a [[Clifford rotation]] in three dimensions using the three dimensional Clifford algebra, [[Cl(3)]]. First, consider the result of crossing a vector with a bivector. Starting with an arbitrary Clifford vector,
$$
v = v^i \sigma_i = v^1 \sigma_1 + v^2 \sigma_2 + v^3 \sigma_3
$$
and, for example, a "small" bivector in the $\sigma_1 \sigma_2$ plane,
$$
B = \epsilon \sigma_{12}
$$
their [[cross product|antisymmetric bracket]] gives
\begin{eqnarray}
B \times v &=& \epsilon \lp v^1 \sigma_{12} \times \sigma_1 + v^2 \sigma_{12} \times \sigma_2 + v^3 \sigma_{12} \times \sigma_3 \rp \\
 &=& \epsilon \lp - v^1 \sigma_2 + v^2 \sigma_1 \rp
\end{eqnarray}
This new vector, $B \times v$, is perpendicular to $v$, and in the plane of $B$. This "small" vector is the one that needs to be added to $v$ in order to rotate it a small amount clockwise in the plane of $B$:
$$
v' \simeq v + B \times v \simeq \lp 1 + \frac{\ep}{2} \sigma_{12} \rp v \lp 1 - \frac{\ep}{2} \sigma_{12} \rp
$$
where the "$\simeq$" holds to first order in $\epsilon$. Infinitesimal rotations like this one can be combined, with $\ep$ equated to an amplitude devided by a large integer, to give a finite rotation,
\begin{eqnarray}
v' &=& \lim_{N \rightarrow \infty} \lp 1+ \frac{\th}{2N} \sigma_{12} \rp^N v \lp 1- \frac{\th}{2N} \sigma_{12} \rp^N \\
 &=& e^{\frac{\th}{2} \sigma_{12}} v e^{-\frac{\th}{2} \sigma_{12}} = U v U^-
\end{eqnarray}
using the "limit" definition for [[exponentiation]]. This is an exact expression for the rotation of a vector by a bivector. In three dimensions an arbitrary bivector, $B$, can be written as
$$
B = \theta b
$$
with a scalar amplitude, $\theta$, multiplying a unit bivector encoding the orientation, $bb=-1$. The exponential can then be written, via exponentiation of the bivector, as:
$$
U = e^{\frac{1}{2} B} = \cos(\frac{\th}{2}) + b \sin(\frac{\th}{2})
$$
An arbitrary rotation in any plane can be expressed efficiently as $v' = UvU^-$. For example, a rotation of an arbitrary vector by $B=\theta \sigma_{12}$ gives (using some trig identities) :
\begin{eqnarray}
v' &=& e^{\frac{1}{2} B} v e^{-\frac{1}{2} B} \\
&=& \lp \cos(\frac{\th}{2}) + \sigma_{12} \sin(\frac{\th}{2}) \rp \lp v^1 \sigma_1 + v^2 \sigma_2 + v^3 \sigma_3 \rp \lp \cos(\frac{\th}{2}) - \sigma_{12} \sin(\frac{\th}{2}) \rp \\
&=& \lp v^1 \cos(\theta) + v^2 \sin(\theta) \rp \sigma_1 + \lp v^2 \cos(\theta) - v^1 \sin(\theta) \rp \sigma_2 + v^3 \sigma_3
\end{eqnarray}
This is widely considered to be pretty neat, and useful as a general method of expressing and calculating rotations. It is equivalent to employing rotation matrices, but generally more intuitive.

A rotation matrix is a 3x3 special [[orthogonal]] matrix (an element of the [[special orthogonal group]], $SO(3)$) that transforms one set of basis vectors into another. This equates to the Clifford way of doing a rotation as:
$$
\sigma'_i = \sigma_j L^j{}_i = U \sigma_i U^-
$$
For any rotation encoded by $U$ (which, as the exponential of a bivector, also represents an arbitrary [[SU(2)]] element), the corresponding rotation matrix elements may be explicitly calculated using the [[scalar part operator|Clifford grade]] as
$$
L^j{}_i = \left< \sigma^j U \sigma_i U^- \right>
$$
$SU(2)$ elements of inequivalent sign, $U$ and $-U$, generate equivalent rotations. In this way, $SU(2)$ is a double cover (and the universal cover) of $SO(3)$.
It is sometimes useful to factor the co[[frame]] matrix, $\lp e_i\rp^\al$, into a ''conformal scalar'', $s$, times the ''special coframe matrix'',
\begin{eqnarray}
\lp e_i\rp^\al &=& s \lp e^s_i\rp^\al\\
\f{e} = s \f{e^s}
\end{eqnarray}
such that the frame [[determinant]] depends only on the conformal scalar
\[ \ll e \rl = \det \lp e_i\rp^\al = s^n \]
and the special coframe matrix is restricted to satisfy
\[ \ll e^s \rl = \det \lp e^s_i\rp^\al = 1 \]
This factorization allows separate consideration of this conformal factor, $s$, which carries [[units]] of time, $T$.

Ref:
*L. Smolin
**[[The quantization of unimodular gravity and the cosmological constant problem|http://arxiv.org/abs/0904.4841v1]]
*Philip D. Mannheim
**[[Making the Case for Conformal Gravity|http://arxiv.org/abs/1101.2186]]
The ''special orthogonal group'' of order $n$, $G=SO(n)$, is the [[simple]], compact, connected, $\ha n (n-1)$ dimensional [[Lie group]] of [[orthogonal]] $n \times n$ real matrices with unit [[determinant]]. It's Lie algebra, $so(n)$, is the [[spin Lie algebra]].

The ''special unitary group'' of order $n$, $G=SU(n)$, is the [[simple]], compact, connected, $(n^2-1)$ dimensional [[Lie group]] of [[unitary]] $n \times n$ complex matrices with unit [[determinant]].
Ref:
*Ali H. Chamseddine and Alain Connes
**[[Quantum Gravity Boundary Terms from Spectral Action|http://arxiv.org/abs/0705.1786]]
**[[Gravity and the standard model with neutrino mixing|http://arxiv.org/abs/hep-th/0610241]]
**[[The Spectral Action Principle|http://arxiv.org/abs/hep-th/9606001]]
An [[Ehresmann principal bundle connection]] is a [[vector valued 1-form|vector valued form]],
$$
\f{\ve{\cal A}} = \f{A^B} \ve{\xi^L_B} + \f{\ve{\cal I}}
$$
with an interesting [[spectral decomposition|eigen]]. The the eigenequations for the eigenvectors and eigenforms are solved by:
\begin{eqnarray}
\ve{\xi^L_B} \f{\ve{\cal A}} &=& \ve{\xi^L_B}\\
\ve{h_i} \f{\ve{\cal A}} &=& 0\\
\f{\ve{\cal A}} \f{dx^i} &=& 0\\
\f{\ve{\cal A}} \lp \f{\xi_L^B} + \f{A^B} \rp &=& \lp \f{\xi_L^B} + \f{A^B} \rp\\
\end{eqnarray}
with the horizontal eigenvectors, $\ve{h_i} = \ve{\pa_i} - A_i{}^B \ve{\xi^L_B} \in \ve{\De_H}$. These eigenvectors and eigenforms are chosen to satisfy the normality conditions,
\begin{eqnarray}
\ve{h_i} \f{dx^j} &=& \de_i^j\\
\ve{h_i} \lp \f{\xi_L^B} + \f{A^B} \rp &=& 0\\
\ve{\xi^L_B} \f{dx^j} &=& 0\\
\ve{\xi^L_B} \lp \f{\xi_L^C} + \f{A^C} \rp &=& \de_B^C
\end{eqnarray}
and the resulting ''spectral decomposition of the Ehresmann principal bundle connection'' is
$$
\f{\ve{\cal A}} = \lp \f{\xi_L^B} + \f{A^B} \rp \ve{\xi^L_B} 
$$
<<ListTagged speculative>>
In flat three dimensional space, the most common coordinates are the ''cartesian coordinates'', $(x^1 = x, x^2 = y, x^3 = z)$. The second most common coordinates are ''spherical coordinates'', $(r,\th,\ph)$, describing the distance, $r$, from some "center" point, the angle, $\th$, from the z-axis, and the angle, $\ph$, from the x-axis. In terms of these ''angular'' variables, $(a^1 = r, a^2 = \th, a^3 = \ph)$, the equivalent  cartesian coordinates are
\begin{eqnarray}
x^1 &=& r \sin(\th) \cos(\ph) \\
x^2 &=& r \sin(\th) \sin(\ph) \\
x^3 &=& r \cos(\th) 
\end{eqnarray}
Elements, $T_A$, of the ''spin [[Lie algebra]]'', $spin(n)$, equivalent to the [[special orthogonal group]] Lie algebra, $so(n)$, may be represented by $\ha n (n-1)$  antisymmetric $n \times n$ matrices, $T_A \sim J_{ij}$, antisymmetric in the $ij$ labels, or alternatively by [[bivectors|Clifford basis elements]], $T_A \sim \ga_{\al \be}$, of the $Cl(n,0)$ [[Clifford algebra]]. The bivectors satisfy the [[Clifford basis product identities]]:
$$
\lb \ga_{\al \be}, \ga_{\ga \de} \rb = 2 \left\{ - \et_{\al \ga} \ga_{\be \de} + \et_{\al \de} \ga_{\be \ga} + \et_{\be \ga} \ga_{\al \de} - \et_{\be \de} \ga_{\al \ga} \right\} = C_{\lb{\al \be}\rb \lb \ga \de \rb}{}^{\lb \ep \up \rb} \ga_{\ep \up}
$$
giving the $spin(n)$ structure constants,
$$
C_{\lb{\al \be}\rb \lb \ga \de \rb}{}^{\lb \ep \up \rb}
= 2 \left\{ - \et_{\al \ga} \de^{\lb\ep \up\rb}_{\be \de} + \et_{\al \de} \de^{\lb\ep \up\rb}_{\be \ga} + \et_{\be \ga} \de^{\lb\ep \up\rb}_{\al \de} - \et_{\be \de} \de^{\lb\ep \up\rb}_{\al \ga} \right\}
$$
with $\et_{\al \ga} = \de_{\al \ga}$ for $Cl(n,0)$. The [[Killing form]] for $spin(n)$ is
$$
g_{\lb \al \be \rb \lb \ga \de \rb} = C_{\lb \al \be \rb \lb \ep \up \rb}{}^{\lb \ze \et \rb} C_{\lb \ga \de \rb \lb \ze \et \rb}{}^{\lb \ep \up \rb}
= 8 \lp n-2 \rp \lp \et_{\al \de} \et_{\be \ga} - \et_{\al \ga} \et_{\be \de} \rp
$$

The Lie algebra of the ''spin group of mixed signature'', $spin(p,q)=so(p,q)$, is similarly described by the bivectors of $Cl(p,q)$. 
The ''spin connection'', a Clifford bivector valued 1-form,
$$
\f{\om} = \f{dx^i} \ha \om_i{}^{\al \be} \ga_{\al \be} \in \f{Cl^2}
$$
serves as the  [[Clifford connection]] for the [[Clifford vector bundle]] or for the [[graded Clifford bundle|Clifford vector bundle]], since the structure group for both of these is the group of [[Clifford rotation]]s. The [[vector bundle covariant derivative|vector bundle connection]] acting on the [[Clifford basis vectors]] for the fiber of either bundle is 
$$
\f{\na} \ga_\al = \f{\om} \times \ga_\al = \f{\om}^\be{}_\al \ga_\be 
$$
This gives the covariant derivative acting on the [[Clifford basis elements]] of the graded Clifford bundle as the [[cross product|antisymmetric bracket]] of the connection with the basis element,
$$
\f{\na} \ga_{\al \dots \be} = \f{\om} \times \ga_{\al \dots \be} 
$$
and the covariant derivative for a section of either bundle,
$$
\f{\na} C = \f{d} C + \f{\om} \times C 
$$
This covariant derivative necessarily preserves [[Clifford grade]]. The ''Clifford vector bundle covariant derivative'' acting on any [[Clifform]] is
$$
\f{\na} \nf{C} = \f{d} \nf{C} + \f{\om} \times \nf{C} 
$$

Through the identification of orthonormal basis vectors with Clifford basis vectors, $\ve{e_\al} \leftrightarrow \ga_\al$, via the [[frame]], $\ve{e_\al} \f{e} = \ga_\al$, the spin connection coefficients equal the [[tangent bundle spin connection|tangent bundle connection]] coefficients, $\om_i{}^\be{}_\al = w_i{}^\be{}_\al$, since the defining equation,
$$
\f{\na} \ve{e_\al} = \f{w}^\be{}_\al \ve{e_\be}
$$
must agree with the similar equation for the covariant derivative of Clifford basis elements.
refs:
*[[Conrady and Freidel - Path integral representation of spin foam models of 4d gravity|http://arxiv.org/abs/0806.4640]]
**good recent review
*Thiemann, p464
*Rovelli, p249
*[[Freidel and Krasnov - Spin Foam Models and the Classical Action Principle|http://arxiv.org/abs/hep-th/9807092v2]]
*[[Freidel and Starodubtsev - Quantum gravity in terms of topological observables|http://arxiv.org/abs/hep-th/0501191v2]]
*[[Freidel and Krasnov - A New Spin Foam Model for 4d Gravity|http://arxiv.org/abs/0708.1595v1]]
*[[Baez - An Introduction to Spin Foam Models of Quantum Gravity and BF Theory|http://arxiv.org/abs/gr-qc/9905087]]
**good review
*[[Baez - Spin Foam Models|http://xxx.lanl.gov/abs/gr-qc/9709052]]
**one of the first introductions
*[[Baez - Spin Networks, Spin Foams, and Quantum Gravity|http://math.ucr.edu/home/baez/foam/]]
**collection of lecture tiddlers
*[[Baez - Towards a Spin Foam Model of Quantum Gravity|http://www.youtube.com/watch?v=cVfE6aK57S8]]
**youtube and pdf tiddlers from Loops 05
*[[Perez - Spin Foam Models for Quantum Gravity|http://arxiv.org/abs/gr-qc/0301113]]
**review paper
*[[Conrady - Geometric spin foams, Yang-Mills theory and background-independent models|http://arxiv.org/abs/gr-qc/0504059]]
**recent review, related to lattice gauge theory
A ''spin [[gauge transformation]]'' is a rotation of the [[Clifford basis elements]] for a [[Clifford vector bundle]] or [[graded Clifford bundle|Clifford vector bundle]]. The change may be induced by the action of an arbitrary, position dependent [[Clifford rotation]],
$$
\ga'_\al = U \ga_\al U^- = \ga_\be L^\be{}_\al
$$
This is an active transformation of bundle elements, and transforms any Clifford valued field (section), $\Ph$, to
$$
\Ph' = U \Ph U^-
$$
The transformation law for the [[spin connection]] under this [[Clifford gauge transformation]] is
$$
\f{\om'} = U \f{\om} U^- - 2 \lp \f{d} U \rp U^-
$$
For an infinitesimal gauge transformation, $U \simeq 1 + \ha B$, parameterized by an arbitrary Clifford bivector valued field, $B$, the connection changes to
$$
\f{\om'} \simeq \f{\om} - \f{d} B - \ha \f{\om} B + \ha B \f{\om} = \f{\om} - \f{\na} B
$$
giving the change $\de \f{\om} = - \f{\na} B$.

The spin gauge transformation also results in the rotation of the [[frame]],
$$
\f{e'} = U \f{e} U^- \simeq \f{e} + B \times \f{e}
$$
equivalent to a rotation of the constituent frame basis 1-forms and orthonormal basis vectors.
A ''spin group'', $G=Spin(n)$, is the double cover of the [[special orthogonal group]]. The two groups have the same [[spin Lie algebra]], $spin(n) = so(n)$.
good new paper of tiddler, by Levine and Terno:
[[Reconstructing Quantum Geometry from Quantum Information: Area Renormalisation, Coarse-Graining and Entanglement on Spin Networks|http://arxiv.org/abs/gr-qc/0603008]]
The fermions of the [[SO(10)]] GUT live in a sixteen-dimensional complex positive [[chiral]] spinor representation space of the $spin(10) = so(10)$ Lie algebra. Using a [[chiral]] [[Clifford matrix representation]], a Clifford bivector element of so(10), $B = \ha B^{\al \be} \ga_{\al \be}$, acts on a positive [[spinor]], $\ps = \ps^a Q^+_a$, as
$$
B \ps = \ha B^{\al \be} (\ga^+_{\al \be})^b{}_a  \Ps^a Q^+_b
$$
The Lie algebra has a five-dimensional Cartan subalgebra, so each fermion weight corresponds to five charges, $u, v, x, y, z$ of a $spin(10)$ [[spinor]]:
|                        | $u$ | $v$ | $x$ | $y$ | $z$ |
| $\nu_e$          | $-$  | $+$ | $-$ | $-$ | $-$ |
| $\bar{\nu}_e$ | $-$ | $-$ | $+$ | $+$ | $+$ |
| $e$                 | $+$ | $-$ | $-$ | $-$ | $-$ |
| $\bar{e}$         | $+$ | $+$ | $+$ | $+$ | $+$ |
| $u^r$              | $-$ | $+$ | $-$ | $+$ | $+$ |
| $\bar{u}^r$      | $-$ | $-$ | $+$ | $-$ | $-$ |
| $d^r$              | $+$ | $-$ | $-$ | $+$ | $+$ |
| $\bar{d}^r$      | $+$ | $+$ | $+$ | $-$ | $-$ |
| $u^g$              | $-$ | $+$ | $+$ | $-$ | $+$ |
| $\bar{u}^g$     | $-$ | $-$ | $-$ | $+$ | $-$ |
| $d^g$              | $+$ | $-$ | $+$ | $-$ | $+$ |
| $\bar{d}^g$      | $+$ | $+$ | $-$ | $+$ | $-$ |
| $u^b$              | $-$ | $+$ | $+$ | $+$ | $-$ |
| $\bar{u}^b$      | $-$ | $-$ | $-$ | $-$ | $+$ |
| $d^b$              | $+$ | $-$ | $+$ | $+$ | $-$ |
| $\bar{d}^b$      | $+$ | $+$ | $-$ | $-$ | $+$ |
The $x, y, z$ charges combine to give two [[su(3)]] color charges, $g_3, g_8$, and ''Baryon minus Lepton number'', $B= (x+y+z)/3$. The $u, v$ charges combine to give weak charge, $W = (v-u)/2$, and ''weaker charge'', $W' = (v+u)/2$, which contributes to give hypercharge, $Y=W'+B$, or electric charge, $Q = v+B$, matching the weak, hyper, and color charges of [[one generation of fermions]]. This can be seen in a weight diagram of the [[Spin(10) GUT]]. Note that, being in a chiral representation space, all particles have an odd number of positive weights. Also, each particle can be identified by its $v,x,y,z$ weights, without $u$, and these can be interpreted as all of the four weights of a $spin(8)$ spinor.
The [[spin Lie algebra]] $spin(4,4)$ has a nice representation as [[Cl(4,4)]] bivectors, which can be represented as $16 \times 16$ matrices. This 28-dimensional Lie algebra has [[triality]] outer automorphisms. To describe an automorphism, once may choose a Cartan subalgebra with all compact or all non-compact generators. One such Cartan subalgebra, using the matrix rep of Cl(4,4), is spanned by:
\begin{eqnarray}
\Ga_{12} &=& i \,  \si^P_0 \otimes \si^P_0 \otimes \si^P_3 \otimes \si^P_2 \\
\Ga_{38} &=& i \, \si^P_3 \otimes \si^P_2 \otimes \si^P_3 \otimes \si^P_0 \\
\Ga_{46} &=& -i \, \si^P_0 \otimes \si^P_2 \otimes \si^P_0 \otimes \si^P_0 \\
\Ga_{57} &=& -i \, \si^P_0 \otimes \si^P_0 \otimes \si^P_0 \otimes \si^P_2
\end{eqnarray}
A typical triality automorphism transforms these generators to
$$
\lb \begin{array}{c}
\Ga'_{12} \\
\Ga'_{38} \\
\Ga'_{46} \\
\Ga'_{57}
\end{array} \rb
=
\lb \begin{array}{cccc}
\ha & \ha & \ha & -\ha \\
\ha & \ha & -\ha & \ha \\
-\ha & \ha & -\ha & -\ha \\
-\ha & \ha & \ha & \ha
\end{array} \rb
\lb \begin{array}{c}
\Ga_{12} \\
\Ga_{38} \\
\Ga_{46} \\
\Ga_{57}
\end{array} \rb
$$
and also similarly transforms the other generators, roots, and weights.

With the above matrix representation, $spin(4,4)$ acts on 8-dimensional positive and negative real [[chiral]] [[spinor]]s. If we interpret the real $\Ga_{57} = j$ generator as our complex structure, and the spinors to be composed of alternating real and imaginary parts, then the Cartan subalgebra acting on a positive complex spinor is:
\begin{eqnarray}
\Ga^+_{12} &=& -j \, \si^P_0 \otimes \si^P_3 \\
\Ga^+_{38} &=& i \, \si^P_2 \otimes \si^P_3 \\
\Ga^+_{46} &=& -i \, \si^P_2 \otimes \si^P_0 \\
\Ga^+_{57} &=& j
\end{eqnarray}
A ''spinor'', $\Psi$, is an element of the minimal left ideal of a [[Clifford algebra]], $I_L \in Cl$. Each ''minimal left ideal'', $I_L$, is the smallest subset such that for all $\Psi \in I_L$ and $A \in Cl$ we have $A \Psi \in I_L$. Using a [[Clifford matrix representation]], each minimal left ideal corresponds to a collumn of the matrix, so a spinor, $\Ps = \Ps^a Q_a$, is simply represented as a ''matrix collumn'' on which the Clifford algebra elements, represented as matrices, such as the bivector $B = \ha B^{\al \be} \ga_{\al \be}$, act from the left.
$$
B \Ps =  \ha B^{\al \be} \ga_{\al \be} \Ps^a Q_a = \ha B^{\al \be} \Ps^a Q_b (\ga_{\al \be})^b{}_c \de_a^c = \ha B^{\al \be} \Ps^a Q_b (\ga_{\al \be})^b{}_a
$$
In terms of matrix components, this gives
$$
\lp B \Psi \rp^b = \ha B^{\al \be} (\ga_{\al \be})^b{}_a \Ps^a
$$

A spinor, written as a collumn of numbers, may be real if the Clifford matrix representation is real, or [[chiral]] if the rep is chiral. A ''Grassmann spinor'' may be written as a collumn of real or complex [[Grassmann number]]s. 

Geometrically, a ''spinor field'' is a section of a [[fiber bundle]] [[associated]] to a [[principal bundle]], with the elements of the structure group, in the appropriate representation, acting on the spinor from the left.

Ref:
http://en.wikipedia.org/wiki/Spinor
The ''spinor covariant derivative'' is the [[covariant derivative]] of a [[spinor]] field,
$$
\f{\nabla} \ps = \f{\pa} \ps + \ha \f{\om} \ps
$$
in which $\f{\pa}$ is the [[partial derivative]] and $\f{\om}$ is the [[spin connection]] acting on the spinor via [[Clifford algebra]] multiplication.
The ''split-octonions'' are an eight-dimensional [[division algebra]], spanned by eight basis elements, $e_0,...,e_7$. The split-octonion identity element is $e_0=1$, and the other seven can be thought of as different imaginary directions, with the first three squaring to $-1$ and the last four to $+1$. Split-octonionic multiplication is non-commutative and, unusually, non-associative. The multiplication table for the basis elements is
|            | $e_0$ | $e_1$ | $e_2$ | $e_3$ | $e_4$ | $e_5$ | $e_6$ | $e_7$ |
| $e_0$ | $e_0$ | $e_1$ | $e_2$ | $e_3$ | $e_4$ | $e_5$ | $e_6$ | $e_7$ |
| $e_1$ | $e_1$ | $-e_0$ | $e_3$ | $-e_2$ | $-e_5$ | $e_4$ | $-e_7$ | $e_6$ |
| $e_2$ | $e_2$ | $-e_3$ | $-e_0$ | $e_1$ | $-e_6$ | $e_7$ | $e_4$ | $-e_5$ |
| $e_3$ | $e_3$ | $e_2$ | $-e_1$ | $-e_0$ | $-e_7$ | $-e_6$ | $e_5$ | $e_4$ |
| $e_4$ | $e_4$ | $e_5$ | $e_6$ | $e_7$ | $e_0$ | $e_1$ | $e_2$ | $e_3$ |
| $e_5$ | $e_5$ | $-e_4$ | $-e_7$ | $e_6$ | $-e_1$ | $e_0$ | $e_3$ | $-e_2$ |
| $e_6$ | $e_6$ | $e_7$ | $-e_4$ | $-e_5$ | $-e_2$ | $-e_3$ | $e_0$ | $e_1$ |
| $e_7$ | $e_7$ | $-e_6$ | $e_5$ | $-e_4$ | $-e_3$ | $e_2$ | $-e_1$ | $e_0$ |
which can be written using a ''split-octonion multiplication coefficient matrix'' as
$$
e_a e_b = M_{ab}{}^c e_c
$$
so, for example, $e_1 e_2 = e_4$ and $M_{12}{}^4 = 1$. ''Split-octonionic conjugation'' is the same as [[octonion]]ic conjugation,
$$
e_{\os{0}}=e_0 \;\;\;\;\; e_{\os{1}}=-e_1 \;\;\;\; ... \;\;\;\; e_{\os{7}}=-e_7 
$$
and satisfies $\widetilde{(e_a e_b)} = e_{\os{b}} e_{\os{a}}$, so we have $M_{ab}{}^c = M_{\os{b} \os{a}}{}^{\os{c}}$. 

The split-octonion metric is defined as
$$
n_{ab}=(e_a, e_b) = \ha ( e_a e_{\os{b}} + e_b e_{\os{a}} )
$$
and is $n_{ab}=diag(1, 1, 1, 1, -1, -1, -1, -1)$, "split" between four positive and four negative, unlike the octonion metric. This metric can be used to raise or lower split-octonion indices.
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