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style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="gravity">Gravity</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>, <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></strong></p> <p><strong>Formalism</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+Lie+algebra">Poincaré Lie algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Poincar%C3%A9+Lie+algebra">super Poincaré Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a>, <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a></p> </li> <li> <p>(<a class="existingWikiWord" href="/nlab/show/super-Cartan+geometry">super</a>-)<a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+manifold">pseudo</a>-<a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a></p> </li> </ul> </li> </ul> <p><strong>Definition</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Einstein-Hilbert+action">Einstein-Hilbert action</a>, <a class="existingWikiWord" href="/nlab/show/Einstein%27s+equations">Einstein's equations</a>, <a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/first-order+formulation+of+gravity">first-order formulation of gravity</a>, <a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity+as+a+BF-theory">gravity as a BF-theory</a>, <a class="existingWikiWord" href="/nlab/show/Plebanski+formulation+of+gravity">Plebanski formulation of gravity</a>, <a class="existingWikiWord" href="/nlab/show/teleparallel+gravity">teleparallel gravity</a></p> </li> </ul> </li> </ul> <p><strong>Spacetime configurations</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/black+hole+spacetime">black hole spacetime</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Schwarzschild+spacetime">Schwarzschild spacetime</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kerr+spacetime">Kerr spacetime</a></p> </li> </ul> </li> </ul> <p><strong>Properties</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein mechanism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bekenstein-Hawking+entropy">Bekenstein-Hawking entropy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+second+law+of+thermodynamics">generalized second law of thermodynamics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cosmic+censorship+hypothesis">cosmic censorship hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weak+gravity+conjecture">weak gravity conjecture</a></p> </li> </ul> <p><strong>Spacetimes</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/standard+model+of+cosmology">standard model of cosmology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a></p> </li> </ul> <div> <table><thead><tr><th><strong><a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a> <a class="existingWikiWord" href="/nlab/show/spacetimes">spacetimes</a></strong></th><th>vanishing <a class="existingWikiWord" href="/nlab/show/angular+momentum">angular momentum</a></th><th><a class="existingWikiWord" href="/nlab/show/positive+number">positive</a> <a class="existingWikiWord" href="/nlab/show/angular+momentum">angular momentum</a></th></tr></thead><tbody><tr><td style="text-align: left;"><strong>vanishing <a class="existingWikiWord" href="/nlab/show/charge">charge</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Schwarzschild+spacetime">Schwarzschild spacetime</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Kerr+spacetime">Kerr spacetime</a></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/positive+number">positive</a> <a class="existingWikiWord" href="/nlab/show/charge">charge</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Reissner-Nordstrom+spacetime">Reissner-Nordstrom spacetime</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Kerr-Newman+spacetime">Kerr-Newman spacetime</a></td></tr> </tbody></table> </div> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/asymptotically+flat+spacetime">asymptotically flat spacetime</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravitational+wave">gravitational wave</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/anti+de+Sitter+spacetime">anti de Sitter spacetime</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/FRW+spacetime">FRW spacetime</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Taub-NUT+spacetime">Taub-NUT spacetime</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Einstein+manifold">Einstein manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pp-wave+spacetime">pp-wave spacetime</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/KK-monopole">KK-monopole</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Malament%E2%80%93Hogarth+spacetime">Malament–Hogarth spacetime</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+spacetime">super spacetime</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Minkowski+spacetime">super Minkowski spacetime</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/black+brane">black brane</a></p> </li> </ul> <p><strong>Quantum theory</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+gravity">quantum gravity</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/graviton">graviton</a>, <a class="existingWikiWord" href="/nlab/show/gravitino">gravitino</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a></p> </li> </ul> </li> </ul> </div></div> <h4 id="supergeometry">Super-Geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a></strong> and (<a class="existingWikiWord" href="/nlab/show/synthetic+differential+supergeometry">synthetic</a> ) <strong><a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a></strong></p> <h2 id="background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/graded+object">graded object</a></p> </li> </ul> <h2 id="introductions">Introductions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+superalgebra">geometry of physics – superalgebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+supergeometry">geometry of physics – supergeometry</a></p> </li> </ul> <h2 id="superalgebra">Superalgebra</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/super+commutative+monoid">super commutative monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+abelian+group">super abelian group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+ring">super ring</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supercommutative+ring">supercommutative ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exterior+ring">exterior ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Clifford+ring">Clifford ring</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+module">super module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+vector+space">super vector space</a>, <a class="existingWikiWord" href="/nlab/show/SVect">SVect</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+algebra">super algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supercommutative+algebra">supercommutative algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exterior+algebra">exterior algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Clifford+algebra">Clifford algebra</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/superdeterminant">superdeterminant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">super Poincare Lie algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+complex+of+super+vector+spaces">chain complex of super vector spaces</a> (<a class="existingWikiWord" href="/nlab/show/model+structure+on+chain+complexes+of+super+vector+spaces">model structure</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+graded-commutative+superalgebra">differential graded-commutative superalgebra</a> (<a class="existingWikiWord" href="/nlab/show/model+structure+on+differential+graded-commutative+superalgebras">model structure</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+L-infinity+algebra">super L-infinity algebra</a></p> </li> </ul> <h2 id="supergeometry">Supergeometry</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/superpoint">superpoint</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Cartesian+space">super Cartesian space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supermanifold">supermanifold</a>, <a class="existingWikiWord" href="/nlab/show/SDiff">SDiff</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/NQ-supermanifold">NQ-supermanifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+vector+bundle">super vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+supermanifold">complex supermanifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+supermanifold">Euclidean supermanifold</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+spacetime">super spacetime</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Minkowski+spacetime">super Minkowski spacetime</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integration+over+supermanifolds">integration over supermanifolds</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Berezin+integral">Berezin integral</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Lie+group">super Lie group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Lie+group">super translation group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Euclidean+group">super Euclidean group</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+%E2%88%9E-groupoid">super ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+formal+smooth+%E2%88%9E-groupoid">super formal smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+line+2-bundle">super line 2-bundle</a></p> </li> </ul> <h2 id="supersymmetry">Supersymmetry</h2> <p><a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/division+algebra+and+supersymmetry">division algebra and supersymmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">super Poincare Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supermultiplet">supermultiplet</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BPS+state">BPS state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/M-theory+super+Lie+algebra">M-theory super Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/type+II+super+Lie+algebra">type II super Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a>, <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a></p> </li> </ul> <h2 id="supersymmetric_field_theory">Supersymmetric field theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/superfield">superfield</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supersymmetric+quantum+mechanics">supersymmetric quantum mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adinkra">adinkra</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">super Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/gauged+supergravity">gauged supergravity</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/superstring+theory">superstring theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a></p> </li> </ul> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometric+model+for+elliptic+cohomology">geometric model for elliptic cohomology</a></li> </ul> <div> <p> <a href="/nlab/edit/supergeometry+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="fields_and_quanta">Fields and quanta</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> and <a class="existingWikiWord" href="/nlab/show/fundamental+particle">particles</a> in <a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a></strong></p> <p><strong>and in the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a></strong>:</p> <p><strong><a class="existingWikiWord" href="/nlab/show/force">force</a> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> <a class="existingWikiWord" href="/nlab/show/gauge+field">gauge</a> <a class="existingWikiWord" href="/nlab/show/bosons">bosons</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/photon">photon</a> - <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a> (<a class="existingWikiWord" href="/nlab/show/abelian+group">abelian</a> <a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/W-boson">W</a>, <a class="existingWikiWord" href="/nlab/show/Z-boson">Z</a>, <a class="existingWikiWord" href="/nlab/show/B-boson">B-boson</a> - <a class="existingWikiWord" href="/nlab/show/electroweak+field">electroweak field</a> (<a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gluon">gluon</a> - <a class="existingWikiWord" href="/nlab/show/strong+nuclear+force">strong nuclear force</a> (<a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/graviton">graviton</a> - <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infraparticle">infraparticle</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/scalar+field">scalar</a> <a class="existingWikiWord" href="/nlab/show/bosons">bosons</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Higgs+boson">Higgs boson</a>, <a class="existingWikiWord" href="/nlab/show/inflaton">inflaton</a> (<a class="existingWikiWord" href="/nlab/show/scalar+field">scalar field</a>)</li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/matter">matter</a> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> <a class="existingWikiWord" href="/nlab/show/fermions">fermions</a></strong> (<a class="existingWikiWord" href="/nlab/show/spinors">spinors</a>, <a class="existingWikiWord" href="/nlab/show/Dirac+fields">Dirac fields</a>)</p> <div> <table><thead><tr><th><strong><a class="existingWikiWord" href="/nlab/show/flavor+%28particle+physics%29">flavors</a> of <a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental</a> <a class="existingWikiWord" href="/nlab/show/fermions">fermions</a> in the</strong> <br /> <strong><a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a>:</strong></th><th></th><th></th><th></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/generation+of+fermions">generation of fermions</a></td><td style="text-align: left;">1st generation</td><td style="text-align: left;">2nd generation</td><td style="text-align: left;">3d generation</td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/quarks">quarks</a></strong> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math>)</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">up-type</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/up+quark">up quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/charm+quark">charm quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/top+quark">top quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;">down-type</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/down+quark">down quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strange+quark">strange quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/bottom+quark">bottom quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/leptons">leptons</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">charged</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/electron">electron</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/muon">muon</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/tauon">tauon</a></td></tr> <tr><td style="text-align: left;">neutral</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/electron+neutrino">electron neutrino</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/muon+neutrino">muon neutrino</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/tau+neutrino">tau neutrino</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/bound+states">bound states</a>:</strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/mesons">mesons</a></strong></td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/light+mesons">light mesons</a></strong>: <br /> <a class="existingWikiWord" href="/nlab/show/pion">pion</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">u d</annotation></semantics></math>) <br /> <a class="existingWikiWord" href="/nlab/show/%CF%81-meson">ρ-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">u d</annotation></semantics></math>) <br /> <a class="existingWikiWord" href="/nlab/show/%CF%89-meson">ω-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">u d</annotation></semantics></math>) <br /> <a class="existingWikiWord" href="/nlab/show/f1-meson">f1-meson</a> <br /> <a class="existingWikiWord" href="/nlab/show/a1-meson">a1-meson</a></td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/strange+quark">strange</a>-mesons</strong>: <br /> <a class="existingWikiWord" href="/nlab/show/%CF%95-meson">ϕ-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mover><mi>s</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">s \bar s</annotation></semantics></math>), <br /> <a class="existingWikiWord" href="/nlab/show/kaon">kaon</a>, <a class="existingWikiWord" href="/nlab/show/K%2A-meson">K*-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">u s</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">d s</annotation></semantics></math>) <br /> <a class="existingWikiWord" href="/nlab/show/eta-meson">eta-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>u</mi><mo>+</mo><mi>d</mi><mi>d</mi><mo>+</mo><mi>s</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">u u + d d + s s</annotation></semantics></math>) <br /> <br /> <strong><a class="existingWikiWord" href="/nlab/show/charm+quark">charmed</a> <a class="existingWikiWord" href="/nlab/show/heavy+mesons">heavy mesons</a></strong>: <br /> <a class="existingWikiWord" href="/nlab/show/D-meson">D-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>c</mi></mrow><annotation encoding="application/x-tex"> u c</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>c</mi></mrow><annotation encoding="application/x-tex">d c</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mi>c</mi></mrow><annotation encoding="application/x-tex">s c</annotation></semantics></math>) <br /> <a class="existingWikiWord" href="/nlab/show/J%2F%CF%88-meson">J/ψ-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mover><mi>c</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">c \bar c</annotation></semantics></math>)</td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/bottom+quark">bottom</a> <a class="existingWikiWord" href="/nlab/show/heavy+mesons">heavy mesons</a></strong>: <br /> <a class="existingWikiWord" href="/nlab/show/B-meson">B-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">q b</annotation></semantics></math>) <br /> <a class="existingWikiWord" href="/nlab/show/%CF%92-meson">ϒ-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mover><mi>b</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">b \bar b</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/baryons">baryons</a></strong></td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/nucleons">nucleons</a></strong>: <br /> <a class="existingWikiWord" href="/nlab/show/proton">proton</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>u</mi><mi>u</mi><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(u u d)</annotation></semantics></math> <br /> <a class="existingWikiWord" href="/nlab/show/neutron">neutron</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>u</mi><mi>d</mi><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(u d d)</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> </tbody></table> </div> <p>(also: <a class="existingWikiWord" href="/nlab/show/antiparticles">antiparticles</a>)</p> <p><strong><a class="existingWikiWord" href="/nlab/show/effective+field+theory">effective particles</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Goldstone+bosons">Goldstone bosons</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/hadrons">hadrons</a></strong> (<a class="existingWikiWord" href="/nlab/show/bound+states">bound states</a> of the above <a class="existingWikiWord" href="/nlab/show/quarks">quarks</a>)</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/meson">meson</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/scalar+meson">scalar meson</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%CF%83-meson">σ-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pion">pion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/kaon">kaon</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D-meson">D-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/B-meson">B-meson</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+meson">vector meson</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%CF%89-meson">ω-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%CF%81-meson">ρ-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/f1-meson">f1-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/a1-meson">a1-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/b1-meson">b1-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/h1-meson">h1-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%2A-meson">K*-meson</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tensor+meson">tensor meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quarkonium">quarkonium</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/charmonium">charmonium</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%CF%92-meson">ϒ-meson</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exotic+meson">exotic meson</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/XYZ+meson">XYZ meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tetraquark">tetraquark</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/baryon">baryon</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/nucleon">nucleon</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/proton">proton</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/neutron">neutron</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chemical+element">chemical element</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/carbon">carbon</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nitrogen">nitrogen</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lambda+baryon">Lambda baryon</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pentaquark">pentaquark</a></p> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/solitons">solitons</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Skyrmion">Skyrmion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/caloron">caloron</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/instanton">instanton</a></p> </li> </ul> <p><strong>in <a class="existingWikiWord" href="/nlab/show/grand+unified+theory">grand unified theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/leptoquark">leptoquark</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Z%27-boson">Z'-boson</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/MSSM">minimally extended supersymmetric standard model</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/superpartners">superpartners</a></strong></p> <p>bosinos:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/gravitino">gravitino</a> - <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> of <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> (<a class="existingWikiWord" href="/nlab/show/Rarita-Schwinger+field">Rarita-Schwinger field</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gaugino">gaugino</a> - <a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">super Yang-Mills field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gluino">gluino</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/sfermions">sfermions</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/squark">squark</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/dark+matter">dark matter</a> candidates</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/WIMP">WIMP</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/axion">axion</a></p> </li> </ul> <p><strong>Exotica</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/preon">preon</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/graviphoton">graviphoton</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dilaton">dilaton</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monopole">monopole</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dual+graviton">dual graviton</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/giant+graviton">giant graviton</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/auxiliary+fields">auxiliary fields</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/ghost+field">ghost field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/antifield">antifield</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/antighost+field">antighost field</a>, <a class="existingWikiWord" href="/nlab/show/Nakanishi-Lautrup+field">Nakanishi-Lautrup field</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <ul> <li><a href='#SuperCartanGeometry'>As a gauge theory – Super Cartan geometry</a></li> <li><a href='#SolutionsWithGlobalSupersymmetry'>Solutions with global supersymmetry</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#as_a_background_for_greenschwarz_models'>As a background for Green-Schwarz <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-models</a></li> <li><a href='#UDuality'>Scalar moduli spaces and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>-duality</a></li> <li><a href='#exceptional_geometry'>Exceptional geometry</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#phenomenology'>Phenomenology</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#renormalization'>Renormalization</a></li> <li><a href='#UDualityReferences'>U-duality</a></li> <li><a href='#gauged_supergravity'>Gauged supergravity</a></li> <li><a href='#chernsimons_supergravity'>Chern-Simons supergravity</a></li> <li><a href='#History'>History</a></li> <li><a href='#related'>Related</a></li> <li><a href='#PhenomenologyReferences'>Phenomenology</a></li> <ul> <li><a href='#dark_matter'>Dark matter</a></li> <li><a href='#dark_energy'>Dark energy</a></li> <li><a href='#cosmic_inflation'>Cosmic inflation</a></li> <li><a href='#standard_model_of_particle_physics'>Standard model of particle physics</a></li> </ul> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>A <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> of <strong>supergravity</strong> is similar to the theory of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>, but where (in <a class="existingWikiWord" href="/nlab/show/first-order+formulation+of+gravity">first order formulation</a>) the latter is given by an <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> (the <a class="existingWikiWord" href="/nlab/show/Einstein-Hilbert+action">Einstein-Hilbert action</a> functional) on the space of <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection</a>s (over <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>) with values in the <a class="existingWikiWord" href="/nlab/show/Poincare+Lie+algebra">Poincare Lie algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔦𝔰𝔬</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{iso}(n,1)</annotation></semantics></math>, supergravity is defined by an extension of this to an action functional on the space of connections with values in the <a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">super Poincare Lie algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔰𝔦𝔰𝔬</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{siso}(n,1)</annotation></semantics></math>. One says that supergravity is the theory of <em>local</em> (Poincaré) <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a> in the same sense that ordinary <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> is the theory of “local Poincaré-symmetry”. These are <a class="existingWikiWord" href="/nlab/show/gauge+theories">gauge theories</a> for the <a class="existingWikiWord" href="/nlab/show/Poincare+Lie+algebra">Poincare Lie algebra</a> and the <a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">super Poincare Lie algebra</a>, respectively, in that the <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a> is a <a class="existingWikiWord" href="/nlab/show/Cartan+connection">Cartan connection</a> for the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>o</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>↪</mo><mi>𝔰𝔦𝔰𝔬</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">o(n,1) \hookrightarrow \mathfrak{siso}(n,1)</annotation></semantics></math>:</p> <p>if we write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔰𝔦𝔰𝔬</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{siso}(n,1)</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/semidirect+product">semidirect product</a> of the translation Lie algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{(n,1)}</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/special+orthogonal+Lie+algebra">special orthogonal Lie algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔰𝔬</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{so}(n,1)</annotation></semantics></math> and a <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a> <a class="existingWikiWord" href="/nlab/show/representation">representation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math>, then locally a connection is a <a class="existingWikiWord" href="/nlab/show/Lie+algebra+valued+1-form">Lie algebra valued 1-form</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>:</mo><mi>T</mi><mi>X</mi><mo>→</mo><mi>𝔰𝔦𝔰𝔬</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> A : T X \to \mathfrak{siso}(n,1) </annotation></semantics></math></div> <p>that decomposes into three components, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mi>Ω</mi><mo>,</mo><mi>Ψ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A = (E, \Omega, \Psi)</annotation></semantics></math>:</p> <ul> <li> <p>a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo>,</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{n,1}</annotation></semantics></math>-valued 1-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> – the <a class="existingWikiWord" href="/nlab/show/vielbein">vielbein</a></p> <p>(this encodes the <a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+metric">pseudo-Riemannian metric</a> and hence the field of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>);</p> </li> <li> <p>a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔰𝔬</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{so}(n,1)</annotation></semantics></math>-valued 1-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math> – called the <a class="existingWikiWord" href="/nlab/show/spin+connection">spin connection</a>;</p> </li> <li> <p>a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math>-valued 1-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ψ</mi></mrow><annotation encoding="application/x-tex">\Psi</annotation></semantics></math> – called the <a class="existingWikiWord" href="/nlab/show/gravitino">gravitino</a> field.</p> </li> </ul> <p>Typically in fact the field content of supergravity is larger, in that a field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is really an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra-valued+differential+form">∞-Lie algebra-valued differential form</a> with values in an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra">∞-Lie algebra</a> such as the <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a> (<a href="#DAuriaFreCastellani">DAuriaFreCastellani</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔰𝔲𝔤𝔯𝔞</mi><mo stretchy="false">(</mo><mn>10</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{sugra}(10,1)</annotation></semantics></math>. Specifically such a field</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>:</mo><mi>T</mi><mi>X</mi><mo>→</mo><mi>𝔰𝔲𝔤𝔯𝔞</mi><mo stretchy="false">(</mo><mn>10</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> A : T X \to \mathfrak{sugra}(10,1) </annotation></semantics></math></div> <p>has one more component</p> <ul> <li>a 3-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> – the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a>.</li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>s on the space of such connections that are parameterized by the elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math> are called <a class="existingWikiWord" href="/nlab/show/supersymmetries">supersymmetries</a>.</p> <div> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/geometry">geometric</a> context</th><th><a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a></th><th><a class="existingWikiWord" href="/nlab/show/stabilizer+subgroup">stabilizer subgroup</a></th><th>local model <a class="existingWikiWord" href="/nlab/show/space">space</a></th><th>local <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a></th><th>global <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a></th><th><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a></th><th><a class="existingWikiWord" href="/nlab/show/first+order+formulation+of+gravity">first order formulation of gravity</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>/<a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> (<a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>↪</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">H \hookrightarrow G</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> (“<a class="existingWikiWord" href="/nlab/show/coset+space">coset space</a>”) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G/H</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Klein+geometry">Klein geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Cartan+connection">Cartan connection</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">examples</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Euclidean+group">Euclidean group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Iso</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Iso(d)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rotation+group">rotation group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(d)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>d</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^d</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Euclidean+geometry">Euclidean geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Riemannian+geometry">Riemannian geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/affine+connection">affine connection</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Euclidean+gravity">Euclidean gravity</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+group">Poincaré group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Iso</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Iso(d-1,1)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lorentz+group">Lorentz group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(d-1,1)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d-1,1}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lorentzian+geometry">Lorentzian geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+geometry">pseudo-Riemannian geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/spin+connection">spin connection</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Einstein+gravity">Einstein gravity</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/anti+de+Sitter+group">anti de Sitter group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(d-1,2)</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(d-1,1)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/anti+de+Sitter+spacetime">anti de Sitter spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>AdS</mi> <mi>d</mi></msup></mrow><annotation encoding="application/x-tex">AdS^d</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/AdS+gravity">AdS gravity</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/de+Sitter+group">de Sitter group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(d,1)</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(d-1,1)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/de+Sitter+spacetime">de Sitter spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>dS</mi> <mi>d</mi></msup></mrow><annotation encoding="application/x-tex">dS^d</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/deSitter+gravity">deSitter gravity</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+algebraic+group">linear algebraic group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/parabolic+subgroup">parabolic subgroup</a>/<a class="existingWikiWord" href="/nlab/show/Borel+subgroup">Borel subgroup</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/flag+variety">flag variety</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/parabolic+geometry">parabolic geometry</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/conformal+group">conformal group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(d,t+1)</annotation></semantics></math></td><td style="text-align: left;">conformal parabolic subgroup</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M%C3%B6bius+space">Möbius space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mi>d</mi><mo>,</mo><mi>t</mi></mrow></msup></mrow><annotation encoding="application/x-tex">S^{d,t}</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/conformal+geometry">conformal geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/conformal+connection">conformal connection</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/conformal+gravity">conformal gravity</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+Lie+group">super Lie group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> (<a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>↪</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">H \hookrightarrow G</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> (“<a class="existingWikiWord" href="/nlab/show/coset+space">coset space</a>”) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G/H</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+Klein+geometry">super Klein geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+Cartan+geometry">super Cartan geometry</a></td><td style="text-align: left;">Cartan <a class="existingWikiWord" href="/nlab/show/superconnection">superconnection</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">examples</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+Poincar%C3%A9+group">super Poincaré group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+Minkowski+spacetime">super Minkowski spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mi>N</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d-1,1\vert N}</annotation></semantics></math></td><td style="text-align: left;">Lorentzian <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/superconnection">superconnection</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+anti+de+Sitter+group">super anti de Sitter group</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+anti+de+Sitter+spacetime">super anti de Sitter spacetime</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher differential geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/smooth+2-group">smooth 2-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-monomorphism">2-monomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">H \to G</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+quotient">homotopy quotient</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G//H</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Klein+2-geometry">Klein 2-geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Cartan+2-geometry">Cartan 2-geometry</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cohesive">cohesive</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/n-monomorphism">∞-monomorphism</a> (i.e. any <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">H \to G</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+quotient">homotopy quotient</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G//H</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/higher+Klein+geometry">higher Klein geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/higher+Cartan+geometry">higher Cartan geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/higher+Cartan+connection">higher Cartan connection</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">examples</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/extended+superspacetime">extended super Minkowski spacetime</a></td><td style="text-align: left;"></td><td style="text-align: left;">extended supergeometry</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/higher+gauge+theory">higher</a> <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>: <a class="existingWikiWord" href="/nlab/show/type+II+supergravity">type II</a>, <a class="existingWikiWord" href="/nlab/show/heterotic+supergravity">heterotic</a>, <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11d</a></td></tr> </tbody></table> </div> <p>The condition of <a class="existingWikiWord" href="/nlab/show/gauge+invariance">gauge invariance</a> of an action functional on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔰𝔦𝔰𝔬</mi></mrow><annotation encoding="application/x-tex">\mathfrak{siso}</annotation></semantics></math>-connections is considerably more restrictive than for one on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔦𝔰𝔬</mi></mrow><annotation encoding="application/x-tex">\mathfrak{iso}</annotation></semantics></math>-connections. For instance there is, under mild assumptions, a <em>unique</em> maximally supersymmetric supergravity extension of the ordinary <a class="existingWikiWord" href="/nlab/show/Einstein-Hilbert+action">Einstein-Hilbert action</a> on a 4-dimensional manifold. This in turn is obtained from the <em>unique</em> (under mild assumptions) maximally supersymmetric supergravity action functional on a (10,1)-dimensional <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> by thinking of the 4-dimensional action function as being a <a class="existingWikiWord" href="/nlab/show/dimensional+reduction">dimensional reduction</a> of the 11-dimensional one.</p> <p>This uniqueness (under mild conditions) is one reason for interest in supergravity theories. Another important reason is that supergravity theories tend to remove some of the problems that are encountered when trying to realize <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> as a <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a>. Originally there had been high hopes that the maximally supersymmetric supergravity theory in 4-dimensions is fully <a class="existingWikiWord" href="/nlab/show/renormalizable">renormalizable</a>. This couldn’t be shown computationally – until recently: triggered by new insights recently there there has been lots of renewed activity on the renormalizability of maximal supergravity.</p> <h3 id="SuperCartanGeometry">As a gauge theory – Super Cartan geometry</h3> <p>Ordinary <a class="existingWikiWord" href="/nlab/show/Einstein+gravity">Einstein gravity</a> has a natural formulation in terms of <a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a> for the inclusion of the <a class="existingWikiWord" href="/nlab/show/Lorentz+Lie+algebra">Lorentz Lie algebra</a> into the <a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+Lie+algebra">Poincaré Lie algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔬</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>↪</mo><mi>ℑ𝔰𝔬</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{o}(d-1,1) \hookrightarrow \mathfrak{Iso}(\mathbb{R}^{d-1,1})</annotation></semantics></math>. In this <a class="existingWikiWord" href="/nlab/show/first+order+formulation+of+gravity">first order formulation of gravity</a> a <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> configuration is a <a class="existingWikiWord" href="/nlab/show/Cartan+connection">Cartan connection</a> with such <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a>.</p> <p>This perspective directly generalizes to <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a> and yields the <em>superspace formulation</em> of theories of supegravity – <em><a class="existingWikiWord" href="/nlab/show/super+Cartan+geometry">super Cartan geometry</a></em>.</p> <p>After picking a <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">d\in \mathbb{N}</annotation></semantics></math> and writing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℑ𝔰𝔬</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{Iso}(\mathbb{R}^{d-1,1})</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+Lie+algebra">Poincaré Lie algebra</a>, then a choice of “number of supersymmetries” is a choice of <a href="spin+representation#RealIrreducibleSpinRepresentationInLorentzSignature">real spin representation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math>. Then the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℑ𝔰𝔬</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mi>N</mi></mrow></msup><mo stretchy="false">)</mo><mo>≔</mo><msub><munder><mrow><mi>ℑ𝔰𝔬</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><mo>⏟</mo></munder> <mi>even</mi></msub><mo>⊕</mo><msub><munder><mi>N</mi><mo>⏟</mo></munder> <mi>odd</mi></msub></mrow><annotation encoding="application/x-tex"> \mathfrak{Iso}(\mathbb{R}^{d-1,1|N}) \coloneqq \underbrace{\mathfrak{Iso}(\mathbb{R}^{d-1,1})}_{even} \oplus \underbrace{N}_{odd} </annotation></semantics></math></div> <p>regarded as a <a class="existingWikiWord" href="/nlab/show/super+vector+space">super vector space</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math> in odd degree becomes a <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a> by letting the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>even</mi><mo>,</mo><mi>odd</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[even,odd]</annotation></semantics></math> bracket be given by the defining <a class="existingWikiWord" href="/nlab/show/action">action</a> and by letting the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>odd</mi><mo>,</mo><mi>odd</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[odd,odd]</annotation></semantics></math> bracket be given by a canonically induced bilinear and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔬</mi></mrow><annotation encoding="application/x-tex">\mathfrak{o}</annotation></semantics></math>-equivariant pairing – the <a class="existingWikiWord" href="/nlab/show/super+Poincar%C3%A9+Lie+algebra">super Poincaré Lie algebra</a>. This still canonical contains the <a class="existingWikiWord" href="/nlab/show/Lorentz+Lie+algebra">Lorentz Lie algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔬</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{o}(\mathbb{R}^{d-1,1})</annotation></semantics></math> and the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mi>N</mi></mrow></msup><mo>≔</mo><mi>ℑ𝔰𝔬</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mi>N</mi></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>𝔬</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbb{R}^{d-1,1|N} \coloneqq \mathfrak{Iso}(\mathbb{R}^{d-1,1|N})/\mathfrak{o}(\mathbb{R}^{d-1,1}) </annotation></semantics></math></div> <p>is called <a class="existingWikiWord" href="/nlab/show/super+Minkowski+spacetime">super Minkowski spacetime</a> (equipped with its <a class="existingWikiWord" href="/nlab/show/super+translation+Lie+algebra">super translation Lie algebra</a> structure).</p> <p>From this, a <a class="existingWikiWord" href="/nlab/show/super-Cartan+geometry">super-Cartan geometry</a> is defined in direct analogy to the Cartan formulation of Riemannian geometry</p> <table><thead><tr><th>(<a class="existingWikiWord" href="/nlab/show/higher+Cartan+geometry">higher</a>-)<a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔥</mi></mrow><annotation encoding="application/x-tex">\mathfrak{h}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi><mo stretchy="false">/</mo><mi>𝔥</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}/\mathfrak{h}</annotation></semantics></math></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Einstein+gravity">Einstein gravity</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℑ𝔰𝔬</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{Iso}(\mathbb{R}^{d-1,1})</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔬</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{o}(d-1,1)</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d-1,1}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℑ𝔰𝔬</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mi>N</mi></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{Iso}(\mathbb{R}^{d-1,1\vert N})</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔬</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{o}(d-1,1)</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mi>N</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d-1,1\vert N}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℑ𝔰𝔬</mi><mo stretchy="false">(</mo><msup><mover><mi>ℝ</mi><mo>^</mo></mover> <mrow><mn>10</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mi>N</mi><mo>=</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{Iso}(\widehat{\mathbb{R}}^{10,1\vert N=1})</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔬</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{o}(d-1,1)</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mover><mi>ℝ</mi><mo>^</mo></mover> <mrow><mn>10</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mi>N</mi><mo>=</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\widehat{\mathbb{R}}^{10,1\vert N=1}</annotation></semantics></math></td></tr> </tbody></table> <p>Indeed, all the traditional literature on supergravity (e.g. (<a href="#CastellaniDAuriaFre91">Castellani-D’Auria-Fré 91</a>)) is phrased, more or less explicitly, in terms of <a class="existingWikiWord" href="/nlab/show/Cartan+connections">Cartan connections</a> for the inclusion of the <a class="existingWikiWord" href="/nlab/show/Lorentz+group">Lorentz group</a> into the <a class="existingWikiWord" href="/nlab/show/super+Poincar%C3%A9+group">super Poincaré group</a> this way, this being the formalization of what physicists mean when saying that they pass to “local supersymmetry”.</p> <p>One subtlety to take care of is that this makes <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> a <a class="existingWikiWord" href="/nlab/show/super-spacetime">super-spacetime</a> locally modeled on <a class="existingWikiWord" href="/nlab/show/super+Minkowski+spacetime">super Minkowski spacetime</a>. But the resulting theory is supposed to be a field theory on an ordinary <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> locally modeled on ordinary <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>. This is enforced by a further constraint on the super-Cartan connection which forces it to be determined by the bosonic manifold underlying the given supermanifold. This constraint is variously known as the <em>superspace constraints</em> or as <em><a href="D%27Auria-Fre+formulation+of+supergravity#Rheonomy">rheonomy</a></em> .</p> <p>The other subtlety to take care of is that a key aspect of higher dimensional <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> theories is that their <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> content necessarily includes, in addition to the <a class="existingWikiWord" href="/nlab/show/graviton">graviton</a> and the <a class="existingWikiWord" href="/nlab/show/gravitino">gravitino</a>, higher <a class="existingWikiWord" href="/nlab/show/differential+n-form">differential n-form</a> fields, notably the 2-form <a class="existingWikiWord" href="/nlab/show/B-field">B-field</a> of 10-dimensional <a class="existingWikiWord" href="/nlab/show/type+II+supergravity">type II supergravity</a> and <a class="existingWikiWord" href="/nlab/show/heterotic+supergravity">heterotic supergravity</a> as well as the 3-form <a class="existingWikiWord" href="/nlab/show/C-field">C-field</a> of <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a>.</p> <p>This means that these higher dimensional supergravity theories are not in fact entirely described by super-<a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a> – but by super-<a class="existingWikiWord" href="/nlab/show/higher+Cartan+geometry">higher Cartan geometry</a>.</p> <p>This follows a key insight due to (<a href="#DAuriaFreRegge80">D’Auria-Fré-Regge 80</a>, <a href="#DAuriaFre82">D’Auria-Fré 82</a>) – the <em><a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a></em> – that the “tensor multiplet” fields of higher dimensional supergravity theories as above are naturally brought into the previous perspective if only one allows more general <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebras">Chevalley-Eilenberg algebras</a>.</p> <p>Namely, we may add to the above CE-algebra</p> <ul> <li>a single generator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">c_3</annotation></semantics></math> of degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>3</mn><mo>,</mo><mi>even</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(3,even)</annotation></semantics></math></li> </ul> <p>and extend the differential to that by the formula</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>CE</mi></msub><mspace width="thinmathspace"></mspace><msub><mi>c</mi> <mn>3</mn></msub><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mover><mi>ψ</mi><mo stretchy="false">¯</mo></mover><msup><mi>Γ</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msup><mo>∧</mo><mi>ψ</mi><mo>∧</mo><msub><mi>e</mi> <mi>a</mi></msub><mo>∧</mo><msub><mi>e</mi> <mi>b</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> d_{CE} \, c_3 = \frac{1}{2}\bar \psi \Gamma^{a b} \wedge \psi \wedge e_a \wedge e_b \,. </annotation></semantics></math></div> <p>This still squares to zero due to the remarkable property of 11d <a class="existingWikiWord" href="/nlab/show/super+Minkowski+spacetime">super Minkowski spacetime</a> by which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mover><mi>ψ</mi><mo stretchy="false">¯</mo></mover><msup><mi>Γ</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msup><mo>∧</mo><mi>ψ</mi><mo>∧</mo><msub><mi>e</mi> <mi>a</mi></msub><mo>∧</mo><msub><mi>e</mi> <mi>b</mi></msub><mo>∈</mo><msup><mi>CE</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><mi>ℑ𝔰𝔬</mi><mo stretchy="false">(</mo><mn>10</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mi>N</mi><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\frac{1}{2}\bar \psi \Gamma^{a b} \wedge \psi \wedge e_a \wedge e_b \in CE^4(\mathfrak{Iso}(10,1|N=1))</annotation></semantics></math> is a representative of an exception <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super</a>-<a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a> class. (The collection of all these exceptional classes constitutes what is known as the <em><a class="existingWikiWord" href="/nlab/show/brane+scan">brane scan</a></em>).</p> <p>In the textbook (<a href="#CastellaniDAuriaFre91">Castellani-D’Auria-Fré 91</a>) a beautiful algorithm for constructing and handling higher supergravity theories based on such generalized CE-algebras is presented, but it seems fair to say that the authors struggle a bit with the right mathematical perspective to describe what is really happening here.</p> <p>But from a modern perspective this becomes crystal clear: these generalized CE algebras are CE-algebras not of <a class="existingWikiWord" href="/nlab/show/Lie+algebras">Lie algebras</a> but of <a class="existingWikiWord" href="/nlab/show/strong+homotopy+Lie+algebras">strong homotopy Lie algebras</a>, hence of <a class="existingWikiWord" href="/nlab/show/L-infinity+algebras">L-infinity algebras</a>, in fact of <a class="existingWikiWord" href="/nlab/show/Lie+n-algebras">Lie (p+1)-algebras</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p+1)</annotation></semantics></math> the degree of the relevant differential form field.</p> <p>Specifically, me may write the above generalized CE-algebra with the extra degree-3 generator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">c_3</annotation></semantics></math> as the CE-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔪</mi><mn>2</mn><mi>𝔟𝔯𝔞𝔫𝔢</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CE(\mathfrak{m}2\mathfrak{brane}) </annotation></semantics></math></p> <p>of the <em><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔪</mi><mn>2</mn><mi>𝔟𝔯𝔞𝔫𝔢</mi></mrow><annotation encoding="application/x-tex">\mathfrak{m}2\mathfrak{brane}</annotation></semantics></math>.</p> <p>Now a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mover><mo>⟵</mo><mrow></mrow></mover><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔪</mi><mn>2</mn><mi>𝔟𝔯𝔞𝔫𝔢</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>A</mi></mrow><annotation encoding="application/x-tex"> \Omega^\bullet(U) \stackrel{}{\longleftarrow} CE(\mathfrak{m}2\mathfrak{brane}) \;\colon\; A </annotation></semantics></math></div> <p>encodes <a class="existingWikiWord" href="/nlab/show/graviton">graviton</a> and <a class="existingWikiWord" href="/nlab/show/gravitino">gravitino</a> fields as above, but in addition it encodes a 3-form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>3</mn></msub><mo>≔</mo><mi>A</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>Ω</mi> <mrow><mo stretchy="false">(</mo><mn>3</mn><mo>,</mo><mi>even</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> C_3 \coloneqq A(c_3) \in \Omega^{(3,even)}(U) </annotation></semantics></math></div> <p>whose <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>4</mn></msub><mo>=</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msub><mi>C</mi> <mn>3</mn></msub><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mover><mi>Ψ</mi><mo stretchy="false">¯</mo></mover><msup><mi>Γ</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msup><mo>∧</mo><mi>Ψ</mi><mo>∧</mo><msub><mi>E</mi> <mi>a</mi></msub><mo>∧</mo><msub><mi>E</mi> <mi>b</mi></msub></mrow><annotation encoding="application/x-tex"> G_4 = \mathbf{d}C_3 + \frac{1}{2}\bar \Psi \Gamma^{a b} \wedge \Psi \wedge E_a \wedge E_b </annotation></semantics></math></div> <p>satisfies a modified <a class="existingWikiWord" href="/nlab/show/Bianchi+identity">Bianchi identity</a>, crucial for the theory of <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a> (<a href="#DAuriaFre82">D’Auria-Fré 82</a>).</p> <p>So this collection of differential form data is no longer a <a class="existingWikiWord" href="/nlab/show/Lie+algebra+valued+differential+form">Lie algebra valued differential form</a>, it is an <a class="existingWikiWord" href="/nlab/show/L-infinity+algebra+valued+differential+form">L-infinity algebra valued differential form</a>, with values in the <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a>.</p> <p>The quotient</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mover><mi>ℝ</mi><mo>^</mo></mover> <mrow><mn>10</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mi>N</mi><mo>=</mo><mn>1</mn></mrow></msup><mo>≔</mo><mi>𝔤</mi><mo stretchy="false">/</mo><mi>𝔥</mi><mo>=</mo><mi>𝔪</mi><mn>2</mn><mi>𝔟𝔯𝔞𝔫𝔢</mi><mo stretchy="false">/</mo><mi>𝔬</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mn>10</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mi>N</mi><mo>=</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \widehat{\mathbb{R}}^{10,1|N=1} \coloneqq \mathfrak{g}/\mathfrak{h} = \mathfrak{m}2\mathfrak{brane} / \mathfrak{o}(\mathbb{R}^{10,1|N=1}) </annotation></semantics></math></div> <p>is known as <em><a class="existingWikiWord" href="/nlab/show/extended+super+Minkowski+spacetime">extended super Minkowski spacetime</a></em>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> of this is a <a class="existingWikiWord" href="/nlab/show/smooth+infinity-group">smooth 3-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> which receives a map from the <a class="existingWikiWord" href="/nlab/show/Lorentz+group">Lorentz group</a>.</p> <p>This means that a global description of the geometry which (<a href="#CastellaniDAuriaFre91">Castellani-D’Auria-Fré 91</a>) discuss locally on <a class="existingWikiWord" href="/nlab/show/charts">charts</a> has to be a higher kind of Cartan geometry which is locally modeled not just on <a class="existingWikiWord" href="/nlab/show/cosets">cosets</a>, but on the <a class="existingWikiWord" href="/nlab/show/homotopy+quotients">homotopy quotients</a> of (<a class="existingWikiWord" href="/nlab/show/smooth+infinity-group">smooth</a>, <a class="existingWikiWord" href="/nlab/show/smooth+super+infinity-groupoid">supergeometric</a>, …) <a class="existingWikiWord" href="/nlab/show/infinity-groups">infinity-groups</a> – <a class="existingWikiWord" href="/nlab/show/higher+Cartan+geometry">higher Cartan geometry</a>.</p> <h3 id="SolutionsWithGlobalSupersymmetry">Solutions with global supersymmetry</h3> <p>A solution to the bosonic <a class="existingWikiWord" href="/nlab/show/Einstein+equations">Einstein equations</a> of ordinary <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> – some <a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a> – has a <em>global symmetry</em> if it has a <a class="existingWikiWord" href="/nlab/show/Killing+vector">Killing vector</a>.</p> <p>Accordingly, a configuration that solves the supergravity <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange equations</a> is a <em><a class="existingWikiWord" href="/nlab/show/global+supersymmetry">global supersymmetry</a></em> if it has a <a class="existingWikiWord" href="/nlab/show/Killing+spinor">Killing spinor</a>: a <a class="existingWikiWord" href="/nlab/show/covariantly+constant+spinor">covariantly constant spinor</a>.</p> <p>Here the notion of <a class="existingWikiWord" href="/nlab/show/covariant+derivative">covariant derivative</a> includes the usual <a class="existingWikiWord" href="/nlab/show/Levi-Civita+connection">Levi-Civita connection</a>, but also in general <a class="existingWikiWord" href="/nlab/show/torsion">torsion</a> components and contributions from other <a class="existingWikiWord" href="/nlab/show/background+gauge+fields">background gauge fields</a> such as a <a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a> and the <a class="existingWikiWord" href="/nlab/show/RR-field">RR-field</a>s in <a class="existingWikiWord" href="/nlab/show/type+II+supergravity">type II supergravity</a> or <a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic supergravity</a>.</p> <p>Of particular interest to phenomenologists around the turn of the millennium (but maybe less so today with new experimental evidence) has been in solutions of spacetime manifolds of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>M</mi> <mn>4</mn></msup><mo>×</mo><msup><mi>Y</mi> <mn>6</mn></msup></mrow><annotation encoding="application/x-tex">M^4 \times Y^6</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>M</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">M^4</annotation></semantics></math> the locally observed <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> (that plays a role as the background for all available particle accelerator experiments) and a small closed 6-dimensional <a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Y</mi> <mn>6</mn></msup></mrow><annotation encoding="application/x-tex">Y^6</annotation></semantics></math>.</p> <p>In the absence of further fields besides gravity, the condition that such a configuration has precisely one <a class="existingWikiWord" href="/nlab/show/Killing+spinor">Killing spinor</a> and hence precisely one <a class="existingWikiWord" href="/nlab/show/global+supersymmetry">global supersymmetry</a> turns out to be precisely that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Y</mi> <mn>6</mn></msup></mrow><annotation encoding="application/x-tex">Y^6</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifold">Calabi-Yau manifold</a>. This is where all the interest into these manifolds in <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> comes from. (Notice though that nothing in the theory itself demands such a compactification. It is only the phenomenological assumption of the factorized spacetime compactification together with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">N = 1</annotation></semantics></math> supersymmetry that does so.)</p> <p>More generally, in the presence of other <a class="existingWikiWord" href="/nlab/show/background+gauge+field">background gauge field</a>s, the Calabi-Yau condition here is deformed. One also speaks of <a class="existingWikiWord" href="/nlab/show/generalized+Calabi-Yau+space">generalized Calabi-Yau space</a>s. (For instance (<a href="#GMPT">GMPT05</a>)).</p> <p>For more see</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/supersymmetry+and+Calabi-Yau+manifolds">supersymmetry and Calabi-Yau manifolds</a></li> </ul> <h2 id="properties">Properties</h2> <h3 id="as_a_background_for_greenschwarz_models">As a background for Green-Schwarz <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-models</h3> <p>The <a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a> of those theories of supergravity which qualify as <a class="existingWikiWord" href="/nlab/show/target+spaces">target spaces</a> for <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+action+functional">Green-Schwarz action functional</a> <a class="existingWikiWord" href="/nlab/show/sigma+models">sigma models</a> (e.g. 10d <a class="existingWikiWord" href="/nlab/show/heterotic+supergravity">heterotic supergravity</a> for the <a class="existingWikiWord" href="/nlab/show/heterotic+string">heterotic string</a> and 10d <a class="existingWikiWord" href="/nlab/show/type+II+supergravity">type II supergravity</a> for the <a class="existingWikiWord" href="/nlab/show/type+II+string">type II string</a>) are supposed to be equivalent to those <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-models being well defined (the <a class="existingWikiWord" href="/nlab/show/WZW-model">WZW-model</a> term being well defined, hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>-symmetry being in effect). See at <em><a href="Green-Schwarz+action+functional#ReferencesSupergravityBackgroundEquationsOfMotion">Green-Schwarz action – References – Supergravity equations of motion</a></em> for pointers.</p> <h3 id="UDuality">Scalar moduli spaces and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>-duality</h3> <p>The <a class="existingWikiWord" href="/nlab/show/compact+space">compact</a> <a class="existingWikiWord" href="/nlab/show/exceptional+Lie+groups">exceptional Lie groups</a> form a series</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub><mo>,</mo><msub><mi>E</mi> <mn>7</mn></msub><mo>,</mo><msub><mi>E</mi> <mn>6</mn></msub></mrow><annotation encoding="application/x-tex"> E_8, E_7, E_6 </annotation></semantics></math></div> <p>which is usefully thought of to continue as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>5</mn></msub><mo>:</mo><mo>=</mo><mi>Spin</mi><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo><mo>,</mo><msub><mi>E</mi> <mn>4</mn></msub><mo>:</mo><mo>=</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo><mo>,</mo><msub><mi>E</mi> <mn>3</mn></msub><mo>:</mo><mo>=</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mo>×</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E_5 := Spin(10), E_4 := SU(5), E_3 := SU(3) \times SU(2) \,. </annotation></semantics></math></div> <p>Supergravity theories are controled by the corresponding <a class="existingWikiWord" href="/nlab/show/split+real+forms">split real forms</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mrow><mn>8</mn><mo stretchy="false">(</mo><mn>8</mn><mo stretchy="false">)</mo></mrow></msub><mo>,</mo><msub><mi>E</mi> <mrow><mn>7</mn><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo></mrow></msub><mo>,</mo><msub><mi>E</mi> <mrow><mn>6</mn><mo stretchy="false">(</mo><mn>6</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex"> E_{8(8)}, E_{7(7)}, E_{6(6)} </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mrow><mn>5</mn><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></mrow></msub><mo>:</mo><mo>=</mo><mi>Spin</mi><mo stretchy="false">(</mo><mn>5</mn><mo>,</mo><mn>5</mn><mo stretchy="false">)</mo><mo>,</mo><msub><mi>E</mi> <mrow><mn>4</mn><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></msub><mo>:</mo><mo>=</mo><mi>SL</mi><mo stretchy="false">(</mo><mn>5</mn><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>E</mi> <mrow><mn>3</mn><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></msub><mo>:</mo><mo>=</mo><mi>SL</mi><mo stretchy="false">(</mo><mn>3</mn><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo><mo>×</mo><mi>SL</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E_{5(5)} := Spin(5,5), E_{4(4)} := SL(5, \mathbb{R}), E_{3(3)} := SL(3, \mathbb{R}) \times SL(2, \mathbb{R}) \,. </annotation></semantics></math></div> <p>For instance the <a class="existingWikiWord" href="/nlab/show/scalar+fields">scalar fields</a> in the field <a class="existingWikiWord" href="/nlab/show/supermultiplet">supermultiplet</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>3</mn><mo>≤</mo><mi>d</mi><mo>≤</mo><mn>11</mn></mrow><annotation encoding="application/x-tex">3 \leq d \leq 11</annotation></semantics></math>-dimensional supergravity have <a class="existingWikiWord" href="/nlab/show/moduli+spaces">moduli spaces</a> parameterized by the <a class="existingWikiWord" href="/nlab/show/homogeneous+spaces">homogeneous spaces</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mrow><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">/</mo><msub><mi>K</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex"> E_{n(n)}/ K_n </annotation></semantics></math></div> <p>for</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>11</mn><mo>−</mo><mi>d</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> n = 11 - d \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">K_n</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/maximal+compact+subgroup">maximal compact subgroup</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mrow><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">E_{n(n)}</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mn>8</mn></msub><mo>≃</mo><mi>Spin</mi><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo><mo>,</mo><msub><mi>K</mi> <mn>7</mn></msub><mo>≃</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>8</mn><mo stretchy="false">)</mo><mo>,</mo><msub><mi>K</mi> <mn>6</mn></msub><mo>≃</mo><mi>Sp</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> K_8 \simeq Spin(16), K_7 \simeq SU(8), K_6 \simeq Sp(4) </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mn>5</mn></msub><mo>≃</mo><mi>Spin</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo><mo>×</mo><mi>Spin</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo><mo>,</mo><msub><mi>K</mi> <mn>4</mn></msub><mo>≃</mo><mi>Spin</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo><mo>,</mo><msub><mi>K</mi> <mn>3</mn></msub><mo>≃</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>×</mo><mi>SO</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> K_5 \simeq Spin(5) \times Spin(5), K_4 \simeq Spin(5), K_3 \simeq SU(2) \times SO(2) \,. </annotation></semantics></math></div> <p>Therefore <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mrow><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">E_{n(n)}</annotation></semantics></math> acts as a <a class="existingWikiWord" href="/nlab/show/global+symmetry">global symmetry</a> on the supergravity fields.</p> <p>This is no longer quite true for their <a class="existingWikiWord" href="/nlab/show/UV-completion">UV-completion</a> by the corresponding <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">compactifications</a> of <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> (e.g. <a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a> for <a class="existingWikiWord" href="/nlab/show/type+II+supergravity">type II supergravity</a>, etc.). Instead, on these a <a class="existingWikiWord" href="/nlab/show/discrete+group">discrete subgroup</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mrow><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo>↪</mo><msub><mi>E</mi> <mrow><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex"> E_{n(n)}(\mathbb{Z}) \hookrightarrow E_{n(n)} </annotation></semantics></math></div> <p>acts as global symmetry. This is called the <strong><a class="existingWikiWord" href="/nlab/show/U-duality">U-duality</a></strong> group of the supergravity theory (see there for more).</p> <p>It has been argued that this pattern should continue in some way further to the remaining values <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>d</mi><mo><</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">0 \leq d \lt 3</annotation></semantics></math>, with “<a class="existingWikiWord" href="/nlab/show/Kac-Moody+groups">Kac-Moody groups</a>” corresponding to the <a class="existingWikiWord" href="/nlab/show/Kac-Moody+algebras">Kac-Moody algebras</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝔢</mi> <mn>9</mn></msub><mo>,</mo><msub><mi>𝔢</mi> <mn>10</mn></msub><mo>,</mo><msub><mi>𝔢</mi> <mn>11</mn></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathfrak{e}_9, \mathfrak{e}_10, \mathfrak{e}_{11} \,. </annotation></semantics></math></div> <p>Continuing in the other direction to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">d = 10</annotation></semantics></math> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n = 1</annotation></semantics></math>) connects to the <a class="existingWikiWord" href="/nlab/show/T-duality">T-duality</a> group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(d,d,\mathbb{Z})</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a>.</p> <p>See the references (<a href="#UDualityReferences">below</a>).</p> <div> <table><thead><tr><th></th><th><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a> (<a class="existingWikiWord" href="/nlab/show/split+real+form">split real form</a>)</th><th><a class="existingWikiWord" href="/nlab/show/T-duality">T-duality</a> group (via toroidal <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a>)</th><th><a class="existingWikiWord" href="/nlab/show/U-duality">U-duality</a></th><th>maximal <a class="existingWikiWord" href="/nlab/show/gauged+supergravity">gauged supergravity</a></th><th></th></tr></thead><tbody><tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SL</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SL(2,\mathbb{R})</annotation></semantics></math></td><td style="text-align: left;">1</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SL%282%2CZ%29"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>SL</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mi>ℤ</mi> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">SL(2,\mathbb{Z})</annotation> </semantics> </math></a> <a class="existingWikiWord" href="/nlab/show/S-duality">S-duality</a></td><td style="text-align: left;">D=10 <a class="existingWikiWord" href="/nlab/show/type+IIB+supergravity">type IIB supergravity</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/special+linear+group">SL</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo><mo>×</mo></mrow><annotation encoding="application/x-tex">(2,\mathbb{R}) \times</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/orthogonal+group">O</a>(1,1)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SL%282%2CZ%29"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>SL</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mi>ℤ</mi> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">SL(2,\mathbb{Z})</annotation> </semantics> </math></a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>×</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\times \mathbb{Z}_2</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D%3D9+supergravity">D=9 supergravity</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/special+unitary+group">SU</a>(3)<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>×</mo></mrow><annotation encoding="application/x-tex">\times</annotation></semantics></math> SU(2)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/special+linear+group">SL</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>3</mn><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo><mo>×</mo><mi>SL</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(3,\mathbb{R}) \times SL(2,\mathbb{R})</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>;</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(2,2;\mathbb{Z})</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SL</mi><mo stretchy="false">(</mo><mn>3</mn><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo>×</mo><mi>SL</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SL(3,\mathbb{Z})\times SL(2,\mathbb{Z})</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D%3D8+supergravity">D=8 supergravity</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/special+unitary+group">SU</a>(5)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SL</mi><mo stretchy="false">(</mo><mn>5</mn><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SL(5,\mathbb{R})</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>3</mn><mo>,</mo><mn>3</mn><mo>;</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(3,3;\mathbb{Z})</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SL</mi><mo stretchy="false">(</mo><mn>5</mn><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SL(5,\mathbb{Z})</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D%3D7+supergravity">D=7 supergravity</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/spin+group">Spin</a>(10)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>5</mn><mo>,</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(5,5)</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>4</mn><mo>,</mo><mn>4</mn><mo>;</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(4,4;\mathbb{Z})</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>5</mn><mo>,</mo><mn>5</mn><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(5,5,\mathbb{Z})</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D%3D6+supergravity">D=6 supergravity</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/E%E2%82%86">E₆</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mrow><mn>6</mn><mo stretchy="false">(</mo><mn>6</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">E_{6(6)}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>5</mn><mo>,</mo><mn>5</mn><mo>;</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(5,5;\mathbb{Z})</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mrow><mn>6</mn><mo stretchy="false">(</mo><mn>6</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E_{6(6)}(\mathbb{Z})</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D%3D5+supergravity">D=5 supergravity</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/E%E2%82%87">E₇</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mrow><mn>7</mn><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">E_{7(7)}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>6</mn><mo>,</mo><mn>6</mn><mo>;</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(6,6;\mathbb{Z})</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mrow><mn>7</mn><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E_{7(7)}(\mathbb{Z})</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D%3D4+supergravity">D=4 supergravity</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/E%E2%82%88">E₈</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mrow><mn>8</mn><mo stretchy="false">(</mo><mn>8</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">E_{8(8)}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>7</mn><mo>,</mo><mn>7</mn><mo>;</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(7,7;\mathbb{Z})</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mrow><mn>8</mn><mo stretchy="false">(</mo><mn>8</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E_{8(8)}(\mathbb{Z})</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D%3D3+supergravity">D=3 supergravity</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/E%E2%82%89">E₉</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mrow><mn>9</mn><mo stretchy="false">(</mo><mn>9</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">E_{9(9)}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>8</mn><mo>,</mo><mn>8</mn><mo>;</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(8,8;\mathbb{Z})</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mrow><mn>9</mn><mo stretchy="false">(</mo><mn>9</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E_{9(9)}(\mathbb{Z})</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D%3D2+supergravity">D=2 supergravity</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/E%E2%82%88">E₈</a>-<a class="existingWikiWord" href="/nlab/show/equivariant+elliptic+cohomology">equivariant elliptic cohomology</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/E%E2%82%81%E2%82%80">E₁₀</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mrow><mn>10</mn><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">E_{10(10)}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>;</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(9,9;\mathbb{Z})</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mrow><mn>10</mn><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E_{10(10)}(\mathbb{Z})</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/E%E2%82%81%E2%82%81">E₁₁</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mrow><mn>11</mn><mo stretchy="false">(</mo><mn>11</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">E_{11(11)}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>10</mn><mo>,</mo><mn>10</mn><mo>;</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(10,10;\mathbb{Z})</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mrow><mn>11</mn><mo stretchy="false">(</mo><mn>11</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E_{11(11)}(\mathbb{Z})</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> </tbody></table> <p>(<a href="U-duality#HullTownsend94">Hull-Townsend 94, table 1, table 2</a>)</p> </div> <h3 id="exceptional_geometry">Exceptional geometry</h3> <p>For the moment see the remarks/references on supergravity at <em><a class="existingWikiWord" href="/nlab/show/exceptional+geometry">exceptional geometry</a></em> and <em><a class="existingWikiWord" href="/nlab/show/exceptional+generalized+geometry">exceptional generalized geometry</a></em>.</p> <h2 id="examples">Examples</h2> <p>The usual <a class="existingWikiWord" href="/nlab/show/folklore">folklore</a> is that for supergravity Lagrangians “of ordinary type” it turns out that</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">N = 1</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a></li> </ul> <p>is the highest dimension possible, but see also at</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/12-dimensional+supergravity">12-dimensional supergravity</a></li> </ul> <p>for some qualification.</p> <p>All lower dimensional theories in this class appear as <a class="existingWikiWord" href="/nlab/show/KK-compactifications">KK-compactifications</a> of this theory or are <a class="existingWikiWord" href="/nlab/show/deformations">deformations</a> of such:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/10-dimensional+supergravity">10-dimensional supergravity</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/type+II+supergravity">type II supergravity</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/type+IIA+supergravity">type IIA supergravity</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/massive+type+IIA+supergravity">massive type IIA supergravity</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/type+IIB+supergravity">type IIB supergravity</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/heterotic+supergravity">heterotic supergravity</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/9-dimensional+supergravity">9-dimensional supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/8-dimensional+supergravity">8-dimensional supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/7-dimensional+supergravity">7-dimensional supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/6-dimensional+supergravity">6-dimensional supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/5-dimensional+supergravity">5-dimensional supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/4-dimensional+supergravity">4-dimensional supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/3-dimensional+supergravity">3-dimensional supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-dimensional+supergravity">2-dimensional supergravity</a></p> </li> </ul> <p>In dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1+0)</annotation></semantics></math> supergravity coupled to <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> fields is the <a class="existingWikiWord" href="/nlab/show/spinning+particle">spinning particle</a>.</p> <p>In dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1+1)</annotation></semantics></math> supergravity coupled to <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> fields is the <a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>/<a class="existingWikiWord" href="/nlab/show/NSR+superstring">NSR superstring</a>.</p> <p>In non-Lorenzian signature it is also possible to consider</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/12-dimensional+supergravity">12-dimensional supergravity</a></li> </ul> <h2 id="phenomenology">Phenomenology</h2> <p>Discussion of evidence for supergravity from <a class="existingWikiWord" href="/nlab/show/experiment">experiment</a>/<a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a> includes the following:</p> <p>in (<a href="#DalianisFarakos15">Dalianis-Farakos 15</a>) it is argued that the <a class="existingWikiWord" href="/nlab/show/Starobinsky+model+of+cosmic+inflation">Starobinsky model of cosmic inflation</a>, which is strongly preferred by experiment, further improves after embedding into supergravity.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Rarita-Schwinger+field">Rarita-Schwinger field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauged+supergravity">gauged supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/magic+supergravity">magic supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topologically+twisted+supergravity">topologically twisted supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravitino+condensation">gravitino condensation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/duality+in+physics">duality in physics</a>, <a class="existingWikiWord" href="/nlab/show/duality+in+string+theory">duality in string theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/special+geometry">special geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/torsion+constraints+of+supergravity">torsion constraints of supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dragon%27s+theorem">Dragon's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a></p> </li> <li> <p><a href="string+theory+FAQ#DoesSTPredictSupersymmetry">string theory FAQ – Does string theory predict supersymmetry?</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cosmic+inflation">cosmic inflation</a> and <a class="existingWikiWord" href="/nlab/show/supersymmetry+breaking">supersymmetry breaking</a> via <a class="existingWikiWord" href="/nlab/show/higher+curvature+corrections">higher curvature corrections</a> of supergravity are discussed in the context of the <a class="existingWikiWord" href="/nlab/show/Starobinsky+model+of+cosmic+inflation">Starobinsky model of cosmic inflation</a></p> </li> </ul> <div> <p><strong>Table of <a class="existingWikiWord" href="/nlab/show/brane">branes</a> appearing in <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>/<a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a></strong> (for classification see at <em><a class="existingWikiWord" href="/nlab/show/brane+scan">brane scan</a></em>).</p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/brane">brane</a></th><th>in <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></th><th><a class="existingWikiWord" href="/nlab/show/charge">charge</a>d under <a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a></th><th>has <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> theory</th></tr></thead><tbody><tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/black+brane">black brane</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/higher+gauge+field">higher gauge field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SCFT">SCFT</a></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+II+supergravity">type II</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/RR-field">RR-field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">super Yang-Mills theory</a></td></tr> <tr><td style="text-align: left;"><strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>D</mi><mo>=</mo><mn>2</mn><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(D = 2n)</annotation></semantics></math></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+IIA+supergravity">type IIA</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D%28-2%29-brane">D(-2)-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D0-brane">D0-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/BFSS+matrix+model">BFSS matrix model</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D2-brane">D2-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D4-brane">D4-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D%3D5+super+Yang-Mills+theory">D=5 super Yang-Mills theory</a> with <a class="existingWikiWord" href="/nlab/show/Khovanov+homology">Khovanov homology</a> <a class="existingWikiWord" href="/nlab/show/observables">observables</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D6-brane">D6-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D%3D7+super+Yang-Mills+theory">D=7 super Yang-Mills theory</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D8-brane">D8-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>D</mi><mo>=</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(D = 2n+1)</annotation></semantics></math></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+IIB+supergravity">type IIB</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D%28-1%29-brane">D(-1)-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D1-brane">D1-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;">2d <a class="existingWikiWord" href="/nlab/show/CFT">CFT</a> with <a class="existingWikiWord" href="/nlab/show/BH+entropy">BH entropy</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D3-brane">D3-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/N%3D4+D%3D4+super+Yang-Mills+theory">N=4 D=4 super Yang-Mills theory</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D5-brane">D5-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D7-brane">D7-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D9-brane">D9-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28p%2Cq%29-string">(p,q)-string</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/D25-brane">D25-brane</a>)</td><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/bosonic+string+theory">bosonic string theory</a>)</td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/NS-brane">NS-brane</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/supergravity">type I, II, heterotic</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-connection</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/string">string</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/B2-field">B2-field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2d+SCFT">2d SCFT</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/NS5-brane">NS5-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/B6-field">B6-field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/little+string+theory">little string theory</a></td></tr> <tr><td style="text-align: left;"><strong>D-brane for <a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/A-brane">A-brane</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/B-brane">B-brane</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/M-brane">M-brane</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11D SuGra</a>/<a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-connection</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/C3-field">C3-field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/ABJM+theory">ABJM theory</a>, <a class="existingWikiWord" href="/nlab/show/BLG+model">BLG model</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/C6-field">C6-field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-superconformal+QFT">6d (2,0)-superconformal QFT</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M9-brane">M9-brane</a>/<a class="existingWikiWord" href="/nlab/show/O-plane">O9-plane</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M-wave">M-wave</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/topological+M2-brane">topological M2-brane</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/topological+M-theory">topological M-theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/C3-field">C3-field</a> on <a class="existingWikiWord" href="/nlab/show/G%E2%82%82-manifold">G₂-manifold</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/topological+M5-brane">topological M5-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/C6-field">C6-field</a> on <a class="existingWikiWord" href="/nlab/show/G%E2%82%82-manifold">G₂-manifold</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/S-brane">S-brane</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SM2-brane">SM2-brane</a>,<br /><a class="existingWikiWord" href="/nlab/show/membrane+instanton">membrane instanton</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M5-brane+instanton">M5-brane instanton</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D3-brane+instanton">D3-brane instanton</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/solitons">solitons</a></strong> on <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-superconformal+QFT">6d (2,0)-superconformal QFT</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/self-dual+string">self-dual string</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/self-dual+higher+gauge+field">self-dual</a> <a class="existingWikiWord" href="/nlab/show/B-field">B-field</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/3-brane+in+6d">3-brane in 6d</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <h3 id="general">General</h3> <p>Early review:</p> <ul> <li id="Nieuwenhuizen81"> <p><a class="existingWikiWord" href="/nlab/show/Peter+van+Nieuwenhuizen">Peter van Nieuwenhuizen</a>, <em>Supergravity</em>, Physics Reports, <strong>68,</strong> (1981) 189-398 [<a href="https://doi.org/10.1016/0370-1573(81)90157-5">doi:10.1016/0370-1573(81)90157-5</a>]</p> </li> <li id="DuffNilssonPope86"> <p><a class="existingWikiWord" href="/nlab/show/Mike+Duff">Mike Duff</a>, <a class="existingWikiWord" href="/nlab/show/Bengt+Nilsson">Bengt Nilsson</a>, <a class="existingWikiWord" href="/nlab/show/Christopher+Pope">Christopher Pope</a>, <em>Kaluza-Klein supergravity</em>, Physics Reports <strong>130</strong> 1–2 (1986) 1-142 [<a href="https://inspirehep.net/record/229417">spire:229417</a>, <a href="https://doi.org/10.1016/0370-1573(86)90163-8">doi:10.1016/0370-1573(86)90163-8</a>]</p> <blockquote> <p>(emphasis on <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+compactification">Kaluza-Klein compactification</a>)</p> </blockquote> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yuri+Manin">Yuri Manin</a>, §5.7 in: <em><a class="existingWikiWord" href="/nlab/show/Gauge+Field+Theory+and+Complex+Geometry">Gauge Field Theory and Complex Geometry</a></em>, Grundlehren der Mathematischen Wissenschaften <strong>289</strong>, Springer (1988) [<a href="https://doi.org/10.1007/978-3-662-07386-5">doi:10.1007/978-3-662-07386-5</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoshiaki+Tanii">Yoshiaki Tanii</a>, <em>Introduction to Supergravities in Diverse Dimensions</em>, in <em><a href="https://inspirehep.net/conferences/971516">YITP Workshop on Supersymmetry</a></em>, Kyoto (1996) [<a href="https://arxiv.org/abs/hep-th/9802138">arXiv:hep-th/9802138</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bernard+de+Wit">Bernard de Wit</a>, <a class="existingWikiWord" href="/nlab/show/Jan+Louis">Jan Louis</a>, <em>Supersymmetry and Dualities in various dimensions</em>, NATO Sci. Ser. C <strong>520</strong> (1999) 33-101 [<a href="https://arxiv.org/abs/hep-th/9801132">arXiv:hep-th/9801132</a>, <a href="https://inspirehep.net/literature/453367">inspire:453367</a>]</p> </li> </ul> <p>Textbook accounts:</p> <ul> <li id="CastellaniDAuriaFre91"> <p><a class="existingWikiWord" href="/nlab/show/Leonardo+Castellani">Leonardo Castellani</a>, <a class="existingWikiWord" href="/nlab/show/Riccardo+D%27Auria">Riccardo D'Auria</a>, <a class="existingWikiWord" href="/nlab/show/Pietro+Fr%C3%A9">Pietro Fré</a>, <em><a class="existingWikiWord" href="/nlab/show/Supergravity+and+Superstrings+-+A+Geometric+Perspective">Supergravity and Superstrings - A Geometric Perspective</a></em>, World Scientific (1991)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Steven+Weinberg">Steven Weinberg</a>, <em>Supergravity</em>, Section 31 in: <em>The quantum theory of fields. Vol. 3: Supersymmetry</em>, Cambridge University Press (2000) [<a href="https://www.cambridge.org/ae/universitypress/subjects/physics/theoretical-physics-and-mathematical-physics/quantum-theory-fields-volume-3?format=AR&isbn=9780521670555">ISBN:9781139632638</a>, <a href="https://inspirehep.net/literature/527189">spire:527189</a>, <a href="http://www.stat.ucla.edu/~ywu/research/documents/weinberg3.pdf">pdf</a>]</p> <blockquote> <p>“Gravity exists, so if there is any truth to supersymmetry then any realistic supersymmetry theory must eventually be enlarged to a supersymmetric theory of matter and gravitation, known as supergravity. Supersymmetry without supergravity is not an option, though it may be a good approximation at energies far below the Planck scale.”</p> </blockquote> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Freedman">Daniel Freedman</a>, <a class="existingWikiWord" href="/nlab/show/Antoine+Van+Proeyen">Antoine Van Proeyen</a>: <em>Supergravity</em>, Cambridge University Press (2012) [<a href="https://doi.org/10.1017/CBO9781139026833">doi:10.1017/CBO9781139026833</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pietro+Fr%C3%A9">Pietro Fré</a>, Ch 6 in: <em>Black Holes, Cosmology and Introduction to Supergravity</em>, volume 2 of: <em>Gravity, a Geometrical Course</em>, Springer (2013) [<a href="https://doi.org/10.1007/978-94-007-5443-0">doi:10.1007/978-94-007-5443-0</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Michel+Rausch+de+Traubenberg">Michel Rausch de Traubenberg</a>, Mauricio Valenzuela, <em>A Supergravity Primer – From Geometrical Principles to the Final Lagrangian</em>, World Scientific (2020) [<a href="https://doi.org/10.1142/11557">doi:10.1142/11557</a>]</p> </li> </ul> <p>Collection of original articles:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Abdus+Salam">Abdus Salam</a>, <a class="existingWikiWord" href="/nlab/show/Ergin+Sezgin">Ergin Sezgin</a> (eds.): <em>Supergravities in Diverse Dimensions</em>, Elsevier & World Scientific (1990) [<a href="https://doi.org/10.1142/0277">doi:10.1142/0277</a>]</li> </ul> <p>Survey:</p> <ul> <li> <p>Florian Domingo, <a class="existingWikiWord" href="/nlab/show/Michel+Rausch+de+Traubenberg">Michel Rausch de Traubenberg</a>, <em>Supergravity: Application in Particle Physics</em>, in <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Quantum+Gravity">Handbook of Quantum Gravity</a></em>, Springer (2023) [<a href="https://arxiv.org/abs/2209.12541">arXiv:2209.12541</a>, <a href="https://doi.org/10.1007/978-981-19-3079-9">doi:10.1007/978-981-19-3079-9</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ergin+Sezgin">Ergin Sezgin</a>, <em>Survey of supergravities</em>, in: <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Quantum+Gravity">Handbook of Quantum Gravity</a></em>, Springer (2023) [<a href="https://arxiv.org/abs/2312.06754">arXiv:2312.06754</a>, <a href="https://doi.org/10.1007/978-981-19-3079-9">doi:10.1007/978-981-19-3079-9</a>]</p> </li> </ul> <p>Lecture notes:</p> <ul> <li> <p>P. Binetruy, G. Girardi, R. Grimm, <em>Supergravity couplings: a geometric formulation</em>, Phys.Rept.343:255-462,2001 (<a href="http://arxiv.org/abs/hep-th/0005225">arXiv:hep-th/0005225</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Friedemann+Brandt">Friedemann Brandt</a>, <em>Lectures on supergravity</em>, Fortsch. Phys. <strong>50</strong> (2002) 1126-1172 [<a href="http://arxiv.org/abs/hep-th/0204035">arXiv:hep-th/0204035</a>, <A href=“https://doi.org/10.1002/1521-3978(200210)50:10/11%3C1126::AID-PROP1126%3E3.0.CO;2-B”>doi:10.1002/1521-3978(200210)50:10/11%3C1126::AID-PROP1126%3E3.0.CO;2-B</a> ]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bernard+de+Wit">Bernard de Wit</a>, <em>Supergravity</em> (<a href="http://arxiv.org/abs/hep-th/0212245">arXiv:hep-th/0212245</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Antoine+Van+Proeyen">Antoine Van Proeyen</a>, <em>Structure of supergravity theories</em> (<a href="http://arxiv.org/abs/hep-th/0301005">arXiv:hep-th/0301005</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Joachim+Gomis">Joachim Gomis</a>, <em>Three lectures on Supergravity</em> (<a href="http://www.fis.puc.cl/~jalfaro/supergravity/Three%20Lectures%20on%20Supergravity.pdf">pdf slides</a>)</p> </li> </ul> <p>Further surveys:</p> <ul> <li id="Duff04"><a class="existingWikiWord" href="/nlab/show/Michael+Duff">Michael Duff</a>, <em>The status of local supersymmetry</em>, in: <em>From Quarks to Black Holes: Progress in understanding the logic of Nature</em>, World Scientific (2005) 60-116 [<a href="http://arxiv.org/abs/hep-th/0403160">arXiv:hep-th/0403160</a>, <a href="https://doi.org/10.1142/9789812701794_0004">doi:10.1142/9789812701794_0004</a>]</li> </ul> <p>A fair bit of detail on <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a> and on supergravity is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Pierre+Deligne">P. Deligne</a>, <a class="existingWikiWord" href="/nlab/show/Pavel+Etingof">P. Etingof</a>, <a class="existingWikiWord" href="/nlab/show/Dan+Freed">D.S. Freed</a>, L. Jeffrey, <a class="existingWikiWord" href="/nlab/show/David+Kazhdan">D. Kazhdan</a>, J. Morgan, D.R. Morrison, <a class="existingWikiWord" href="/nlab/show/Edward+Witten">E. Witten</a>, eds. <em><a class="existingWikiWord" href="/nlab/show/Quantum+Fields+and+Strings">Quantum Fields and Strings</a>, A course for mathematicians</em>, 2 vols. Amer. Math. Soc. Providence 1999. (<a href="http://www.math.ias.edu/qft">web version</a>)</li> </ul> <p>The original article that introduced the <a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a> is</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Riccardo+D%27Auria">Riccardo D'Auria</a>, <a class="existingWikiWord" href="/nlab/show/Pietro+Fr%C3%A9">Pietro Fré</a>, <em><a class="existingWikiWord" href="/nlab/files/GeometricSupergravity.pdf" title="Geometric Supergravity in D=11 and its hidden supergroup">Geometric Supergravity in D=11 and its hidden supergroup</a></em>, Nuclear Physics B201 (1982) 101-140</li> </ul> <p>The underlying <a class="existingWikiWord" href="/nlab/show/super+Cartan+geometry">super Cartan geometry</a> is made fully explicit in:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Konstantin+Eder">Konstantin Eder</a>, <em>Super Cartan geometry and the super Ashtekar connection</em> (<a href="https://arxiv.org/abs/2010.09630">arXiv:2010.09630</a>)</li> </ul> <p>A compendium, of relevant <a class="existingWikiWord" href="/nlab/show/action+functionals">action functionals</a> and <a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a> is in</p> <ul> <li>M. J. D. Hamilton, <em>The field and Killing spinor equations of M-theory and type IIA/IIB supergravity in coordinate-free notation</em> (<a href="http://arxiv.org/abs/1607.00327">arXiv:1607.00327</a>)</li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a>-solutions:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Riccardo+D%27Auria">Riccardo D'Auria</a>, <a class="existingWikiWord" href="/nlab/show/Pietro+Fre">Pietro Fre</a>, <em>BPS black holes in supergravity</em>, (<a href="http://arxiv.org/abs/hep-th/9812160">hep-th/9812160</a>)</p> </li> <li> <p>Antonio Gallerati, <em>Constructing black hole solutions in supergravity theories</em> (<a href="https://arxiv.org/abs/1905.04104">arXiv:1905.04104</a>)</p> </li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/pure+spinor">pure spinor</a>-techniques:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Martin+Cederwall">Martin Cederwall</a>, <em>Pure spinors in classical and quantum supergravity</em>, in <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Quantum+Gravity">Handbook of Quantum Gravity</a></em> (2023) [<a href="https://arxiv.org/abs/2210.06141">arXiv:2210.06141</a>]</li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/higher+curvature+corrections">higher curvature corrections</a>:</p> <ul> <li>Mehmet Ozkan, Yi Pang, <a class="existingWikiWord" href="/nlab/show/Ergin+Sezgin">Ergin Sezgin</a>, <em>Higher Derivative Supergravities in Diverse Dimensions</em> [<a href="https://arxiv.org/abs/2401.08945">arXiv:2401.08945</a>]</li> </ul> <h3 id="renormalization">Renormalization</h3> <ul> <li>S. Deser, J.H. Kay, K.S. Stelle, <em>Renormalizability Properties of Supergravity</em>, Phys Rev Lett 38, 527 (1977) (reproduced as <a href="http://arxiv.org/abs/1506.03757">arXiv:1506.03757</a>)</li> </ul> <h3 id="UDualityReferences">U-duality</h3> <p>Some basic facts are recalled in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacques+Distler">Jacques Distler</a>, <em>Split real forms</em> (<a href="http://golem.ph.utexas.edu/~distler/blog/archives/001213.html">blog post</a>).</li> </ul> <p>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mrow><mn>7</mn><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">E_{7(7)}</annotation></semantics></math>-symmetry was first discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Bernard+de+Wit">Bernard de Wit</a>, <a class="existingWikiWord" href="/nlab/show/Hermann+Nicolai">Hermann Nicolai</a>, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><mn>11</mn></mrow><annotation encoding="application/x-tex">D = 11</annotation></semantics></math> Supergravity With Local <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>8</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(8)</annotation></semantics></math> Invariance</em>, Nucl. Phys. <p>B 274, 363 (1986)</p> </li> </ul> <p>and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mrow><mn>8</mn><mo stretchy="false">(</mo><mn>8</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">E_{8(8)}</annotation></semantics></math> in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hermann+Nicolai">Hermann Nicolai</a>, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><mn>11</mn></mrow><annotation encoding="application/x-tex">D = 11</annotation></semantics></math> Supergravity with Local <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(16)</annotation></semantics></math> Invariance</em> , Phys. Lett. B 187, 316 (1987).</p> </li> <li> <p>K. Koepsell, <a class="existingWikiWord" href="/nlab/show/Hermann+Nicolai">Hermann Nicolai</a>, <a class="existingWikiWord" href="/nlab/show/Henning+Samtleben">Henning Samtleben</a>, <em>An exceptional geometry for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>11</mn></mrow><annotation encoding="application/x-tex">d = 11</annotation></semantics></math> supergravity?</em>, Class. Quant. Grav. 17, 3689 (2000) (<a href="http://arxiv.org/abs/hep-th/0006034">arXiv:hep-th/0006034</a>).</p> </li> </ul> <p>The discrete quantum subgroups were discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Chris+Hull">Chris Hull</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Townsend">Paul Townsend</a>, <em>Unity of Superstring Dualities</em> Nucl.Phys.B438:109-137 (1995) (<a href="http://arxiv.org/abs/hep-th/9410167">arXiv:hep-th/9410167</a>)</li> </ul> <p>which also introduced the term “U-duality”.</p> <p>Review and further discusssion is in</p> <ul> <li>Shun’ya Mizoguchi, Germar Schroeder, <em>On Discrete U-duality in M-theory</em>, Class.Quant.Grav. 17 (2000) 835-870 (<a href="http://arxiv.org/abs/hep-th/9909150">arXiv:hep-th/9909150</a>)</li> </ul> <p>A careful discussion of the topology of the U-duality groups is in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Arjan+Keurentjes">Arjan Keurentjes</a>, <em>The topology of U-duality (sub-)groups</em> (<a href="http://arxiv.org/abs/hep-th/0309106">arXiv:hep-th/0309106</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Arjan+Keurentjes">Arjan Keurentjes</a>, <em>U-duality (sub-)groups and their topology</em> (<a href="http://arxiv.org/abs/hep-th/0312134">arXiv:hep-th/0312134</a>)</p> </li> </ul> <p>A discussion in the context of <a class="existingWikiWord" href="/nlab/show/generalized+complex+geometry">generalized complex geometry</a> / <a class="existingWikiWord" href="/nlab/show/exceptional+generalized+complex+geometry">exceptional generalized complex geometry</a> is in</p> <ul> <li> <p>Paulo Pires Pacheco, <a class="existingWikiWord" href="/nlab/show/Daniel+Waldram">Daniel Waldram</a>, <em>M-theory, exceptional generalised geometry and superpotentials</em> (<a href="http://arxiv.org/abs/0804.1362">arXiv:0804.1362</a>)</p> </li> <li> <p>Nicholas Houston, <em>Supergravity and Generalized Geometry</em> Thesis (2010) (<a href="https://workspace.imperial.ac.uk/theoreticalphysics/Public/MSc/Dissertations/2010/Nicholas%20Houston%20Dissertation.pdf">pdf</a>)</p> </li> </ul> <p>The case of “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>10</mn></msub></mrow><annotation encoding="application/x-tex">E_{10}</annotation></semantics></math>” is discussed in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Thibault+Damour">Thibault Damour</a>, <a class="existingWikiWord" href="/nlab/show/Marc+Henneaux">Marc Henneaux</a>, <a class="existingWikiWord" href="/nlab/show/Hermann+Nicolai">Hermann Nicolai</a>, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E(10)</annotation></semantics></math> and a ‘small tension expansion’ of M</em></p> <p>theory_, Phys. Rev. Lett. 89, 221601 (2002) (<a href="http://arxiv.org/abs/hep-th/0207267">arXiv:hep-th/0207267</a>);</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Axel+Kleinschmidt">Axel Kleinschmidt</a>, <a class="existingWikiWord" href="/nlab/show/Hermann+Nicolai">Hermann Nicolai</a>, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E(10)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(9,9)</annotation></semantics></math> invariant supergravity</em>, JHEP 0407,</p> <p>041 (2004) (<a href="http://arxiv.org/abs/hep-th/0407101">arXiv:hep-th/0407101</a>)</p> </li> </ul> <p>and that of “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>11</mn></msub></mrow><annotation encoding="application/x-tex">E_{11}</annotation></semantics></math>” in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Peter+West">Peter West</a>, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>11</mn></msub></mrow><annotation encoding="application/x-tex">E_{11}</annotation></semantics></math> and M-theory</em>, Class. Quant. Grav. 18, 4443 (2001) (<a href="http://arxiv.org/abs/hep-th/0104081">arXiv:hep-th/0104081</a>).</li> </ul> <p>General discussion of the <a class="existingWikiWord" href="/nlab/show/Kac-Moody+groups">Kac-Moody groups</a> arising in this context is for instance in</p> <ul> <li>Philipp Fleig, <a class="existingWikiWord" href="/nlab/show/Axel+Kleinschmidt">Axel Kleinschmidt</a>, <em>Eisenstein series for infinite-dimensional U-duality groups</em> (<a href="http://arxiv.org/abs/1204.3043">arXiv:1204.3043</a>)</li> </ul> <h3 id="gauged_supergravity">Gauged supergravity</h3> <ul> <li>Natxo Alonso-Alberca; and Tomáas Ortín, <em>Gauged/Massive supergravities in diverse dimensions</em> (<a href="http://digital.csic.es/bitstream/10261/38952/1/ARTICULOS302400%5B1%5D.pdf">pdf</a>)</li> </ul> <h3 id="chernsimons_supergravity">Chern-Simons supergravity</h3> <p>A survey of the <a class="existingWikiWord" href="/nlab/show/Chern-Simons+gravity">Chern-Simons gravity</a>-style action functionals for supergravity is in</p> <ul id="Zanelli"> <li><a class="existingWikiWord" href="/nlab/show/Jorge+Zanelli">Jorge Zanelli</a>, <em>Lecture notes on Chern-Simons (super-)gravities</em> (<a href="http://arxiv.org/abs/hep-th/0502193">arXiv:0502193</a>)</li> </ul> <h3 id="History">History</h3> <p>The idea of supergravity was proposed by</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Dmitry+Volkov">Dmitry Volkov</a>, V.P. Akulov, <em>Possible universal neutrino interaction</em>, ZhETF Pis. Red. (JETP Letters, AIP translation), 16, n.11 (1972) 621 (<a href="http://www.jetpletters.ac.ru/ps/1766/article_26864.shtml">pdf</a>)</li> </ul> <p>followed up by the first model of supergravity (in nonlinear realization) constructed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Dmitry+Volkov">Dmitry Volkov</a>, <a class="existingWikiWord" href="/nlab/show/Vyacheslav+Soroka">Vyacheslav Soroka</a>, <em>Higgs effect for Goldstone particles with spin 1/2</em>, ZhETF Pis. Red. (JETP Letters, AIP translation), 18, n.8 (1973) 529 (<a href="https://www.jetpletters.ac.ru/ps/1568/article_24038.shtml">pdf</a>)</li> </ul> <p>However, the term “supergravity” was coined later by <a href="#FreedmanNieuwenhuizenFerrara76">Freedman, Nieuwenhuizen, Ferrara 76</a>, whose work on supergravity is regarded as foundational for the subject.</p> <p>This early history is discussed in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Steven+Duplij">Steven Duplij</a>, <em>Supergravity was discovered by D.V. Volkov and V.A. Soroka in 1973, wasn’t it?</em>, East Eur. J. Phys., v3, p. 81-82 (2019) (<a href="https://arxiv.org/abs/1910.03259">arXiv:1910.03259</a>)</p> </li> <li id="Kuzenko21"> <p>Sergei M. Kuzenko, <em>Local supersymmetry: Variations on a theme by Volkov and Soroka</em> (<a href="https://arxiv.org/abs/2110.12835">arXiv:2110.12835</a>)</p> </li> </ul> <p>Supergravity, in the guise of <a class="existingWikiWord" href="/nlab/show/4d+supergravity">4d supergravity</a>, was first found (constructed) in</p> <ul> <li id="FreedmanNieuwenhuizenFerrara76"><a class="existingWikiWord" href="/nlab/show/Daniel+Freedman">Daniel Freedman</a>, <a class="existingWikiWord" href="/nlab/show/Peter+van+Nieuwenhuizen">Peter van Nieuwenhuizen</a>, <a class="existingWikiWord" href="/nlab/show/Sergio+Ferrara">Sergio Ferrara</a>, <em>Progress toward a theory of supergravity</em>, Phys. Rev. D13 (1976) 3214 (<a href="https://doi.org/10.1103/PhysRevD.13.3214">doi.org/10.1103/PhysRevD.13.3214</a>)</li> </ul> <p>Accounts of the early history include the following:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Sergio+Ferrara">Sergio Ferrara</a>, <em>Supergravity and the quest for a unified theory</em> (<a href="https://arxiv.org/abs/hep-th/9405065">arxiv:hep-th/9405065</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dmitry+Volkov">Dmitry Volkov</a>, <em>Supergravity before 1976</em>,</p> <p>International Conference on the History of Original Ideas and Basic Discoveries in Particle Physics, Erice, 1994 (<a href="http://link.springer.com/chapter/10.1007/978-1-4613-1147-8_34">Springer, Chapter</a> or <a href="http://arxiv.org/abs/hep-th/9410024">arXiv:hep-th/9410024</a>)</p> </li> <li> <p>R. Arnowitt, <a class="existingWikiWord" href="/nlab/show/Ali+Chamseddine">Ali Chamseddine</a>, Pran Nath, <em>The Development of Supergravity Grand Unification: Circa 1982-85</em> (<a href="http://arxiv.org/abs/1206.3175">arXiv:1206.3175</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dmitry+Volkov">Dmitry Volkov</a>, <em>Supergravity before and after 1976. The story of goldstonions</em>, in [Concise Encyclopedia of Supersymmetry,]</p> <p>S. Duplij, J. Bagger, W. Siegel (Eds.), Springer, 2004](http://www.springer.com/gp/book/9781402013386) or <a href="http://arxiv.org/abs/hep-th/9404153">arXiv:hep-th/9404153</a></p> </li> <li> <p>David Appell, <em>When supergravity was born</em>, 2012 (<a href="http://www.davidappell.com/articles/PWSep12appell-supergravity.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Peter+van+Nieuwenhuizen">Peter van Nieuwenhuizen</a>, <em>Aspects of supergravity</em>, 2014 (<a href="http://media.scgp.stonybrook.edu/presentations/20140109_vanNieuwenhuizen.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Vyacheslav+Soroka">Vyacheslav Soroka</a>, <em>The Sources of Supergravity</em> in <a href="http://www.worldscientific.com/doi/10.1142/9789812385505_0011">The Supersymmetric World. The Beginnings of the Theory, G. Kane and M. Shifman (Eds.), World Scientific, 2000</a> or <a href="https://arxiv.org/abs/hep-th/0203171">arXiv:hep-th/0203171</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sergio+Ferrara">Sergio Ferrara</a>, A. Sagnotti, <em>Supergravity at 40: Reflections and Perspectives</em>, Based in part on the talk delivered by S. F. at the “Infeld Colloquium and Discrete”, in Warsaw, on December 1 2016, and on a joint CERN Courier article. Dedicated to <a class="existingWikiWord" href="/nlab/show/John+Schwarz">John Schwarz</a> on the occasion of his 75-th birthday (<a href="https://arxiv.org/abs/1702.00743">arXiv:1702.00743</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Stanley+Deser">Stanley Deser</a>, <em>A brief history of Supergravity: the first three weeks</em> (<a href="https://arxiv.org/abs/1704.05886">arXiv:1704.05886</a>)</p> </li> </ul> <p>See also <em><a href="supersymmetry#ReferencesHistory">supersymmetry – History</a></em>.</p> <h3 id="related">Related</h3> <p>Further physics monographs on supergravity include</p> <ul> <li> <p>I. L. Buchbinder, S. M. Kuzenko, <em>Ideas and methods of supersymmetry and supergravity; or A walk through superspace</em>, <a href="http://books.google.com/books?isbn=0750305061">googB</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Julius+Wess">Julius Wess</a>, Jonathan Bagger, <em>Supersymmetry and supergravity</em>, 1992</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Steven+Weinberg">Steven Weinberg</a>, <em>Quantum theory of fields</em>, volume III: supersymmetry</p> </li> <li> <p><a href="http://www.springer.com/gp/book/9781402013386">Concise Encyclopedia of Supersymmetry, S. Duplij, J. Bagger, W. Siegel (Eds.), Springer, 2004</a>, <a href="http://ivv5hpp.uni-muenster.de/u/douplii/susy/SUSYEnc_Story.pdf">SUSY story</a> narrated by its founders</p> </li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/Cauchy+problem">Cauchy problem</a> for classical solutions of simple supergravity has been discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yvonne+Choquet-Bruhat">Yvonne Choquet-Bruhat</a>, <em><a class="existingWikiWord" href="/nlab/show/The+Cauchy+Problem+in+Classical+Supergravity">The Cauchy Problem in Classical Supergravity</a></em>, Letters in Mathematical Physics 7 (1983) 459-467. 0377</li> </ul> <p>A canonical textbook reference for the role of Calabi-Yau manifolds in compactifications of 10-dimensional <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> is volume II, starting on page 1091 in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/P.+Deligne">P. Deligne</a>, <a class="existingWikiWord" href="/nlab/show/P.+Etingof">P. Etingof</a>, <a class="existingWikiWord" href="/nlab/show/D.+Freed">D. Freed</a>, L. Jeffrey, <a class="existingWikiWord" href="/nlab/show/D.+Kazhdan">D. Kazhdan</a>, J. Morgan, D. R. Morrison and <a class="existingWikiWord" href="/nlab/show/E.+Witten">E. Witten</a>, eds. <em><a class="existingWikiWord" href="/nlab/show/Quantum+Fields+and+Strings">Quantum Fields and Strings</a>, A course for mathematicians</em>, 2 vols. Amer. Math. Soc. Providence 1999. (<a href="http://www.math.ias.edu/qft">web version</a>)</li> </ul> <p>Discussion of solutions with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">N = 1</annotation></semantics></math> global supersymmetry left and their relation to Calabi-Yau compactifications are for instance in</p> <ul> <li id="GMPT"><a class="existingWikiWord" href="/nlab/show/Mariana+Gra%C3%B1a">Mariana Graña</a>, <a class="existingWikiWord" href="/nlab/show/Ruben+Minasian">Ruben Minasian</a>, Michela Petrini, <a class="existingWikiWord" href="/nlab/show/Alessandro+Tomasiello">Alessandro Tomasiello</a>, <em>Generalized structures of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">N=1</annotation></semantics></math> vacua</em>, <a href="http://arxiv.org/abs/hep-th/0505212">arXiv:hep-th/0505212</a></li> </ul> <h3 id="PhenomenologyReferences">Phenomenology</h3> <p>On supergravity <a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a>:</p> <h4 id="dark_matter">Dark matter</h4> <p>Discussion of the <a class="existingWikiWord" href="/nlab/show/gravitino">gravitino</a> as a <a class="existingWikiWord" href="/nlab/show/dark+matter">dark matter</a> candidate:</p> <ul> <li id="EllisOlive10"><a class="existingWikiWord" href="/nlab/show/John+Ellis">John Ellis</a>, <a class="existingWikiWord" href="/nlab/show/Keith+Olive">Keith Olive</a>, <em>Supersymmetric Dark Matter Candidates</em> (<a href="http://arxiv.org/abs/1001.3651">arXiv:1001.3651</a>)</li> </ul> <p id="NicolaiMeissner"> A proposal for super-heavy <a class="existingWikiWord" href="/nlab/show/gravitinos">gravitinos</a> as <a class="existingWikiWord" href="/nlab/show/dark+matter">dark matter</a>, by embedding <a class="existingWikiWord" href="/nlab/show/D%3D4+N%3D8+supergravity">D=4 N=8 supergravity</a> into <a class="existingWikiWord" href="/nlab/show/E10">E10</a>-<a class="existingWikiWord" href="/nlab/show/U-duality">U-duality</a>-invariant <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a>:</p> <ul> <li id="MeissnerNicolai18a"> <p><a class="existingWikiWord" href="/nlab/show/Krzysztof+A.+Meissner">Krzysztof A. Meissner</a>, <a class="existingWikiWord" href="/nlab/show/Hermann+Nicolai">Hermann Nicolai</a>: <em>Standard Model Fermions and Infinite-Dimensional R-Symmetries</em>, Phys. Rev. Lett. <strong>121</strong> 091601 (2018) [<a href="https://arxiv.org/abs/1804.09606">arXiv:1804.09606</a>, <a href="https://doi.org/10.1103/PhysRevLett.121.091601">doi:10.1103/PhysRevLett.121.091601</a>]</p> </li> <li id="MeissnerNicolai18b"> <p><a class="existingWikiWord" href="/nlab/show/Krzysztof+A.+Meissner">Krzysztof A. Meissner</a>, <a class="existingWikiWord" href="/nlab/show/Hermann+Nicolai">Hermann Nicolai</a>, <em>Planck Mass Charged Gravitino Dark Matter</em>, Phys. Rev. D 100, 035001 (2019) (<a href="https://arxiv.org/abs/1809.01441">arXiv:1809.01441</a>)</p> </li> </ul> <p>following the proposal towards the end of</p> <ul> <li id="GellMann83"> <p><a class="existingWikiWord" href="/nlab/show/Murray+Gell-Mann">Murray Gell-Mann</a>, introductory talk at <em><a href="https://en.wikipedia.org/wiki/Shelter_Island_Conference">Shelter Island II</a></em>, 1983 (<a class="existingWikiWord" href="/nlab/files/Gell-Mann_ShelterIslandII_1983.pdf" title="pdf">pdf</a>)</p> <p>in: <em>Shelter Island II: Proceedings of the 1983 Shelter Island Conference on Quantum Field Theory and the Fundamental Problems of Physics</em>. MIT Press. pp. 301–343. ISBN 0-262-10031-2.</p> </li> </ul> <p>Further discussion:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Krzysztof+A.+Meissner">Krzysztof A. Meissner</a>, <a class="existingWikiWord" href="/nlab/show/Hermann+Nicolai">Hermann Nicolai</a>, <em>Supermassive gravitinos and giant primordial black holes</em> (<a href="https://arxiv.org/abs/2007.11889">arXiv:2007.11889</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Krzysztof+A.+Meissner">Krzysztof A. Meissner</a>, <a class="existingWikiWord" href="/nlab/show/Hermann+Nicolai">Hermann Nicolai</a>, <em>Evidence for a stable supermassive gravitino with charge <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mo stretchy="false">/</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">2/3</annotation></semantics></math>?</em> [<a href="https://arxiv.org/abs/2303.09131">arXiv:2303.09131</a>]</p> </li> <li> <p>Adrianna Kruk, Michał Lesiuk, <a class="existingWikiWord" href="/nlab/show/Krzysztof+A.+Meissner">Krzysztof A. Meissner</a>, <a class="existingWikiWord" href="/nlab/show/Hermann+Nicolai">Hermann Nicolai</a>: <em>Signatures of supermassive charged gravitinos in liquid scintillator detectors</em> [<a href="https://arxiv.org/abs/2407.04883">arXiv:2407.04883</a>]</p> </li> </ul> <h4 id="dark_energy">Dark energy</h4> <p>On supergravity <a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a> in relation to <a class="existingWikiWord" href="/nlab/show/dark+energy">dark energy</a>/<a class="existingWikiWord" href="/nlab/show/cosmological+constant">cosmological constant</a>:</p> <ul> <li>Y. Aghababaie, <a class="existingWikiWord" href="/nlab/show/Cliff+P.+Burgess">Cliff P. Burgess</a>, S.L. Parameswaran, <a class="existingWikiWord" href="/nlab/show/Fernando+Quevedo">Fernando Quevedo</a>: <em>Towards a Naturally Small Cosmological Constant from Branes in 6D Supergravity</em>, Nucl. Phys. B <strong>680</strong> (2004) 389-414 [<a href="https://arxiv.org/abs/hep-th/0304256">arXiv:hep-th/0304256</a>, <a href="https://doi.org/10.1016/j.nuclphysb.2003.12.015">doi:10.1016/j.nuclphysb.2003.12.015</a>]</li> </ul> <h4 id="cosmic_inflation">Cosmic inflation</h4> <p>On supergravity in <a class="existingWikiWord" href="/nlab/show/cosmic+inflation">cosmic inflation</a>:</p> <ul> <li id="DalianisFarakos15">Ioannis Dalianis, <a class="existingWikiWord" href="/nlab/show/Fotis+Farakos">Fotis Farakos</a>, <em>On the initial conditions for inflation with plateau potentials: the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>+</mo><msup><mi>R</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">R + R^2</annotation></semantics></math> (super)gravity case</em> (<a href="http://arxiv.org/abs/1502.01246">arXiv:1502.01246</a>)</li> </ul> <h4 id="standard_model_of_particle_physics">Standard model of particle physics</h4> <p>On the <a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a> of supergravity coupled to the (non-supersymmetric) <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cliff+P.+Burgess">Cliff P. Burgess</a>, <a class="existingWikiWord" href="/nlab/show/Fernando+Quevedo">Fernando Quevedo</a>: <em>Who’s Afraid of the Supersymmetric Dark? The Standard Model vs Low-Energy Supergravity</em>, Fortschr. Physik <strong>70</strong> 7-8 (2022) 2200077 [<a href="https://arxiv.org/abs/2110.13275">arXiv:2110.13275</a>, <a href="https://doi.org/10.1002/prop.202200077">doi:10.1002/prop.202200077</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 27, 2024 at 10:40:51. 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