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value="Search" 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="riemannian_geometry">Riemannian geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/Riemannian+geometry">Riemannian geometry</a></strong></p> <h2 id="basic_definitions">Basic definitions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/metric">metric</a>, <a class="existingWikiWord" href="/nlab/show/isometry">isometry</a>, <a class="existingWikiWord" href="/nlab/show/isometry+group">isometry group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/moduli+space+of+Riemannian+metrics">moduli space of Riemannian metrics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+manifold">pseudo-Riemannian manifold</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lorentzian+manifold">Lorentzian manifold</a>, <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geodesic">geodesic</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geodesic+convexity">geodesic convexity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geodesic+flow">geodesic flow</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Levi-Civita+connection">Levi-Civita connection</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Riemann+curvature">Riemann curvature</a>, <a class="existingWikiWord" href="/nlab/show/torsion+of+a+metric+connection">torsion of a metric connection</a></li> </ul> </li> </ul> <h2 id="further_concepts">Further concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+inner+product">Hodge inner product</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+star+operator">Hodge star operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gradient">gradient</a>, <a class="existingWikiWord" href="/nlab/show/gradient+flow">gradient flow</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+conjecture">Poincaré conjecture</a>-theorem</li> </ul> <h2 id="applications">Applications</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Einstein-Hilbert+action">Einstein-Hilbert action</a>, <a class="existingWikiWord" href="/nlab/show/Einstein+equations">Einstein equations</a>, <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/Riemannian+geometry+-+contents">Edit this sidebar</a> </p> </div></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="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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#properties'>Properties</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometric</a> analog of <a class="existingWikiWord" href="/nlab/show/anti-de+Sitter+spacetime">anti-de Sitter spacetime</a> (<a class="existingWikiWord" href="/nlab/show/product+manifold">times</a> an internal space). By the discussion at <em><a href="supersymmetry#ClassificationSuperconformal">supersymmetry – Classification – Superconformal and super anti de Sitter symmetry</a></em> this includes the following <a class="existingWikiWord" href="/nlab/show/coset">coset</a> <a class="existingWikiWord" href="/nlab/show/superspacetimes">superspacetimes</a> (<a class="existingWikiWord" href="/nlab/show/super+Klein+geometries">super Klein geometries</a>):</p> <table><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><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><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/super+anti+de+Sitter+spacetime">super anti de Sitter spacetime</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math>4<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</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="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mfrac><mrow><mi>OSp</mi><mo stretchy="false">(</mo><mn>8</mn><mo stretchy="false">|</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo></mrow></mfrac><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;\;\;\;\frac{OSp(8\vert 4)}{Spin(3,1) \times \mathrm{O}(7)}\;\;\;\;</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math>5<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</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="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mfrac><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo stretchy="false">|</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>4</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></mrow></mfrac><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;\;\;\;\frac{SU(2,2 \vert 5)}{Spin(4,1)\times \mathrm{O}(5)}\;\;\;\;</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math>7<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</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="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mfrac><mrow><mi>OSp</mi><mo stretchy="false">(</mo><mn>6</mn><mo>,</mo><mn>2</mn><mo stretchy="false">|</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>6</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></mfrac><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;\;\;\;\frac{OSp(6,2 \vert 4)}{Spin(6,1) \times \mathrm{O}(4)}\;\;\;\;</annotation></semantics></math></td></tr> </tbody></table> <p>(Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Osp</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Osp(-\vert-)</annotation></semantics></math> denotes <a class="existingWikiWord" href="/nlab/show/orthosymplectic+super+Lie+groups">orthosymplectic super Lie groups</a>.)</p> <p>E.g. for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">d=4</annotation></semantics></math> this identification [<a href="#DAuriaFré83">D’Auria & Fré 1983</a>] relies on the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>OSp</mi><mo stretchy="false">(</mo><mn>8</mn><mo stretchy="false">|</mo><mn>4</mn><mo>;</mo><mi>ℝ</mi><msub><mo stretchy="false">)</mo> <mi>bos</mi></msub><mo>≃</mo><mi>Sp</mi><mo stretchy="false">(</mo><mn>4</mn><mo>;</mo><mi>ℝ</mi><mo stretchy="false">)</mo><mo>×</mo><mi>O</mi><mo stretchy="false">(</mo><mn>8</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">OSp(8 \vert 4; \mathbb{R})_{bos} \simeq Sp(4;\mathbb{R}) \times O(8)</annotation></semantics></math> and then on the <a class="existingWikiWord" href="/nlab/show/exceptional+isomorphism">exceptional isomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sp</mi><mo stretchy="false">(</mo><mn>4</mn><mo>;</mo><mi>ℝ</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Spin</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sp(4; \mathbb{R}) \simeq Spin(2,3)</annotation></semantics></math> [e.g. <a href="exceptional+isomorphism#Garrett13">Garret 2013, §2.8</a>].</p> <p> <div class='num_remark'> <h6>Remark</h6> <p>On the other hand, <a class="existingWikiWord" href="/nlab/show/de+Sitter+spacetimes">de Sitter spacetimes</a> (instead of anti-de Sitter) do not have a standard extensions to <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a>; but see <a href="https://arxiv.org/abs/1610.01566">arXiv:1610.01566</a>.</p> </div> </p> <h2 id="properties">Properties</h2> <p>Explicit formulas for <a class="existingWikiWord" href="/nlab/show/super+Cartan+geometry">super</a> <a class="existingWikiWord" href="/nlab/show/Cartan+connections">Cartan connections</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>AdS</mi> <mn>4</mn></msub><mo>×</mo><msup><mi>S</mi> <mn>7</mn></msup></mrow><annotation encoding="application/x-tex">AdS_4 \times S^7</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>AdS</mi> <mn>7</mn></msub><mo>×</mo><msup><mi>S</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">AdS_7 \times S^4</annotation></semantics></math> are given in <a href="#dWPPS98">dWPPS 98, p. 156</a>, following <a href="#KalloshRahmfeldRajaraman98">Kallosh, Rahmfeld & Rajaraman 1998</a> and <a href="#ClausKallosh99">Claus & Kallosh 1999</a>, see also <a href="#Claus98">Claus 1998</a> (reviewed in <a href="#Wang23">Wang 2023, §4.4</a>).</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/super+anti+de+Sitter+group">super anti de Sitter group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Minkowski+spacetime">super Minkowski spacetime</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AdS%2FCFT+correspondence">AdS/CFT correspondence</a>, <a class="existingWikiWord" href="/nlab/show/AdS%2FQCD+correspondence">AdS/QCD correspondence</a></p> </li> </ul> <h2 id="references">References</h2> <blockquote> <p>See also the references at <em><a href="Green-Schwarz+action+functional#ReferencesAdSBackgrounds">Green-Schwarz sigma-model – References – AdS backgrounds</a></em>.</p> </blockquote> <p>General discussion:</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>, sections II.2.6, II.3.2-3, II.5, and V.4.4 in: <em><a class="existingWikiWord" href="/nlab/show/Supergravity+and+Superstrings+-+A+Geometric+Perspective">Supergravity and Superstrings - A Geometric Perspective</a></em>, World Scientific (1991) [ch II.2: <a class="existingWikiWord" href="/nlab/files/CastellaniDAuriaFre-ChII2.pdf" title="pdf">pdf</a>, ch II.3: <a class="existingWikiWord" href="/nlab/files/CastellaniDAuriaFre-ChII3.pdf" title="pdf">pdf</a>, ch II.5: <a class="existingWikiWord" href="/nlab/files/CastellaniDAuriaFre-ChII5.pdf" title="pdf">pdf</a>, ch V.4: <a class="existingWikiWord" href="/nlab/files/CastellaniDAuriaFre-ChV4.pdf" title="pdf">pdf</a>]</p> </li> <li id="KalloshRahmfeldRajaraman98"> <p><a class="existingWikiWord" href="/nlab/show/Renata+Kallosh">Renata Kallosh</a>, J. Rahmfeld, <a class="existingWikiWord" href="/nlab/show/Arvind+Rajaraman">Arvind Rajaraman</a>, <em>Near Horizon Superspace</em>, JHEP 9809:002 (1998) [<a href="https://arxiv.org/abs/hep-th/9805217">arXiv:hep-th/9805217</a>, <a href="https://doi.org/10.1088/1126-6708/1998/09/002">doi:10.1088/1126-6708/1998/09/002</a>]</p> </li> <li id="ClausKallosh99"> <p><a class="existingWikiWord" href="/nlab/show/Piet+Claus">Piet Claus</a>, <a class="existingWikiWord" href="/nlab/show/Renata+Kallosh">Renata Kallosh</a>, <em>Superisometries of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>adS</mi><mo>×</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">adS \times S</annotation></semantics></math> Superspace</em>, JHEP 9903:014 (1999) [<a href="https://arxiv.org/abs/hep-th/9812087">arXiv:hep-th/9812087</a>, <a href="https://doi.org/10.1088/1126-6708/1999/03/014">doi:10.1088/1126-6708/1999/03/014</a>]</p> </li> <li id="Claus98"> <p><a class="existingWikiWord" href="/nlab/show/Piet+Claus">Piet Claus</a>, <em>Super M-brane actions in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>adS</mi> <mn>4</mn></msub><mo>×</mo><msup><mi>S</mi> <mn>7</mn></msup></mrow><annotation encoding="application/x-tex">adS_4 \times S^7</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>adS</mi> <mn>7</mn></msub><mo>×</mo><msup><mi>S</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">adS_7 \times S^4</annotation></semantics></math></em>, Phys. Rev. D <strong>59</strong> (1999) 066003 [<a href="http://arxiv.org/abs/hep-th/9809045">arXiv:hep-th/9809045</a>, <a href="https://doi.org/10.1103/PhysRevD.59.066003">doi:10.1103/PhysRevD.59.066003</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Antoine+Van+Proeyen">Antoine Van Proeyen</a>, sections 4.5, 4.7 of: <em>Tools for supersymmetry</em> [<a href="https://arxiv.org/abs/hep-th/9910030">arXiv:hep-th/9910030</a>]</p> </li> <li id="KTM22"> <p><a class="existingWikiWord" href="/nlab/show/Sergei+Kuzenko">Sergei Kuzenko</a>, Gabriele Tartaglino-Mazzucchelli: <em>Supertwistor realisations of AdS superspaces</em>, The European Physical Journal C <strong>82</strong> 2 (2022) 146 [<a href="https://doi.org/10.1140/epjc/s10052-022-10072-y">doi:10.1140/epjc/s10052-022-10072-y</a>, <a href="https://arxiv.org/abs/2108.03907">arXiv:2108.03907</a>]</p> </li> </ul> <p>Specifically concerning super-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>AdS</mi> <mn>4</mn></msub><mo>×</mo><msup><mi>S</mi> <mn>7</mn></msup></mrow><annotation encoding="application/x-tex">AdS_4 \times S^7</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/superspace">super</a>-<a class="existingWikiWord" href="/nlab/show/near+horizon+geometry">near horizon geometry</a> of <a class="existingWikiWord" href="/nlab/show/black+brane">black</a> <a class="existingWikiWord" href="/nlab/show/M2-branes">M2-branes</a> as in <a class="existingWikiWord" href="/nlab/show/AdS4%2FCFT3-duality">AdS4/CFT3-duality</a>):</p> <ul> <li id="DAuriaFré83"> <p><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>Spontaneous generation of symmetry in the spontaneous compactification of <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>, Physics Letters B <strong>121</strong> 2–3 (1983) 141-146 [<a href="https://doi.org/10.1016/0370-2693(83)90903-6">doi:10.1016/0370-2693(83)90903-6</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>, pp. 33 in: <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>(no discussion of <a class="existingWikiWord" href="/nlab/show/superspace">superspace</a>, but of the <a class="existingWikiWord" href="/nlab/show/orthosymplectic+supergroup"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>OSp</mi> <mo stretchy="false">(</mo> <mn>8</mn> <mo stretchy="false">|</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">OSp(8 \vert 4)</annotation> </semantics> </math></a>-<a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a>)</p> </blockquote> </li> <li> <p><a href="#CastellaniDAuriaFre91">Castellani, D’Auria & Fré 1991, §3.1</a></p> </li> <li id="dWPPS98"> <p><a class="existingWikiWord" href="/nlab/show/Bernard+de+Wit">Bernard de Wit</a>, <a class="existingWikiWord" href="/nlab/show/Kasper+Peeters">Kasper Peeters</a>, <a class="existingWikiWord" href="/nlab/show/Jan+Plefka">Jan Plefka</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Sevrin">Alexander Sevrin</a>, <em>The M-theory two-brane in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>AdS</mi> <mn>4</mn></msub><mo>×</mo><msup><mi>S</mi> <mn>7</mn></msup></mrow><annotation encoding="application/x-tex">AdS_4 \times S^7</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>AdS</mi> <mn>7</mn></msub><mo>×</mo><msup><mi>S</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">AdS_7 \times S^4</annotation></semantics></math></em>, Physics Letters B <strong>443</strong> 1-4 (1998) 153-158 [<a href="https://doi.org/10.1016/S0370-2693(98)01340-9">doi:10.1016/S0370-2693(98)01340-9</a>, <a href="https://inspirehep.net/literature/474621">inspire:474621</a>, <a href="https://arxiv.org/abs/hep-th/9808052">arXiv:hep-th/9808052</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gianguido+Dall%27Agata">Gianguido Dall'Agata</a>, Davide Fabbri, Christophe Fraser, <a class="existingWikiWord" href="/nlab/show/Pietro+Fr%C3%A9">Pietro Fré</a>, Piet Termonia, <a class="existingWikiWord" href="/nlab/show/Mario+Trigiante">Mario Trigiante</a>, §4.1 in: <em>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Osp</mi><mo stretchy="false">(</mo><mn>8</mn><mo stretchy="false">|</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Osp(8|4)</annotation></semantics></math> singleton action from the supermembrane</em>, Nucl. Phys. B <strong>542</strong> (1999) 157-194 [<a href="https://arxiv.org/abs/hep-th/9807115">arXiv:hep-th/9807115</a>, <a href="https://doi.org/10.1016/S0550-3213(98)00765-2">doi:10.1016/S0550-3213(98)00765-2</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jaume+Gomis">Jaume Gomis</a>, <a class="existingWikiWord" href="/nlab/show/Dmitri+Sorokin">Dmitri Sorokin</a>, <a class="existingWikiWord" href="/nlab/show/Linus+Wulff">Linus Wulff</a>, <em>The complete <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>AdS</mi> <mn>4</mn></msub><mo>×</mo><mi>ℂ</mi><msup><mi>P</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">AdS_4 \times \mathbb{C}P^3</annotation></semantics></math> superspace for the type IIA superstring and D-branes</em>, JHEP 0903:015 (2009) [<a href="https://arxiv.org/abs/0811.1566">arXiv:0811.1566</a>]</p> </li> </ul> <p>Review:</p> <ul> <li id="Wang23">Zihan Wang, <em>A review on <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 and M2-brane</em>, MSc thesis (2023) [<a href="https://www.imperial.ac.uk/media/imperial-college/research-centres-and-groups/theoretical-physics/msc/dissertations/2023/Zihan-Wang-Dissertation.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Wang-M2Brane.pdf" title="pdf">pdf</a>]</li> </ul> <p>Specifically for the <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a> on super-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>AdS</mi> <mn>5</mn></msub><mo>×</mo><msup><mi>S</mi> <mn>5</mn></msup></mrow><annotation encoding="application/x-tex">AdS_5 \times S^5</annotation></semantics></math>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Igor+A.+Bandos">Igor A. Bandos</a>: <em>Superembedding approach to superstring in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>AdS</mi> <mn>5</mn></msub><mo>×</mo><msup><mi>S</mi> <mn>5</mn></msup></mrow><annotation encoding="application/x-tex">AdS_5 \times S^5</annotation></semantics></math> superspace</em>, in: <em>Fundamental Interactions</em> (2009) 303-334 [<a href="https://arxiv.org/abs/0812.0257">arXiv:0812.0257</a>, <a href="https://doi.org/10.1142/9789814277839_0018">doi:10.1142/9789814277839_0018</a>]</li> </ul> <p>Specifically for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>AdS</mi> <mrow><mn>4</mn><mo stretchy="false">|</mo><mn>4</mn><mi>𝒩</mi></mrow></msup></mrow><annotation encoding="application/x-tex">AdS^{4\vert 4 \mathcal{N}}</annotation></semantics></math>:</p> <ul> <li>Nowar E. Koning, <a class="existingWikiWord" href="/nlab/show/Sergei+M.+Kuzenko">Sergei M. Kuzenko</a>, Emmanouil S. N. Raptakis: <em>The anti-de Sitter supergeometry revisited</em> [<a href="https://arxiv.org/abs/2412.03172">arXiv:2412.03172</a>]</li> </ul> <p>The super 3-cocycle for the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+superstring">Green-Schwarz superstring</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo stretchy="false">|</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>4</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>×</mo><mi>SO</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{SU(2,2 \vert 5)}{Spin(4,1)\times SO(5)}</annotation></semantics></math> is originally due to</p> <ul> <li id="MetsaevTseytlin98"><a class="existingWikiWord" href="/nlab/show/Ruslan+Metsaev">Ruslan Metsaev</a>, <a class="existingWikiWord" href="/nlab/show/Arkady+Tseytlin">Arkady Tseytlin</a>, <em>Type IIB superstring action in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>AdS</mi> <mn>5</mn></msub><mo>×</mo><msup><mi>S</mi> <mn>5</mn></msup></mrow><annotation encoding="application/x-tex">AdS_5 \times S^5</annotation></semantics></math> background</em>, Nucl. Phys.B <strong>533</strong> (1998) 109-126 [<a href="https://arxiv.org/abs/hep-th/9805028">arXiv:hep-th/9805028</a>]</li> </ul> <p>However, a supersymmetric trivialization of this cocycle seems to have been obtained (according to <a href="https://arxiv.org/abs/1808.04470">arxiv:1808.04470, p. 5 and equation (5.5)</a>, but check) in:</p> <ul> <li id="RoibanSiegel00"><a class="existingWikiWord" href="/nlab/show/Radu+Roiban">Radu Roiban</a>, <a class="existingWikiWord" href="/nlab/show/Warren+Siegel">Warren Siegel</a>, <em>Superstrings on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>AdS</mi> <mn>5</mn></msub><mo>×</mo><msup><mi>S</mi> <mn>5</mn></msup></mrow><annotation encoding="application/x-tex">AdS_5 \times S^5</annotation></semantics></math> supertwistor space</em>, JHEP 0011:024 (2000) [<a href="https://arxiv.org/abs/hep-th/0010104">arXiv:hep-th/0010104</a>, <a href="https://doi.org/10.1088/1126-6708/2000/11/024">doi:10.1088/1126-6708/2000/11/024</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on December 9, 2024 at 16:16:58. 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