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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="string_theory">String theory</h4> <div class="hide"><div> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a></strong></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/books+about+string+theory">books about string theory</a></p> </li> </ul> <h3 id="ingredients">Ingredients</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a>,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/CFT">CFT</a>, <a class="existingWikiWord" href="/nlab/show/perturbation+theory">perturbation theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+QFT">effective background QFT</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>, <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a>, <a class="existingWikiWord" href="/nlab/show/quantum+gravity">quantum gravity</a></li> </ul> </li> </ul> <h3 id="critical_string_models">Critical string models</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a>, <a class="existingWikiWord" href="/nlab/show/differential+string+structure">differential string structure</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dual+heterotic+string+theory">dual heterotic string theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+fivebrane+structure">differential fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/type+IIA+string+theory">type IIA string theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/type+IIB+string+theory">type IIB string theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/F-theory">F-theory</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+field+theory">string field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/duality+in+string+theory">duality in string theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/T-duality">T-duality</a>, <a class="existingWikiWord" href="/nlab/show/mirror+symmetry">mirror symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/S-duality">S-duality</a>, <a class="existingWikiWord" href="/nlab/show/electric-magnetic+duality">electric-magnetic duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/U-duality">U-duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open%2Fclosed+string+duality">open/closed string duality</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/AdS%2FCFT+correspondence">AdS/CFT correspondence</a>, <a class="existingWikiWord" href="/nlab/show/holographic+principle">holographic principle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a>, <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%C5%99ava-Witten+theory">Hořava-Witten theory</a></li> </ul> </li> </ul> <h3 id="extended_objects">Extended objects</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/brane">brane</a></p> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/D0-brane">D0-brane</a>, <a class="existingWikiWord" href="/nlab/show/D2-brane">D2-brane</a>, <a class="existingWikiWord" href="/nlab/show/D4-brane">D4-brane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D1-brane">D1-brane</a>, <a class="existingWikiWord" href="/nlab/show/D3-brane">D3-brane</a>, <a class="existingWikiWord" href="/nlab/show/D5-brane">D5-brane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/RR-field">RR-field</a>, <a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/NS-brane">NS-brane</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>, <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/B2-field">B2-field</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/NS5-brane">NS5-brane</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/B6-field">B6-field</a></li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/M-brane">M-brane</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/C3-field">C3-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ABJM+theory">ABJM theory</a>, <a class="existingWikiWord" href="/nlab/show/BLG+model">BLG model</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/C6-field">C6-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a></p> </li> </ul> </li> </ul> </li> </ul> <h3 id="topological_strings">Topological strings</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a>, <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+M-theory">topological M-theory</a></p> </li> </ul> <h2 id="backgrounds">Backgrounds</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/target+space">target space</a>, <a class="existingWikiWord" href="/nlab/show/background+gauge+field">background gauge field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+smooth+cohomology+in+string+theory">twisted smooth cohomology in string theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/landscape+of+string+theory+vacua">landscape of string theory vacua</a></p> </li> </ul> <h2 id="phenomenology">Phenomenology</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/string+phenomenology">string phenomenology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/moduli+stabilization">moduli stabilization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/G%E2%82%82-MSSM">G₂-MSSM</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/string+theory+-+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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</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='#explicit_selfdual_formulations'>Explicit self-dual formulations</a></li> <li><a href='#solutions_and_bps_states'>Solutions and BPS states</a></li> <li><a href='#InTermsOfGeneralizedComplexGeometryReferences'>In terms of (exceptional) generalized complex geometry</a></li> <li><a href='#higher_curvature_corrections'>Higher curvature corrections</a></li> <li><a href='#via_double_field_theory'>Via double field theory</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p><em>Type II supergravity</em> is a <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> in <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> 10 which is the low-energy <a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a> underlying <a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/massive+type+IIA+supergravity">massive type IIA supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/type+II+supergravity+Lie+2-algebra">type II supergravity Lie 2-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/type+II+geometry">type II geometry</a>, <a class="existingWikiWord" href="/nlab/show/generalized+complex+geometry">generalized complex geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/T-duality">T-duality</a>, <a class="existingWikiWord" href="/nlab/show/double+field+theory">double field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/heterotic+supergravity">heterotic 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/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/duality+between+M-theory+and+type+IIA+string+theory">duality between M-theory and type IIA string theory</a></p> </li> <li> <p><a href="string+theory+FAQ#DoesSTPredictSupersymmetry">string theory FAQ – Does string theory predict supersymmetry?</a></p> </li> </ul> <h2 id="References">References</h2> <h3 id="general">General</h3> <p>Original constructions (mostly of <a class="existingWikiWord" href="/nlab/show/type+IIA+supergravity">type IIA</a>):</p> <ul> <li> <p>F. Giani, M. Pernici, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">N=2</annotation></semantics></math> supergravity in ten dimensions</em>, Phys. Rev. D <strong>30</strong> (1984) 325 [<a href="https://doi.org/10.1103/PhysRevD.30.325">doi:10.1103/PhysRevD.30.325</a>]</p> </li> <li> <p>M. Huq, M. A. Namazie, <em>Kaluza-Klein supergravity in ten dimensions</em>, Classical and Quantum Gravity <strong>2</strong> 3 (1985) [<a href="https://iopscience.iop.org/article/10.1088/0264-9381/2/3/007">doi:10.1088/0264-9381/2/3/007</a>]</p> </li> <li> <p>I. C. G. Campbell, <a class="existingWikiWord" href="/nlab/show/Peter+C.+West">Peter C. West</a> <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">N = 2</annotation></semantics></math>, <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> non-chiral supergravity and its spontaneous compactification</em>, Nuclear Physics B <strong>243</strong> 1 (1984) 112-124 [<a href="https://doi.org/10.1016/0550-3213(84)90388-2">doi:10.1016/0550-3213(84)90388-2</a>]</p> </li> </ul> <p>See also the general references at <em><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></em>.</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Joseph+Polchinski">Joseph Polchinski</a>, chapter 12.1 of <em><a class="existingWikiWord" href="/nlab/show/String+Theory">String Theory</a></em> Vol 2: Superstring and beyond_, Cambridge Monographs on Mathematical Physics (1998)</li> </ul> <p>Construction of <a class="existingWikiWord" href="/nlab/show/type+IIA+supergravity">type IIA supergravity</a> via <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a> from <a class="existingWikiWord" href="/nlab/show/11d+supergravity">11d supergravity</a>:</p> <ul> <li> <p>M. Huq, M. A. Namazie, <em>Kaluza-Klein Supergravity In Ten Dimensions</em>, Class. Quantum Grav. 2 (1985) 293 (<a href="https://inspirehep.net/literature/196711">spire:196711</a>)</p> </li> <li id="DFGT08"> <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>, <a class="existingWikiWord" href="/nlab/show/Pietro+Grassi">Pietro Grassi</a>, Trigiante, <em>Pure Spinor Superstrings on Generic type IIA Supergravity Backgrounds</em> <a href="https://arxiv.org/abs/0803.1703">arXiv:0803.1703</a></p> </li> </ul> <h3 id="explicit_selfdual_formulations">Explicit self-dual formulations</h3> <div> <p>Discussion of (<a class="existingWikiWord" href="/nlab/show/Lagrangian+densities">Lagrangian densities</a> for) <a class="existingWikiWord" href="/nlab/show/D%3D10+type+II+supergravity">D=10 type II supergravity</a> with “<a class="existingWikiWord" href="/nlab/show/duality-symmetric+higher+gauge+theory">duality-symmetric</a>”/“democratic”/“<a class="existingWikiWord" href="/nlab/show/pregeometric+RR-fields">pregeometric</a>” for of the <a class="existingWikiWord" href="/nlab/show/RR-fields">RR-fields</a>:</p> <ul> <li id="Dall'AgataLechnerTonin98"> <p><a class="existingWikiWord" href="/nlab/show/Gianguido+Dall%27Agata">Gianguido Dall'Agata</a>, <a class="existingWikiWord" href="/nlab/show/Kurt+Lechner">Kurt Lechner</a>, <a class="existingWikiWord" href="/nlab/show/Mario+Tonin">Mario Tonin</a>, <em><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><mi>IIB</mi></mrow><annotation encoding="application/x-tex">N=IIB</annotation></semantics></math> Supergravity: Lorentz-invariant actions and duality</em>, JHEP 9807:017 (1998) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/hep-th/9806140">arXiv:hep-th/9806140</a>, <a href="https://doi.org/10.1088/1126-6708/1998/07/017">doi:10.1088/1126-6708/1998/07/017</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li id="CremmerJuliaLuPope98"> <p><a class="existingWikiWord" href="/nlab/show/Eugene+Cremmer">Eugene Cremmer</a>, <a class="existingWikiWord" href="/nlab/show/Bernard+Julia">Bernard Julia</a>, H. Lu, <a class="existingWikiWord" href="/nlab/show/Christopher+Pope">Christopher Pope</a>, Section 3 of: <em>Dualisation of Dualities, II: Twisted self-duality of doubled fields and superdualities</em>, Nucl. Phys. B <strong>535</strong>(1998) 242-292 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/hep-th/9806106">arXiv:hep-th/9806106</a>, <a href="https://doi.org/10.1016/S0550-3213(98)00552-5">doi:10.1016/S0550-3213(98)00552-5</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li id="BerhshoeffKalloshOrtinRoestVanProyen01"> <p><a class="existingWikiWord" href="/nlab/show/Eric+Bergshoeff">Eric Bergshoeff</a>, <a class="existingWikiWord" href="/nlab/show/Renata+Kallosh">Renata Kallosh</a>, <a class="existingWikiWord" href="/nlab/show/Tomas+Ortin">Tomas Ortin</a>, <a class="existingWikiWord" href="/nlab/show/Diederik+Roest">Diederik Roest</a>, <a class="existingWikiWord" href="/nlab/show/Antoine+Van+Proeyen">Antoine Van Proeyen</a>, <em>New Formulations of D=10 Supersymmetry and D8-O8 Domain Walls</em>, Class. Quant. Grav. <strong>18</strong> (2001) 3359-3382 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/hep-th/0103233">arXiv:hep-th/0103233</a>, <a href="https://doi.org/10.1088/0264-9381/18/17/303">doi:10.1088/0264-9381/18/17/303</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li id="BandosNurmagambetovSorokin04"> <p><a class="existingWikiWord" href="/nlab/show/Igor+Bandos">Igor Bandos</a>, <a class="existingWikiWord" href="/nlab/show/Alexei+Nurmagambetov">Alexei Nurmagambetov</a>, <a class="existingWikiWord" href="/nlab/show/Dmitri+Sorokin">Dmitri Sorokin</a>, <em>Various Faces of Type IIA Supergravity</em>, Nucl. Phys. B <strong>676</strong> (2004) 189-228 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/hep-th/0307153">arXiv:hep-th/0307153</a>, <a href="https://doi.org/10.1016/j.nuclphysb.2003.10.036">doi:10.1016/j.nuclphysb.2003.10.036</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li id="MkrtchyanValach23"> <p><a class="existingWikiWord" href="/nlab/show/Karapet+Mkrtchyan">Karapet Mkrtchyan</a>, <a class="existingWikiWord" href="/nlab/show/Fridrich+Valach">Fridrich Valach</a>, <em>Democratic actions for type II supergravities</em>, Phys. Rev. D <strong>107</strong> 6 (2023) 066027 [<a href="https://doi.org/10.1103/PhysRevD.107.066027">doi:10.1103/PhysRevD.107.066027</a>, <a href="https://arxiv.org/abs/2207.00626">arXiv:2207.00626</a>]</p> </li> </ul> <p>Enhancement of the <a class="existingWikiWord" href="/nlab/show/self-dual+higher+gauge+field">self-duality</a> constraint on <a class="existingWikiWord" href="/nlab/show/pregeometric+RR-field">pregeometric</a> <a class="existingWikiWord" href="/nlab/show/RR-fields">RR-fields</a> from (<a class="existingWikiWord" href="/nlab/show/twisted+de+Rham+cohomology">twisted</a>) <a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a> to (<a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted</a>) <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a> (under the hypothesized <a class="existingWikiWord" href="/nlab/show/K-theory+classification+of+D-brane+charge">K-theory classification of D-brane charge</a>) in terms of a <a class="existingWikiWord" href="/nlab/show/quadratic+form">quadratic form</a> on <a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a>:</p> <ul> <li id="MooreWitten99"> <p><a class="existingWikiWord" href="/nlab/show/Gregory+Moore">Gregory Moore</a>, <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>Self-Duality, Ramond-Ramond Fields, and K-Theory</em>, JHEP 0005:032 (2000) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="http://arxiv.org/abs/hep-th/9912279">arXiv:hep-th/9912279</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li id="Witten00"> <p><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>Duality Relations Among Topological Effects In String Theory</em>, JHEP 0005:031 (2000) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="http://arxiv.org/abs/hep-th/9912086">arXiv:hep-th/9912086</a>, <a href="https://doi.org/10.1088/1126-6708/2000/05/032">doi:10.1088/1126-6708/2000/05/032</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li id="FreedHopkins00"> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <em>On Ramond-Ramond fields and K-theory</em>, JHEP 0005 (2000) 044 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="http://arxiv.org/abs/hep-th/0002027">arXiv:hep-th/0002027</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li id="Freed02"> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <em><a class="existingWikiWord" href="/nlab/show/Dirac+charge+quantization+and+generalized+differential+cohomology">Dirac charge quantization and generalized differential cohomology</a></em>, Surveys in Differential Geometry <strong>7</strong>, Int. Press (2000) 129-194 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="http://arxiv.org/abs/hep-th/0011220">arXiv:hep-th/0011220</a>, <a href="https://dx.doi.org/10.4310/SDG.2002.v7.n1.a6">doi:10.4310/SDG.2002.v7.n1.a6</a>, <a href="https://inspirehep.net/literature/537392">spire:537392</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> </ul> <p>An indication of a more refined discussion in <a class="existingWikiWord" href="/nlab/show/twisted+differential+K-theory">twisted differential</a> <a class="existingWikiWord" href="/nlab/show/KR-theory">KR-theory</a>:</p> <ul> <li id="DFM09"><a class="existingWikiWord" href="/nlab/show/Jacques+Distler">Jacques Distler</a>, <a class="existingWikiWord" href="/nlab/show/Dan+Freed">Dan Freed</a>, <a class="existingWikiWord" href="/nlab/show/Greg+Moore">Greg Moore</a>, <em>Orientifold Précis</em> in: <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a> (eds.) <em><a class="existingWikiWord" href="/schreiber/show/Mathematical+Foundations+of+Quantum+Field+and+Perturbative+String+Theory">Mathematical Foundations of Quantum Field and Perturbative String Theory</a></em>, Proceedings of Symposia in Pure Mathematics, AMS (2011) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="http://arxiv.org/abs/0906.0795">arXiv:0906.0795</a>, <a href="http://www.ma.utexas.edu/users/dafr/bilbao.pdf">slides</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> <p>See at <em><a class="existingWikiWord" href="/nlab/show/orientifold">orientifold</a></em> for more on this.</p> <p>Expressing the <a class="existingWikiWord" href="/nlab/show/self-dual+higher+gauge+theory">self-duality</a> of <a class="existingWikiWord" href="/nlab/show/pregeometric+RR-fields">pregeometric RR-fields</a> in terms of <a class="existingWikiWord" href="/nlab/show/11d+Chern-Simons+theory">11d Chern-Simons theory</a>:</p> <ul> <li id="BelovMooreII">Dmitriy Belov, <a class="existingWikiWord" href="/nlab/show/Greg+Moore">Greg Moore</a>, <em>Type II Actions from 11-Dimensional Chern-Simons Theories</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="http://arxiv.org/abs/hep-th/0611020">arXiv:hep-th/0611020</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> <p>Some review:</p> <ul> <li id="Szabo12"><a class="existingWikiWord" href="/nlab/show/Richard+Szabo">Richard Szabo</a>, section 3.6 and 4.6 of: <em>Quantization of Higher Abelian Gauge Theory in Generalized Differential Cohomology</em>, ESI 2385 (2012) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="http://arxiv.org/abs/1209.2530">arXiv:1209.2530</a>, <a href="https://www.mat.univie.ac.at/~esiprpr/esi2385.pdf">pdf</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> <p>Discussion in the context of <a class="existingWikiWord" href="/nlab/show/flux+quantization">flux quantization</a> (here: <a class="existingWikiWord" href="/nlab/show/D-brane+charge+quantization+in+K-theory">D-brane charge quantization in K-theory</a>):</p> <ul> <li><a href="duality-symmetric+higher+gauge+theory#SS23">SS23, §3.3</a></li> </ul> </div> <h3 id="solutions_and_bps_states">Solutions and BPS states</h3> <p>Disucssion of <a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a> solutions (see also at <em><a class="existingWikiWord" href="/nlab/show/black+holes+in+string+theory">black holes in string theory</a></em>) includes</p> <ul> <li>U. Gran, <a class="existingWikiWord" href="/nlab/show/Jan+Gutowski">Jan Gutowski</a>, <a class="existingWikiWord" href="/nlab/show/George+Papadopoulos">George Papadopoulos</a>, <em>IIB black hole horizons with five-form flux and extended supersymmetry</em>, JHEP 1109:047,2011 (<a href="http://arxiv.org/abs/1104.2908">arXiv:1104.2908</a>)</li> </ul> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/black+branes">black branes</a> and <a class="existingWikiWord" href="/nlab/show/BPS+states">BPS states</a> for type II supergravity includes</p> <ul> <li> <p>Andrew Callister, <a class="existingWikiWord" href="/nlab/show/Douglas+Smith">Douglas Smith</a>, <em>Topological BPS charges in 10 and 11-dimensional supergravity</em>, Phys. Rev. D78:065042,2008 (<a href="http://arxiv.org/abs/0712.3235">arXiv:0712.3235</a>)</p> </li> <li> <p>Andrew Callister, <a class="existingWikiWord" href="/nlab/show/Douglas+Smith">Douglas Smith</a>, <em>Topological charges in <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> covariant massive 11-dimensional and Type IIB SUGRA</em>, Phys.Rev.D80:125035,2009 (<a href="http://arxiv.org/abs/0907.3614">arXiv:0907.3614</a>)</p> </li> <li> <p>Andrew Callister, <em>Topological BPS charges in 10- and 11-dimensional supergravity</em>, thesis 2010 (<a href="http://inspirehep.net/record/1221591?ln=en">spire</a>)</p> </li> <li> <p>A. A. Golubtsova, V.D. Ivashchuk, <em>BPS branes in 10 and 11 dimensional supergravity</em>, talk at DIAS 2013 (<a href="http://theor.jinr.ru/~diastp/summer13/lectures/Golubtsova.pdf">pdf slides</a>)</p> </li> </ul> <p>Discussion of <em>asymptotic</em> <a class="existingWikiWord" href="/nlab/show/de+Sitter+spacetimes">de Sitter spacetimes</a> from time-dependent <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a> of <a class="existingWikiWord" href="/nlab/show/type+II+supergravity">type II supergravity</a>:</p> <ul> <li>Paul Marconnet, <a class="existingWikiWord" href="/nlab/show/Dimitrios+Tsimpis">Dimitrios Tsimpis</a>, <em>Universal accelerating cosmologies from 10d supergravity</em>, J. High Energ. Phys. <strong>2023</strong> 33 (2023) [<a href="https://arxiv.org/abs/2210.10813">arXiv:2210.10813</a>, <a href="https://doi.org/10.1007/JHEP01(2023)033">doi:10.1007/JHEP01(2023)033</a>]</li> </ul> <p>reviewed in:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Dimitrios+Tsimpis">Dimitrios Tsimpis</a>, <em>Universal accelerating cosmologies from 10d supergravity</em>, talk at <em><a href="/nlab/show/M-Theory+and+Mathematics#2023">M-Theory and Mathematics 2023</a></em>, NYU Abu Dhabi (Jan 2023) [<a href="/nlab/show/M-Theory+and+Mathematics#Tsimpis2023">web</a>]</li> </ul> <h3 id="InTermsOfGeneralizedComplexGeometryReferences">In terms of (exceptional) generalized complex geometry</h3> <p>A relation of the <a class="existingWikiWord" href="/nlab/show/U-duality">U-duality</a> symmetry to <a class="existingWikiWord" href="/nlab/show/generalized+complex+geometry">generalized complex geometry</a> is discussed in</p> <ul> <li> <p>André Coimbra, Charles Strickland-Constable, <a class="existingWikiWord" href="/nlab/show/Daniel+Waldram">Daniel Waldram</a>, <em>Supergravity as Generalised Geometry I: Type II Theories</em> (<a href="http://arxiv.org/abs/1107.1733">arXiv:1107.1733</a>)</p> </li> <li> <p>Paulo Pires Pacheco, Daniel Waldram, <em>M-theory, exceptional generalised geometry and superpotentials</em> (<a href="http://arxiv.org/abs/0804.1362">arXiv:0804.1362</a>)</p> </li> </ul> <p>A thesis reviewing some aspects is</p> <ul> <li>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>)</li> </ul> <h3 id="higher_curvature_corrections">Higher curvature corrections</h3> <p>On <a class="existingWikiWord" href="/nlab/show/higher+curvature+corrections">higher curvature corrections</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/James+Liu">James Liu</a>, <a class="existingWikiWord" href="/nlab/show/Ruben+Minasian">Ruben Minasian</a>, Raffaele Savelli, Andreas Schachner, <em>Type IIB at eight derivatives: insights from Superstrings, Superfields and Superparticles</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/2205.11530">arXiv:2205.11530</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> <h3 id="via_double_field_theory">Via double field theory</h3> <p>Discusdion of type IIA and IIB supergravities via <a class="existingWikiWord" href="/nlab/show/double+field+theory">double field theory</a>:</p> <ul> <li>Olaf Hohm, Seung Ki Kwak, Barton Zwiebach, <em>Unification of Type II Strings and T-duality</em> (<a href="https://arxiv.org/abs/1106.5452">arXiv:1106.5452</a>)</li> </ul> <p>and, to the full order in fermions, in</p> <ul> <li>Imtak Jeon, Kanghoon Lee, Jeong-Hyuck Park, Yoonji Suh, <em>Stringy Unification of Type IIA and IIB Supergravities under N=2 D=10 Supersymmetric Double Field Theory</em> (<a href="https://arxiv.org/abs/1210.5078">arxiv:1210.5078</a>)</li> </ul> <p>Comprehensive discussion in <a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher differential geometry</a>:</p> <ul> <li id="Alfonsi19"> <p><a class="existingWikiWord" href="/nlab/show/Luigi+Alfonsi">Luigi Alfonsi</a>, <em>Global Double Field Theory is Higher Kaluza-Klein Theory</em>, Fortsch. d. Phys. 2020 (<a href="https://arxiv.org/abs/1912.07089">arXiv:1912.07089</a>, <a href="https://doi.org/10.1002/prop.202000010">doi:10.1002/prop.202000010</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Luigi+Alfonsi">Luigi Alfonsi</a>, <em>The puzzle of global Double Field Theory: open problems and the case for a Higher Kaluza-Klein perspective</em> (<a href="https://arxiv.org/abs/2007.04969">arXiv:2007.04969</a>)</p> <p>(relating <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+compactification">Kaluza-Klein compactification</a> on <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundles">principal ∞-bundles</a> to <a class="existingWikiWord" href="/nlab/show/double+field+theory">double field theory</a>, <a class="existingWikiWord" href="/nlab/show/T-folds">T-folds</a>, <a class="existingWikiWord" href="/nlab/show/non-abelian+T-duality">non-abelian T-duality</a>, <a class="existingWikiWord" href="/nlab/show/type+II+geometry">type II geometry</a>, <a class="existingWikiWord" href="/nlab/show/exceptional+geometry">exceptional geometry</a>, …)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on December 21, 2023 at 17:53:41. 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