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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="physics">Physics</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/physics">physics</a></strong>, <a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a>, <a class="existingWikiWord" href="/nlab/show/philosophy+of+physics">philosophy of physics</a></p> <h2 id="surveys_textbooks_and_lecture_notes">Surveys, textbooks and lecture notes</h2> <ul> <li> <p><em><a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">(higher) category theory and physics</a></em></p> </li> <li> <p><em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/books+and+reviews+in+mathematical+physics">books and reviews</a>, <a class="existingWikiWord" href="/nlab/show/physics+resources">physics resources</a></p> </li> </ul> <hr /> <p><a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory (physics)</a>, <a class="existingWikiWord" href="/nlab/show/model+%28physics%29">model (physics)</a></p> <p><a class="existingWikiWord" href="/nlab/show/experiment">experiment</a>, <a class="existingWikiWord" href="/nlab/show/measurement">measurement</a>, <a class="existingWikiWord" href="/nlab/show/computable+physics">computable physics</a></p> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/mechanics">mechanics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/mass">mass</a>, <a class="existingWikiWord" href="/nlab/show/charge">charge</a>, <a class="existingWikiWord" href="/nlab/show/momentum">momentum</a>, <a class="existingWikiWord" href="/nlab/show/angular+momentum">angular momentum</a>, <a class="existingWikiWord" href="/nlab/show/moment+of+inertia">moment of inertia</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dynamics+on+Lie+groups">dynamics on Lie groups</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/rigid+body+dynamics">rigid body dynamics</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lagrangian+mechanics">Lagrangian mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a>, <a class="existingWikiWord" href="/nlab/show/state">state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange equations</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-symplectic+manifold">n-symplectic manifold</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+Lorentzian+manifold">smooth Lorentzian manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/special+relativity">special relativity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/dilaton+gravity">dilaton gravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/classical+field+theory">Classical field theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+physics">classical physics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+mechanics">classical mechanics</a></li> <li><a class="existingWikiWord" href="/nlab/show/waves">waves</a> and <a class="existingWikiWord" href="/nlab/show/optics">optics</a></li> <li><a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a></li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+mechanics">Quantum Mechanics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+mechanics+in+terms+of+dagger-compact+categories">in terms of ∞-compact categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+operator">Hamiltonian operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantization">Quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path+integral">path integral quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semiclassical+approximation">semiclassical approximation</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+field+theory">Quantum Field Theory</a></strong></p> <ul> <li> <p>Axiomatizations</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/AQFT">algebraic QFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+net">local net</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/boson">boson</a>, <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/FQFT">functorial QFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+topological+quantum+field+theory">extended topological quantum field theory</a></p> </li> </ul> </li> </ul> </li> <li> <p>Tools</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-function+theory">geometric ∞-function theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+%28in+particle+phyiscs%29">models</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fields+and+quanta+-+table">fields and quanta</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GUT">Grand Unified Theories</a>, <a class="existingWikiWord" href="/nlab/show/MSSM">MSSM</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/on-shell+recursion">on-shell recursion</a>, <a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></li> </ul> </li> </ul> </li> <li> <p>Structural phenomena</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universality+class">universality class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/instanton">instanton</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spontaneously+broken+symmetry">spontaneously broken symmetry</a></p> </li> <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/integrable+systems">integrable systems</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holonomic+quantum+fields">holonomic quantum fields</a></p> </li> </ul> </li> <li> <p>Types of quantum field thories</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory and elliptic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CFT">CFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</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> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a>, <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>, <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixing</a></p> </li> <li> <p>examples</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a>, <a class="existingWikiWord" href="/nlab/show/QED">QED</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/electric+charge">electric charge</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>, <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spinors+in+Yang-Mills+theory">spinors in Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a></li> <li><a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a></li> <li><a class="existingWikiWord" href="/nlab/show/RR+field">RR field</a></li> <li><a class="existingWikiWord" href="/nlab/show/first-order+formulation+of+gravity">first-order formulation of gravity</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+covariance">general covariance</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity+as+a+BF-theory">gravity as a BF-theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/particle">particle</a>, <a class="existingWikiWord" href="/nlab/show/relativistic+particle">relativistic particle</a>, <a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental particle</a>, <a class="existingWikiWord" href="/nlab/show/spinning+particle">spinning particle</a>, <a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string">string</a>, <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/membrane">membrane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AKSZ+theory">AKSZ theory</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+theory">String Theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+theory+results+applied+elsewhere">string theory results applied elsewhere</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/number+theory+and+physics">number theory and physics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Riemann+hypothesis+and+physics">Riemann hypothesis and physics</a></li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/physicscontents">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='#properties'>Properties</a></li> <ul> <li><a href='#from_dimensional_reduction'>From dimensional reduction</a></li> <li><a href='#from_double_and_exceptional_field_theory'>From double and exceptional field theory</a></li> </ul> <li><a href='#examples'>Examples</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>Theories of <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> in <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>l</mi><mo><</mo><mn>11</mn></mrow><annotation encoding="application/x-tex">l \lt 11</annotation></semantics></math> always contain a global <a class="existingWikiWord" href="/nlab/show/symmetry">symmetry</a> called <a class="existingWikiWord" href="/nlab/show/R-symmetry">R-symmetry</a> (a remnant of the full <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>10</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(10,1)</annotation></semantics></math>-symmetry after <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a>). In some cases this is promoted to a local symmetry, such that there is a <a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a> (<a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a bundle</a>) with coefficients in that group. These are called <em>gauged supergravity</em> theories.</p> <h2 id="properties">Properties</h2> <h3 id="from_dimensional_reduction">From dimensional reduction</h3> <p>In many cases, gauged supergravity theories are obtained by <a class="existingWikiWord" href="/nlab/show/KK-compactification">dimensional reduction</a> from <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a> or <a class="existingWikiWord" href="/nlab/show/type+II+supergravity">type II supergravity</a>, which themselves do not contain a gauge field but higher degree fields (“fluxes”), the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a> and the <a class="existingWikiWord" href="/nlab/show/B-field">B-field</a> respectively. These induce gauged supergravities (e.g. <a href="#Samtleben08">Samtleben 08, figure 1</a>). The <a class="existingWikiWord" href="/nlab/show/gauge+groups">gauge groups</a> are the <a class="existingWikiWord" href="/nlab/show/U-duality">U-duality</a> groups of the compactification (e.g. <a href="#Samtleben08">Samtleben 08, table 1</a>).</p> <h3 id="from_double_and_exceptional_field_theory">From double and exceptional field theory</h3> <p><a class="existingWikiWord" href="/nlab/show/Exceptional+field+theory">Exceptional field theory</a> (but also <a class="existingWikiWord" href="/nlab/show/double+field+theory">double field theory</a>) reduces to <a class="existingWikiWord" href="/nlab/show/gauged+supergravity">gauged supergravity</a> under <a class="existingWikiWord" href="/nlab/show/generalized+Scherk-Schwarz+reduction">generalized Scherk-Schwarz reduction</a>, i.e. a <a class="existingWikiWord" href="/nlab/show/KK-compactification">dimensional reduction</a> combined with <a class="existingWikiWord" href="/nlab/show/U-duality">U-duality</a> (or just <a class="existingWikiWord" href="/nlab/show/T-duality">T-duality</a>) twists.</p> <figure style="margin: 0 0 0 0"> <img src="/nlab/files/gaugedsugraexft.png" width="600px" /> <figcaption style="text-align: center">Picture grabbed from <a href="#Musaev13">Musaev 13</a></figcaption> </figure> <h2 id="examples">Examples</h2> <p>Some examples are discussed at</p> <ul> <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/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> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/tensor+hierarchy">tensor hierarchy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/embedding+tensor">embedding tensor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/double+field+theory">double field theory</a>, <a class="existingWikiWord" href="/nlab/show/exceptional+field+theory">exceptional field theory</a></p> </li> </ul> <h2 id="references">References</h2> <p>Review:</p> <ul> <li id="Roest04"> <p><a class="existingWikiWord" href="/nlab/show/Diederik+Roest">Diederik Roest</a>: <em>M-theory and Gauged Supergravities</em>, Fortsch. Phys. <strong>53</strong> (2005) 119-230 [<a href="http://arxiv.org/abs/hep-th/0408175">arXiv:hep-th/0408175</a>, <a href="https://doi.org/10.1002/prop.200410192">doi:10.1002/prop.200410192</a>]</p> </li> <li> <p>Martin Weidner, <em>Gauged Supergravities in Various Spacetime Dimensions</em>, Fortsch.Phys. <strong>55</strong> 843-945 (2007) [<a href="http://arxiv.org/abs/hep-th/0702084">arXiv:hep-th/0702084</a>]</p> </li> <li id="Samtleben08"> <p><a class="existingWikiWord" href="/nlab/show/Henning+Samtleben">Henning Samtleben</a>, <em>Lectures on Gauged Supergravity and Flux Compactifications</em> [<a href="http://arxiv.org/abs/0808.4076">arXiv:0808.4076</a>]</p> </li> <li> <p>Mario Trigiante, <em>Gauged Supergravities</em>, Physics Reports <strong>680</strong> (2017) 1-175 [<a href="https://arxiv.org/abs/1609.09745">arXiv:1609.09745</a>, <a href="https://doi.org/10.1016/j.physrep.2017.03.001">doi:10.1016/j.physrep.2017.03.001</a>]</p> </li> </ul> <p>Discussion of the origin in <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a> of <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a>/<a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a> is in</p> <ul> <li id="Roest04"><a class="existingWikiWord" href="/nlab/show/Diederik+Roest">Diederik Roest</a>, <em>M-theory and Gauged Supergravities</em>, Fortsch.Phys.53:119-230,2005 (<a href="http://arxiv.org/abs/hep-th/0408175">arXiv:hep-th/0408175</a>)</li> </ul> <p>Maximally gauged <a class="existingWikiWord" href="/nlab/show/D%3D4+supergravity">D=4 supergravity</a> was first discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Bernard+de+Wit">Bernard de Wit</a>, <a class="existingWikiWord" href="/nlab/show/Hermann+Nicolai">Hermann Nicolai</a>, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>8</mn></mrow><annotation encoding="application/x-tex">N = 8</annotation></semantics></math> supergravity</em>, Nucl. Phys. B <strong>208</strong> (1982) 323</li> </ul> <p>and gauged <a class="existingWikiWord" href="/nlab/show/D%3D5+supergravity">D=5 supergravity</a> in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Murat+G%C3%BCnaydin">Murat Günaydin</a>, <a class="existingWikiWord" href="/nlab/show/L.+J.+Romans">L. J. Romans</a> and <a class="existingWikiWord" href="/nlab/show/Nicholas+Warner">Nicholas Warner</a>, <em>Gauged <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>8</mn></mrow><annotation encoding="application/x-tex">N = 8</annotation></semantics></math> Supergravity in Five Dimensions</em>, Phys. Lett. 154B, (1985) 268 (<a href="http://inspirehep.net/record/207663">spire:207663</a>, <a href="https://doi.org/10.1016/0370-2693(85)90361-2">doi:10.1016/0370-2693(85)90361-2</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Murat+G%C3%BCnaydin">Murat Günaydin</a>, <a class="existingWikiWord" href="/nlab/show/L.+J.+Romans">L. J. Romans</a> and <a class="existingWikiWord" href="/nlab/show/Nicholas+Warner">Nicholas Warner</a>, <em>Compact and Non–Compact Gauged Supergravity Theories in Five Dimensions</em>, Nucl. Phys. B272 (1986) 598 (<a href="http://inspirehep.net/record/219727">spire:219727</a>, <a href="https://doi.org/10.1016/0550-3213(86)90237-3">doi:10.1016/0550-3213(86)90237-3</a>)</p> </li> <li> <p>M. Pernici, K. Pilch, <a class="existingWikiWord" href="/nlab/show/Peter+van+Nieuwenhuizen">Peter van Nieuwenhuizen</a>, <em>Gauged <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>8</mn></mrow><annotation encoding="application/x-tex">N=8</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>5</mn></mrow><annotation encoding="application/x-tex">D=5</annotation></semantics></math> Supergravity</em>, Nucl.Phys. B259 (1985) 460 (<a href="https://inspirehep.net/record/16067?ln=en">spire</a>)</p> </li> </ul> <p>Discussion in the context of <a class="existingWikiWord" href="/nlab/show/flux+compactification">flux compactification</a> of <a class="existingWikiWord" href="/nlab/show/type+II+superstring+theory">type II superstring theory</a> includes</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Mariana+Gra%C3%B1a">Mariana Graña</a>, section 5.3 of <em>Flux compactifications in string theory: a comprehensive review</em>, Phys.Rept. 423 (2006) 91-158 (<a href="http://arxiv.org/abs/hep-th/0509003">arXiv:hep-th/0509003</a>)</li> </ul> <p>Discussion in the context of the <a class="existingWikiWord" href="/nlab/show/D%27Auria-Fr%C3%A9+formulation+of+supergravity">D'Auria-Fré formulation of supergravity</a> is in</p> <ul> <li id="Fre05"><a class="existingWikiWord" href="/nlab/show/Pietro+Fr%C3%A9">Pietro Fré</a>, <em>M-theory FDA, Twisted Tori and Chevalley Cohomology</em>, Nucl.Phys.B742:86-123, 2006 (<a href="http://arxiv.org/abs/hep-th/0510068">arXiv:hep-th/0510068</a>)</li> </ul> <p>Discussion related to <a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a> singularities includes</p> <ul> <li>Richard Corrado, Murat Gunaydin, Nicholas P. Warner, Marco Zagermann, <em>Orbifolds and Flows from Gauged Supergravity</em>, Phys.Rev.D65:125024,2002 (<a href="http://arxiv.org/abs/hep-th/0203057">arXiv:hep-th/0203057</a>)</li> </ul> <p>Review of the relation with <a class="existingWikiWord" href="/nlab/show/double+field+theory">double field theory</a> and <a class="existingWikiWord" href="/nlab/show/exceptional+field+theory">exceptional field theory</a>:</p> <ul> <li id="Musaev13"><a class="existingWikiWord" href="/nlab/show/Edvard+Musaev">Edvard Musaev</a>, <em>U-dualities in Type II string theories and M-theory</em>, <em>Symmetry</em> 2019, <em>11</em>(8), 993 (<a href="https://arxiv.org/abs/arXiv:1311.3331">arXiv:1311.3331</a>, <a href="https://doi.org/10.3390/sym11080993">doi:10.3390/sym11080993</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on September 17, 2024 at 09:49:54. 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