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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="quantum_field_theory">Quantum field theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/FQFT">functorial quantum field theory</a></strong></p> <h2 id="contents">Contents</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+category">cobordism category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+cobordism">extended 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/bordism+categories+following+Stolz-Teichner">Riemannian bordism category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+tangle+hypothesis">generalized tangle hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/On+the+Classification+of+Topological+Field+Theories">classification of TQFTs</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functorial+field+theory">functorial field theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+functorial+field+theory">unitary functorial field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+functorial+field+theory">extended functorial field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+field+theory">CFT</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/vertex+operator+algebra">vertex operator algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Reshetikhin-Turaev+model">Reshetikhin-Turaev model</a> / <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/HQFT">HQFT</a></p> </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>, <a class="existingWikiWord" href="/nlab/show/Gromov-Witten+theory">Gromov-Witten theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p>FQFT and <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> <ul> <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</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+models+for+tmf">geometric models for tmf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holographic+principle+of+higher+category+theory">holographic principle of higher category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/holographic+principle">holographic principle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AdS%2FCFT+correspondence">AdS/CFT correspondence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantization+via+the+A-model">quantization via the A-model</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="phyics">Phyics</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> </div> </div> <blockquote> <p>This article is about the general concept of branes. For more specific details see also also at <em><a class="existingWikiWord" href="/nlab/show/super+p-brane">super p-brane</a></em>, <em><a class="existingWikiWord" href="/nlab/show/black+brane">black brane</a></em> and <em><a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a></em>.</p> </blockquote> <hr /> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <ul> <li><a href='#Dbranes'>Boundary conditions or D-branes</a></li> <ul> <li><a href='#in_terms_of_the_algebraic_data_of_the_qft_on_the_worldvolume'>In terms of the algebraic data of the QFT on the worldvolume</a></li> <ul> <li><a href='#in__rational_cft'>In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>d</mi></mrow><annotation encoding="application/x-tex">2d</annotation></semantics></math> rational CFT</a></li> <li><a href='#in__tft'>In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>d</mi></mrow><annotation encoding="application/x-tex">2d</annotation></semantics></math> TFT</a></li> </ul> <li><a href='#in_terms_of_geometric_data_of_the_model_background'>In terms of geometric data of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-model background</a></li> </ul> <li><a href='#fundamental_or_model_branes'>Fundamental or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-model branes</a></li> <li><a href='#the_brane_molecule'>The Brane molecule</a></li> <li><a href='#black_branes'>Black branes</a></li> <li><a href='#defect_branes'>Defect branes</a></li> <li><a href='#exotic_branes'>Exotic branes</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#worldvolume_theories'>Worldvolume theories</a></li> <li><a href='#the_superbrane_scan'>The super-brane scan</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#prehistory'>Prehistory</a></li> <li><a href='#general'>General</a></li> <li><a href='#boundary_conditions__dbranes'>Boundary conditions / D-branes</a></li> <li><a href='#fundamental_branes'>Fundamental branes</a></li> <li><a href='#branes_ending_on_branes'>Branes ending on branes</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>The term <em>brane</em> in formal high energy <a class="existingWikiWord" href="/nlab/show/physics">physics</a>, and in particular in <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a>, refers to entities that one thinks of as physical objects that generalize the notion <em>point particles</em> to higher dimensional objects.</p> <p>The term derives from the word <em>membrane</em> that was originally used to describe 2-dimensional “particles”. When the need was felt to speak also about 3-, 4- and higher dimensional such “particles” the usage “3-brane”, “4-brane” etc. was introduced. Ordinary particles would be 0-branes in this counting, the strings in <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> would be 1-branes and membranes themselves 2-branes.</p> <p>Generally, there are two different incarnations of branes</p> <ol> <li> <p><strong>fundamental <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-branes</strong> (as in “<a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental particle</a>”): these are given by <a class="existingWikiWord" href="/nlab/show/sigma-models">sigma-models</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p+1)</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/worldvolumes">worldvolumes</a> describing propagation of single <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-dimensional objects on certain <a class="existingWikiWord" href="/nlab/show/target+spacetimes">target spacetimes</a>. If these sigma-models are required to exhibit manifest <a class="existingWikiWord" href="/nlab/show/target+spacetime">target spacetime</a> <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a>, then they are <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+sigma+models">Green-Schwarz sigma models</a>, which are classified by a “<a class="existingWikiWord" href="/nlab/show/brane+scan">brane scan</a>” in <a class="existingWikiWord" href="/nlab/show/super+L-infinity+algebra">super L-infinity algebra</a> <a class="existingWikiWord" href="/nlab/show/L-infinity+algebra+cohomology">cohomology</a>;</p> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/black+branes">black p-branes</a></strong> (as in “<a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a>”): these are solutions to <a class="existingWikiWord" href="/nlab/show/field+theories">field theories</a>, typically <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> theories, with <a class="existingWikiWord" href="/nlab/show/singularities">singularities</a> of <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><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p+1)</annotation></semantics></math>. In analogy to how a <a class="existingWikiWord" href="/nlab/show/charged+black+hole">charged black hole</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">p = 0</annotation></semantics></math>) <a class="existingWikiWord" href="/nlab/show/source">sources</a> an <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a> with <a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a> 2-form, so black <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-branes source <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p+2)</annotation></semantics></math>-form <a class="existingWikiWord" href="/nlab/show/higher+gauge+fields">higher gauge fields</a> and hence appear in those <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> theories where such exists.</p> </li> </ol> <p>The idea is that these two concepts match where a <a class="existingWikiWord" href="/nlab/show/condensate">condensate</a> of fundamental <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-branes turns into a black <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-brane.</p> <p>Indeed, the classical <a class="existingWikiWord" href="/nlab/show/no-hair+theorem">no-hair theorem</a> matches <a class="existingWikiWord" href="/nlab/show/fundamental+particles">fundamental particles</a> (i.e. 0-branes) characterized (via the <a class="existingWikiWord" href="/nlab/show/Wigner+classification">Wigner classification</a>) by just their <a class="existingWikiWord" href="/nlab/show/mass">mass</a>, electromagnetic <a class="existingWikiWord" href="/nlab/show/charge">charge</a> and <a class="existingWikiWord" href="/nlab/show/angular+momentum">angular momentum</a> to <a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a> solutions of pure vacuum gravity. Accordingly, it is an old suggestion (<a href="#EinsteinInfeldHoffmann39">Einstein-Infeld-Hoffmann 39</a>) that fundamental particles could be identified with singular solutions of vacuum gravity.</p> <p>This matching generalizes to higher dimensional <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-branes in higher dimensional <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> and there is an exact correspondence between fundamental <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+super+p-branes">Green-Schwarz super p-branes</a> and extremal <a class="existingWikiWord" href="/nlab/show/BPS+solutions">BPS</a> black brane solutions.</p> <p>In <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> there is a third incarnation of branes, known as</p> <ul> <li><strong><a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a></strong>: these are the admissible <a class="existingWikiWord" href="/nlab/show/boundary+conditions">boundary conditions</a> for the 2-dimensional <a class="existingWikiWord" href="/nlab/show/sigma+models">sigma models</a> describing <a class="existingWikiWord" href="/nlab/show/open+strings">open strings</a>.</li> </ul> <p>One envisions that as one passes from <a class="existingWikiWord" href="/nlab/show/perturbative+string+theory">perturbative string theory</a> to the <a class="existingWikiWord" href="/nlab/show/non-perturbative+effect">non-perturbative</a> version of the theory (“<a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a>”) these <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> show <a class="existingWikiWord" href="/nlab/show/back-reaction">back-reaction</a> and turn into the <a class="existingWikiWord" href="/nlab/show/UV-completion">UV-completion</a> of the <a class="existingWikiWord" href="/nlab/show/black+branes">black branes</a> seen in the <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> <a class="existingWikiWord" href="/nlab/show/effective+field+theory">effective field theory</a>. This relation is key in the microscopic computation of <a class="existingWikiWord" href="/nlab/show/black+hole+entropy">black hole entropy</a> for <a class="existingWikiWord" href="/nlab/show/black+holes+in+string+theory">black holes in string theory</a>.</p> <h3 id="Dbranes">Boundary conditions or D-branes</h3> <p>Some words on <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a></p> <h4 id="in_terms_of_the_algebraic_data_of_the_qft_on_the_worldvolume">In terms of the algebraic data of the QFT on the worldvolume</h4> <p>An abstractly defined <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> is a consistent assignment of <a class="existingWikiWord" href="/nlab/show/state">state</a>-space and correlators to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/cobordisms">cobordisms</a> with certain structure (topological structure, conformal structure, Riemannian structure, etc. see <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a>/<a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a>). In an <em>open-closed QFT</em> the cobordisms are allowed to have boundaries. See at <em><a class="existingWikiWord" href="/nlab/show/boundary+field+theory">boundary field theory</a></em> for more on this.</p> <p>In this abstract formulation of QFT a <strong>brane</strong> is a type of data assigned by the QFT to boundaries of cobordisms.</p> <h5 id="in__rational_cft">In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>d</mi></mrow><annotation encoding="application/x-tex">2d</annotation></semantics></math> rational CFT</h5> <p>A well understood class of examples is this one: among all 2-dimensional <a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a>, the case of <em>full rational 2d CFT</em> has been understood completely, using <a class="existingWikiWord" href="/nlab/show/FFRS-formalism">FFRS-formalism</a>. It is then a theorem that full 2-rational CFTs are classified by</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/modular+tensor+category">modular tensor category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> (to be thought of as being the category of representations of the <a class="existingWikiWord" href="/nlab/show/vertex+operator+algebra">vertex operator algebra</a> of the 2d CFT);</p> </li> <li> <p>a special symmetric <a class="existingWikiWord" href="/nlab/show/Frobenius+algebra">Frobenius algebra</a> object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/internalization">internal</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>.</p> </li> </ol> <p>In this formulation a type of <strong>brane</strong> of the theory is precisely an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/module">module</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> (an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/bimodule">bimodule</a> is a <a class="existingWikiWord" href="/nlab/show/bi-brane">bi-brane</a> or <em>defect line</em> ):</p> <p>the 2d cobordisms with boundary on which the theory defined by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">A \in \mathcal{C}</annotation></semantics></math> carry as extra structure on their connected boundary pieces a label given by an equivalence class of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-module in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>. The assignment of the CFT to such a cobordism with boundary is obtained by</p> <ul> <li> <p>triangulating the cobordism,</p> </li> <li> <p>labeling all internal edges by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></p> </li> <li> <p>labelling all boundary pieces by the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-module</p> </li> <li> <p>all vertices where three internal edges meet by the multiplication operation</p> </li> <li> <p>and all points where an internal edge hits a moundary by the corresponding <a class="existingWikiWord" href="/nlab/show/action">action</a> morphism</p> </li> <li> <p>and finally evaluating the resulting <a class="existingWikiWord" href="/nlab/show/string+diagram">string diagram</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>.</p> </li> </ul> <p>So in this abstract algebraic formulation of QFT on the worldvolume, a brane is just the datum assigned by the QFT to the boundary of a cobordism. But abstractly defined QFTs may arise from <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> of <a class="existingWikiWord" href="/nlab/show/sigma+model">sigma model</a>s. This gives these boundary data a geometric interpretation in some space. This we discuss in the next section.</p> <h5 id="in__tft">In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>d</mi></mrow><annotation encoding="application/x-tex">2d</annotation></semantics></math> TFT</h5> <p>Another case where the branes of a QFT are under good mathematical control is <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a>: the <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category">(infinity,1)-category</a>-version of a 2d <a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a>.</p> <p>Particularly the <a class="existingWikiWord" href="/nlab/show/A-model">A-model</a> and the <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a> are well understood.</p> <ul> <li> <p>the branes of the B-model (“B-branes”) form the the <a class="existingWikiWord" href="/nlab/show/stable+%28infinity%2C1%29-category">stable (infinity,1)-category</a> of <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a>es of <a class="existingWikiWord" href="/nlab/show/quasicoherent+sheaves">quasicoherent sheaves</a> on the target space (often considered just in terms of its <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28infinity%2C1%29-category">homotopy category of an (infinity,1)-category</a>, the <a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a> of quasicoherent sheaves);</p> </li> <li> <p>the branes of the A-model form the <a class="existingWikiWord" href="/nlab/show/Fukaya+category">Fukaya category</a> of the target space.</p> </li> </ul> <p>(…)</p> <h4 id="in_terms_of_geometric_data_of_the_model_background">In terms of geometric data of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-model background</h4> <p>An abstractly defined QFT (as a consistent assignment of state spaces and propagators to cobordisms as in <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a>) may be obtained by <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> from <em>geometric data</em> :</p> <p>Such a <em><a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> QFT</em> is the <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> of an <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> on a space of maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\Sigma \to X</annotation></semantics></math> from a cobordism (“worldvolume”) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> to some target space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> that may carry further geometric data such as a <a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemannian metric</a>, or other background <a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a>s.</p> <p>One may therefore try to match the geometric data on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> that encodes the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-model with the algebraic data of the <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a> that results after quantization. This gives a geometric interpretation to many of the otherwise purely abstract algebraic properties of the worldvolume QFT.</p> <p>It turns out that if one checks which geometric data corresponds to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-modules in the above discussion, one finds that these tend to come from structures that look at least roughly like <em>submanifolds</em> of the target space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. And typically these submanifolds themselves carry their own background <a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a> data.</p> <p>A well-understood case is the <a class="existingWikiWord" href="/nlab/show/Wess-Zumino-Witten+model">Wess-Zumino-Witten model</a>: for this the target space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a simple <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">X = G</annotation></semantics></math> and the background field is a <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 2-bundle with connection</a> (a <a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe</a>) on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>, representing the background field that is known as the <a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a>.</p> <p>In this case it turns out that branes for the sigma model on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> are given in the simplest case by conjugacy classes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>⊂</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">D \subset G</annotation></semantics></math> inside the group, and that these carry a <a class="existingWikiWord" href="/nlab/show/twisted+bundle">twisted vector bundle</a> with the twist given by the Kalb-Ramond background bundle. These vector bundles are known in the <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> literature as <em>Chan-Paton vector bundles</em> . The geometric intuition is that a QFT with certain boundary condition comes from a quantization of spaces of maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">\Sigma \to G</annotation></semantics></math> that are restricted to take the boundary of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> to these submanifolds.</p> <p>More generally, one finds that the geometric data that corresponds to the branes in the algebraically defined 2d QFT is given by cocycles in the twisted <a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>. These may be quite far from having a direct interpretation as submanifolds of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>.</p> <p>The case of rational 2d CFT considered so far is only the best understood of a long sequence of other examples. Here the collection of all <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> – identified with the collection of all internal modules over an internal frobenius algebra, forms an ordinary <a class="existingWikiWord" href="/nlab/show/category">category</a>.</p> <p>More generally, at least for 2-dimensional <a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a>s analogous considerations yield not just categories but <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-categories">stable (∞,1)-categories</a> of boundary condition objects. For instance for what is called the <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a> 2-d <a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a> the category of <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> is the <a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a> of <a class="existingWikiWord" href="/nlab/show/coherent+sheaves">coherent sheaves</a> on some Calabi-Yau space.</p> <p>Starting with Kontsevich’s <a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a> reformulation of <a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">mirror symmetry</a> the study of (derived) D-brane categories has become a field in its own right in pure mathematics.</p> <p>… lots of further things to say …</p> <h3 id="fundamental_or_model_branes">Fundamental or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-model branes</h3> <p>In <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> one speaks apart from the <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> also about <strong>fundamental branes</strong> . These are the objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/sigma+model">sigma model</a> themselves.</p> <ul> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n=0</annotation></semantics></math> this describes the ordinary quantum mechanics of a point particles on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. And such point particles are the <em>fundamental particles</em> for instance of the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a>.</p> </li> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n=1</annotation></semantics></math> this describes the quantum propagation of a <a class="existingWikiWord" href="/nlab/show/string+theory">string</a>, and accordingly one speaks of the <em>fundamental string</em> or F1-brane (fundamental 1-brane).</p> </li> <li> <p>For <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> this describes the quantum propagation of a membrane.</p> </li> <li> <p>There are good indications that there is a way to describe heterotic <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> not in terms of fundamental 1-branes but in terms of the <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> of a fundamental 5-brane – the <a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic dual</a> of the 1-brane in 10-dimensions.</p> </li> <li> <p>etc.</p> </li> </ul> <div> <p>The <strong><a class="existingWikiWord" href="/nlab/show/brane">brane</a> scan</strong>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+action+functional">Green-Schwarz type</a> super <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/brane">brane</a> <a class="existingWikiWord" href="/nlab/show/sigma-models">sigma-models</a> (see at <em><a class="existingWikiWord" href="/nlab/show/table+of+branes">table of branes</a></em> for further links and see at <em><a class="existingWikiWord" href="/schreiber/show/The+brane+bouquet">The brane bouquet</a></em> for the full classification):</p> <table><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>=</mo><mi>d</mi></mover></mrow><annotation encoding="application/x-tex">\stackrel{d}{=}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">p =</annotation></semantics></math></th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th><th>9</th></tr></thead><tbody><tr><td style="text-align: left;">11</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M2-brane">M2</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M5-brane">M5</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">10</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D0-brane">D0</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/F1-brane">F1</a>, <a class="existingWikiWord" href="/nlab/show/D1-brane">D1</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D2-brane">D2</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D3-brane">D3</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D4-brane">D4</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/NS5-brane">NS5</a>, <a class="existingWikiWord" href="/nlab/show/D5-brane">D5</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D6-brane">D6</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D7-brane">D7</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D8-brane">D8</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D9-brane">D9</a></td></tr> <tr><td style="text-align: left;">9</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+4-brane+in+9d">*</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">8</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+3-brane+in+8d">*</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">7</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/topological+M2-brane">M2</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>top</mi></msub></mrow><annotation encoding="application/x-tex">{}_{top}</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">6</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/little+string">F1</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>little</mi></msub></mrow><annotation encoding="application/x-tex">{}_{little}</annotation></semantics></math>, <a class="existingWikiWord" href="/nlab/show/self-dual+string">S1</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>sd</mi></msub></mrow><annotation encoding="application/x-tex">{}_{sd}</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/3-brane+in+6d">S3</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">5</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+2-brane+in+5d">*</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">4</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+1-brane+in+4d">*</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+2-brane+in+4d">*</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">3</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+1-brane+in+3d">*</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> </tbody></table> <p>(The first columns follow the <a class="existingWikiWord" href="/nlab/show/exceptional+spinors+table">exceptional spinors table</a>.)</p> <p>The corresponding exceptional <a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebra">super L-∞ algebra</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">cocycles</a> (schematically, without prefactors):</p> <table><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>=</mo><mi>d</mi></mover></mrow><annotation encoding="application/x-tex">\stackrel{d}{=}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">p =</annotation></semantics></math></th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th><th>9</th></tr></thead><tbody><tr><td style="text-align: left;">11</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\Psi^2 E^2</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">sIso</a>(10,1)</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>5</mn></msup><mo>+</mo><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>2</mn></msup><msub><mi>C</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\Psi^2 E^5 + \Psi^2 E^2 C_3</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/m2brane">m2brane</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">10</td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\Psi^2 E^1</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">sIso</a>(9,1)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>B</mi> <mn>2</mn> <mn>2</mn></msubsup><mo>+</mo><msub><mi>B</mi> <mn>2</mn></msub><msup><mi>Ψ</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">B_2^2 + B_2 \Psi^2 + \Psi^2 E^2</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/type+II+supergravity+Lie+2-algebra">StringIIA</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>⋯</mi></mrow><annotation encoding="application/x-tex">\cdots</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/type+II+supergravity+Lie+2-algebra">StringIIB</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>B</mi> <mn>2</mn> <mn>3</mn></msubsup><mo>+</mo><msubsup><mi>B</mi> <mn>2</mn> <mn>2</mn></msubsup><msup><mi>Ψ</mi> <mn>2</mn></msup><mo>+</mo><msub><mi>B</mi> <mn>2</mn></msub><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">B_2^3 + B_2^2 \Psi^2 + B_2 \Psi^2 E^2 + \Psi^2 E^4</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/type+II+supergravity+Lie+2-algebra">StringIIA</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>5</mn></msup></mrow><annotation encoding="application/x-tex">\Psi^2 E^5</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">sIso</a>(9,1)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>B</mi> <mn>2</mn> <mn>4</mn></msubsup><mo>+</mo><mi>⋯</mi><mo>+</mo><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>6</mn></msup></mrow><annotation encoding="application/x-tex">B_2^4 + \cdots + \Psi^2 E^6</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/type+II+supergravity+Lie+2-algebra">StringIIA</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>⋯</mi></mrow><annotation encoding="application/x-tex">\cdots</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/type+II+supergravity+Lie+2-algebra">StringIIB</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>B</mi> <mn>2</mn> <mn>5</mn></msubsup><mo>+</mo><mi>⋯</mi><mo>+</mo><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>8</mn></msup></mrow><annotation encoding="application/x-tex">B_2^5 + \cdots + \Psi^2 E^8</annotation></semantics></math> in <a class="existingWikiWord" href="/nlab/show/type+II+supergravity+Lie+2-algebra">StringIIA</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>⋯</mi></mrow><annotation encoding="application/x-tex">\cdots</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/type+II+supergravity+Lie+2-algebra">StringIIB</a></td></tr> <tr><td style="text-align: left;">9</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\Psi^2 E^4</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">sIso</a>(8,1)</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">8</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\Psi^2 E^3</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">sIso</a>(7,1)</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">7</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\Psi^2 E^2</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">sIso</a>(6,1)</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">6</td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\Psi^2 E^1</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">sIso</a>(5,1)</td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\Psi^2 E^3</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">sIso</a>(5,1)</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">5</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\Psi^2 E^2</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">sIso</a>(4,1)</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">4</td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\Psi^2 E^1</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">sIso</a>(3,1)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\Psi^2 E^2</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">sIso</a>(3,1)</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">3</td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\Psi^2 E^1</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">sIso</a>(2,1)</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> </tbody></table> <h3 id="the_brane_molecule">The Brane molecule</h3> <p>Furthermore, there exists a more general classification of possible supermembranes in spacetime with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> spatial dimensions and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> time dimensions, appearing in (<a href="#BlencoweDuff88">Blencowe-Duff 88</a>). In this sense, the brane scan is but the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">T=1</annotation></semantics></math> branch of the brane molecule. The objects appearing here are expected to be related to other generalizations of string theory. See <a class="existingWikiWord" href="/nlab/show/D%3D12+supergravity">D=12 supergravity</a> and <a class="existingWikiWord" href="/nlab/show/bosonic+M-theory">bosonic M-theory</a>.</p> <p><img src="/nlab/files/branemoleculepc.png" alt="The brane molecule without assuming super Poincare invariance." /></p> <p>Compare:</p> <ul> <li id="BlencoweDuff88"> <p><a class="existingWikiWord" href="/nlab/show/Miles+Blencowe">Miles Blencowe</a>, <a class="existingWikiWord" href="/nlab/show/Mike+Duff">Mike Duff</a>, <em>Supermembranes and the Signature of Space-time</em>, Nucl. Phys. B310 (1988) 387-404 (<a href="inspirehep.net/record/262142">spire:262142</a>, <a href="https://doi.org/10.1016/0550-3213(88)90155-1">10.1016/0550-3213(88)90155-1</a>, <a href="http://inspirehep.net/record/262142/files/cer-000099708.pdf">pdf</a>)</p> </li> <li> <p><em><a class="existingWikiWord" href="/nlab/show/brane+bouquet">brane bouquet</a></em></p> </li> </ul> </div> <h3 id="black_branes">Black branes</h3> <p>See <em><a class="existingWikiWord" href="/nlab/show/black+brane">black brane</a></em>.</p> <h3 id="defect_branes">Defect branes</h3> <p>See <em><a class="existingWikiWord" href="/nlab/show/defect+brane">defect brane</a></em>.</p> <h3 id="exotic_branes">Exotic branes</h3> <p>See <em><a class="existingWikiWord" href="/nlab/show/exotic+brane">exotic brane</a></em>.</p> <h2 id="properties">Properties</h2> <h3 id="worldvolume_theories">Worldvolume theories</h3> <div> <p><strong>Table of <a class="existingWikiWord" href="/nlab/show/brane">branes</a> appearing in <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>/<a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a></strong> (for classification see at <em><a class="existingWikiWord" href="/nlab/show/brane+scan">brane scan</a></em>).</p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/brane">brane</a></th><th>in <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></th><th><a class="existingWikiWord" href="/nlab/show/charge">charge</a>d under <a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a></th><th>has <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> theory</th></tr></thead><tbody><tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/black+brane">black brane</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/higher+gauge+field">higher gauge field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SCFT">SCFT</a></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+II+supergravity">type II</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/RR-field">RR-field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">super Yang-Mills theory</a></td></tr> <tr><td style="text-align: left;"><strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>D</mi><mo>=</mo><mn>2</mn><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(D = 2n)</annotation></semantics></math></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+IIA+supergravity">type IIA</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D%28-2%29-brane">D(-2)-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D0-brane">D0-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/BFSS+matrix+model">BFSS matrix model</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D2-brane">D2-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D4-brane">D4-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D%3D5+super+Yang-Mills+theory">D=5 super Yang-Mills theory</a> with <a class="existingWikiWord" href="/nlab/show/Khovanov+homology">Khovanov homology</a> <a class="existingWikiWord" href="/nlab/show/observables">observables</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D6-brane">D6-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D%3D7+super+Yang-Mills+theory">D=7 super Yang-Mills theory</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D8-brane">D8-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>D</mi><mo>=</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(D = 2n+1)</annotation></semantics></math></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+IIB+supergravity">type IIB</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D%28-1%29-brane">D(-1)-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D1-brane">D1-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;">2d <a class="existingWikiWord" href="/nlab/show/CFT">CFT</a> with <a class="existingWikiWord" href="/nlab/show/BH+entropy">BH entropy</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D3-brane">D3-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/N%3D4+D%3D4+super+Yang-Mills+theory">N=4 D=4 super Yang-Mills theory</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D5-brane">D5-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D7-brane">D7-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D9-brane">D9-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28p%2Cq%29-string">(p,q)-string</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/D25-brane">D25-brane</a>)</td><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/bosonic+string+theory">bosonic string theory</a>)</td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/NS-brane">NS-brane</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/supergravity">type I, II, heterotic</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-connection</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/string">string</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/B2-field">B2-field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2d+SCFT">2d SCFT</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/NS5-brane">NS5-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/B6-field">B6-field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/little+string+theory">little string theory</a></td></tr> <tr><td style="text-align: left;"><strong>D-brane for <a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/A-brane">A-brane</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/B-brane">B-brane</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/M-brane">M-brane</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11D SuGra</a>/<a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-connection</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/C3-field">C3-field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/ABJM+theory">ABJM theory</a>, <a class="existingWikiWord" href="/nlab/show/BLG+model">BLG model</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/C6-field">C6-field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-superconformal+QFT">6d (2,0)-superconformal QFT</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M9-brane">M9-brane</a>/<a class="existingWikiWord" href="/nlab/show/O-plane">O9-plane</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M-wave">M-wave</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/topological+M2-brane">topological M2-brane</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/topological+M-theory">topological M-theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/C3-field">C3-field</a> on <a class="existingWikiWord" href="/nlab/show/G%E2%82%82-manifold">G₂-manifold</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/topological+M5-brane">topological M5-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/C6-field">C6-field</a> on <a class="existingWikiWord" href="/nlab/show/G%E2%82%82-manifold">G₂-manifold</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/S-brane">S-brane</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SM2-brane">SM2-brane</a>,<br /><a class="existingWikiWord" href="/nlab/show/membrane+instanton">membrane instanton</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M5-brane+instanton">M5-brane instanton</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D3-brane+instanton">D3-brane instanton</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/solitons">solitons</a></strong> on <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-superconformal+QFT">6d (2,0)-superconformal QFT</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/self-dual+string">self-dual string</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/self-dual+higher+gauge+field">self-dual</a> <a class="existingWikiWord" href="/nlab/show/B-field">B-field</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/3-brane+in+6d">3-brane in 6d</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> </tbody></table> </div> <h3 id="the_superbrane_scan">The super-brane scan</h3> <p>If the worldvolume QFT of the fundamental branes (for instance the worldsheet 2d<a class="existingWikiWord" href="/nlab/show/CFT">CFT</a> of the string) is required to be a <span class="newWikiWord">supersymmetric QFT<a href="/nlab/new/supersymmetric+QFT">?</a></span>, specifically if the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+action+functional">Green-Schwarz action functional</a> is used only particular combinations of the dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mi>Σ</mi><mo>=</mo><mi>p</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">dim \Sigma = p + 1</annotation></semantics></math> of the worldvolume and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><mi>dim</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">D = dim X</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> are possible.</p> <p>The corresponding table has been called the <strong><a class="existingWikiWord" href="/nlab/show/brane+scan">brane scan</a></strong></p> <p><img src="/nlab/files/branescan.gif" alt="The brane scan" /></p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/black+brane">black brane</a>, <a class="existingWikiWord" href="/nlab/show/solitonic+brane">solitonic brane</a></p> <p><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+sigma-model">fundamental brane</a></p> <p><a class="existingWikiWord" href="/nlab/show/probe+brane">probe brane</a>, <a class="existingWikiWord" href="/nlab/show/flavor+brane">flavor brane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/wrapped+brane">wrapped brane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/brane+intersection">brane intersection</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a>, <a class="existingWikiWord" href="/nlab/show/fractional+D-brane">fractional D-brane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/double+dimensional+reduction">double dimensional reduction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/particle">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/superstring">superstring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/membrane">membrane</a>, <a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a>, <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a>, <a class="existingWikiWord" href="/nlab/show/M-brane">M-brane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/NS5-brane">NS5-brane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/black+brane">black brane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/electric+eigenbrane">electric eigenbrane</a>. <a class="existingWikiWord" href="/nlab/show/magnetic+eigenbrane">magnetic eigenbrane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exotic+brane">exotic brane</a></p> </li> </ul> <div> <table><thead><tr><th>singularity</th><th><a class="existingWikiWord" href="/nlab/show/field+theory+with+singularities">field theory with singularities</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/boundary+condition">boundary condition</a>/<a class="existingWikiWord" href="/nlab/show/brane">brane</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/boundary+field+theory">boundary field theory</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/domain+wall">domain wall</a>/<a class="existingWikiWord" href="/nlab/show/bi-brane">bi-brane</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a></td></tr> </tbody></table> </div><div> <p><strong><a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Chern-Simons+theory">∞-Chern-Simons theory</a> from binary and non-degenerate <a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a></strong></p> <table><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0477fc87f3bab6c93955b29dd7c515f68e3108fd_1"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math></th><th><a class="existingWikiWord" href="/nlab/show/symplectic+Lie+n-algebroid">symplectic Lie n-algebroid</a></th><th><a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integrated</a> <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a> = <a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stack">moduli ∞-stack</a> of <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0477fc87f3bab6c93955b29dd7c515f68e3108fd_2"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-d <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></th><th><a class="existingWikiWord" href="/nlab/show/higher+symplectic+geometry">higher symplectic geometry</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0477fc87f3bab6c93955b29dd7c515f68e3108fd_3"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>d <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></th><th><a class="existingWikiWord" href="/nlab/show/dg-manifold">dg-</a><a class="existingWikiWord" href="/nlab/show/Lagrangian+submanifold">Lagrangian submanifold</a>/ <a class="existingWikiWord" href="/nlab/show/real+polarization">real polarization</a> <a class="existingWikiWord" href="/nlab/show/leaf">leaf</a></th><th>= <a class="existingWikiWord" href="/nlab/show/brane">brane</a></th><th><a class="existingWikiWord" href="/nlab/show/n-module">(n+1)-module</a> of <a class="existingWikiWord" href="/nlab/show/quantum+states">quantum states</a> in <a class="existingWikiWord" href="/nlab/show/codimension">codimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0477fc87f3bab6c93955b29dd7c515f68e3108fd_4"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math></th><th>discussed in:</th></tr></thead><tbody><tr><td style="text-align: left;">0</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lagrangian+submanifold">Lagrangian submanifold</a></td><td style="text-align: left;">–</td><td style="text-align: left;">ordinary <a class="existingWikiWord" href="/nlab/show/space+of+states+%28in+geometric+quantization%29">space of states (in geometric quantization)</a></td><td style="text-align: left;"><em><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a></em></td></tr> <tr><td style="text-align: left;">1</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroid">Poisson Lie algebroid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-plectic+geometry">2-plectic geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Poisson+sigma-model">Poisson sigma-model</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coisotropic+submanifold">coisotropic submanifold</a> (of underlying <a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a>)</td><td style="text-align: left;"><a href="Poisson+sigma-model#Branes">brane of Poisson sigma-model</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-module">2-module</a> = <a class="existingWikiWord" href="/nlab/show/category+of+modules">category of modules</a> over <a class="existingWikiWord" href="/nlab/show/strict+deformation+quantization">strict deformation quantiized</a> <a class="existingWikiWord" href="/nlab/show/algebra+of+observables">algebra of observables</a></td><td style="text-align: left;"><em><a class="existingWikiWord" href="/nlab/show/extended+geometric+quantization+of+2d+Chern-Simons+theory">extended geometric quantization of 2d Chern-Simons theory</a></em></td></tr> <tr><td style="text-align: left;">2</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Courant+Lie+2-algebroid">Courant Lie 2-algebroid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+2-groupoid">symplectic 2-groupoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/3-plectic+geometry">3-plectic geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Courant+sigma-model">Courant sigma-model</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Dirac+structure">Dirac structure</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a> in <a class="existingWikiWord" href="/nlab/show/type+II+geometry">type II geometry</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0477fc87f3bab6c93955b29dd7c515f68e3108fd_5"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+Lie+n-algebroid">symplectic Lie n-algebroid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+infinity-groupoid">symplectic n-groupoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/n-plectic+geometry">(n+1)-plectic geometry</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0477fc87f3bab6c93955b29dd7c515f68e3108fd_6"><semantics><mrow><mi>d</mi><mo>=</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">d = n+1</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/AKSZ+sigma-model">AKSZ sigma-model</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> </tbody></table> <p>(adapted from <a class="existingWikiWord" href="/nlab/show/Some+title+containing+the+words+%22homotopy%22+and+%22symplectic%22%2C+e.g.+this+one">Ševera 00</a>)</p></div> <h2 id="references">References</h2> <h3 id="prehistory">Prehistory</h3> <ul> <li id="EinsteinInfeldHoffmann39"><a class="existingWikiWord" href="/nlab/show/Albert+Einstein">Albert Einstein</a>, <a class="existingWikiWord" href="/nlab/show/Leopold+Infeld">Leopold Infeld</a>, B. Hoffmann, <em>The gravitational equations and the problem of motion</em>, Annals of Mathematics, Vol 39, No. 1, 1938</li> </ul> <p>See also</p> <ul> <li>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Black_hole_electron">Black hole electron</a></em></li> </ul> <p>The terminology “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-brane” originates in</p> <ul> <li id="DuffInamiPopeSezginStelle88"><a class="existingWikiWord" href="/nlab/show/Mike+Duff">Mike Duff</a>, T. Inami, <a class="existingWikiWord" href="/nlab/show/Christopher+Pope">Christopher Pope</a>, <a class="existingWikiWord" href="/nlab/show/Ergin+Sezgin">Ergin Sezgin</a>, <a class="existingWikiWord" href="/nlab/show/Kellogg+Stelle">Kellogg Stelle</a>, <em>Semiclassical Quantization of the Supermembrane</em>, Nucl.Phys. B297 (1988) 515-538 (<a href="http://inspirehep.net/record/247064">spire:247064</a>)</li> </ul> <p>For more history see</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Mike+Duff">Mike Duff</a>, <em>Thirty years of Erice on the brane</em>, Based on lectures at the International Schools of Subnuclear Physics 1987-2017 and the International Symposium “60 Years of Subnuclear Physics at Bologna”, University of Bologna, November 2018 (<a href="https://arxiv.org/abs/1812.11658">arXiv:1812.11658</a>)</li> </ul> <h3 id="general">General</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Pietro+Fr%C3%A9">Pietro Fré</a>, <em>The Branes: Three Viewpoints</em>, Chater 7 in: <em>Gravity, a Geometrical Course</em>, Springer (2013) [<a href="http://inspirehep.net/record/1242195">spire:1242195</a>, <a href="https://doi.org/10.1007/978-94-007-5443-0_7">doi:10.1007/978-94-007-5443-0_7</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Greg+Moore">Greg Moore</a>, <em>What is… a brane?</em>, Notices of the AMS <strong>52</strong> 2 (2005) [<a href="http://www.ams.org/notices/200502/what-is.pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Joan+Sim%C3%B3n">Joan Simón</a>, <em>Brane Effective Actions, Kappa-Symmetry and Applications</em> [<a href="http://arxiv.org/abs/1110.2422">arXiv:1110.2422</a>]</p> </li> <li id="IbanezUranga12"> <p><a class="existingWikiWord" href="/nlab/show/Luis+Ib%C3%A1%C3%B1ez">Luis Ibáñez</a>, <a class="existingWikiWord" href="/nlab/show/Angel+Uranga">Angel Uranga</a>, section 6 of: <em><a class="existingWikiWord" href="/nlab/show/String+Theory+and+Particle+Physics+--+An+Introduction+to+String+Phenomenology">String Theory and Particle Physics – An Introduction to String Phenomenology</a></em>, Cambridge University Press (2012)</p> </li> <li> <p>J. X. Lu: <em>Branes in String/M-Theory</em> [<a href="https://arxiv.org/abs/2502.11575">arXiv:2502.11575</a>]</p> </li> </ul> <h3 id="boundary_conditions__dbranes">Boundary conditions / D-branes</h3> <p>(…)</p> <p>See <a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a>.</p> <p>For exhaustive details on D-branes in 2-dimensional rational <a class="existingWikiWord" href="/nlab/show/CFT">CFT</a> see the references given at</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/FFRS-formalism">FFRS-formalism</a>.</li> </ul> <p>A classical text describing how the physics way to think of D-branes for the <a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a> leads to seeing that they are objects in <a class="existingWikiWord" href="/nlab/show/derived+categories">derived categories</a> (of <a class="existingWikiWord" href="/nlab/show/coherent+sheaves">coherent sheaves</a> for the <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a>) is reviewed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Paul+Aspinwall">Paul Aspinwall</a>, <em>D-Branes on Calabi-Yau Manifolds</em> (<a href="http://arxiv.org/abs/hep-th/0403166">arXiv</a>)</li> </ul> <p>based on</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael+Douglas">Michael Douglas</a>, <em>D-branes, Categories and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">N=1</annotation></semantics></math> Supersymmetry</em>, J.Math.Phys. 42 (2001) 2818-2843 (<a href="https://arxiv.org/abs/hep-th/0011017">arXiv:hep-th/0011017</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Paul+Aspinwall">Paul Aspinwall</a>, Albion Lawrence, <em>Derived Categories and Zero-Brane Stability</em> (<a href="https://arxiv.org/abs/hep-th/0104147">arXiv:hep-th/0104147</a>)</p> </li> </ul> <p>This can to a large extent be read as a dictionary from <a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a> terminology to that of D-brane physics.</p> <p>More recent similar material with the emphasis on the <a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a> aspects is</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Richard+Szabo">Richard Szabo</a>, <em><a class="existingWikiWord" href="/nlab/files/Szabo09.pdf" title="D-Branes and Bivariant K-Theory">D-Branes and Bivariant K-Theory</a></em></li> </ul> <h3 id="fundamental_branes">Fundamental branes</h3> <p>The “<a class="existingWikiWord" href="/nlab/show/brane+scan">brane scan</a>” table showing the consistent dimension pairs for the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+action+functional">Green-Schwarz action functional</a> was depicted in</p> <ul id="Duff"> <li><a class="existingWikiWord" href="/nlab/show/Michael+Duff">Michael Duff</a>, <em>Supermembranes: the first fifteen weeks</em> CERN-TH.4797/87 (<a href="https://iopscience.iop.org/article/10.1088/0264-9381/5/1/023/pdf">scan</a>)</li> </ul> <p>going back to</p> <ul> <li>A. Achúcarro, J. M. Evans, <a class="existingWikiWord" href="/nlab/show/Paul+Townsend">Paul Townsend</a> and D. L. Wiltshire, <em>Super <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-branes</em> Physics Letters B Volume 198, Issue 4, 3 (1987)</li> </ul> <p>Further developments are in</p> <p>More along these lines is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Michael+Duff">Michael Duff</a>, S. Ferrara, <em>Four curious supergravities</em> (<a href="http://arxiv.org/abs/1010.3173">arXiv</a>)</li> </ul> <p>See also <a class="existingWikiWord" href="/nlab/show/division+algebras+and+supersymmetry">division algebras and supersymmetry</a>.</p> <h3 id="branes_ending_on_branes">Branes ending on branes</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/Andrew+Strominger">Andrew Strominger</a>, <em>Open P-Branes</em>, Phys. Lett. B383:44-47,1996 (<a href="http://arxiv.org/abs/hep-th/9512059">arXiv:hep-th/9512059</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on February 18, 2025 at 06:21:11. 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