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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="supergeometry">Super-Geometry</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a> and (<a class="existingWikiWord" href="/nlab/show/synthetic+differential+supergeometry">synthetic</a> ) <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a> <h2 id="background">Background</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/graded+object">graded object</a> </li> </ul> <h2 id="introductions">Introductions</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+superalgebra">geometry of physics – superalgebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+supergeometry">geometry of physics – supergeometry</a> </li> </ul> <h2 id="superalgebra">Superalgebra</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/super+commutative+monoid">super commutative monoid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+abelian+group">super abelian group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+ring">super ring</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/supercommutative+ring">supercommutative ring</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/exterior+ring">exterior ring</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Clifford+ring">Clifford ring</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+module">super module</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+vector+space">super vector space</a>, <a class="existingWikiWord" href="/nlab/show/SVect">SVect</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+algebra">super algebra</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/supercommutative+algebra">supercommutative algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/exterior+algebra">exterior algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Clifford+algebra">Clifford algebra</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/superdeterminant">superdeterminant</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">super Poincare Lie algebra</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/chain+complex+of+super+vector+spaces">chain complex of super vector spaces</a> (<a class="existingWikiWord" href="/nlab/show/model+structure+on+chain+complexes+of+super+vector+spaces">model structure</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/differential+graded-commutative+superalgebra">differential graded-commutative superalgebra</a> (<a class="existingWikiWord" href="/nlab/show/model+structure+on+differential+graded-commutative+superalgebras">model structure</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/super+L-infinity+algebra">super L-infinity algebra</a> </li> </ul> <h2 id="supergeometry">Supergeometry</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/superpoint">superpoint</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+Cartesian+space">super Cartesian space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/supermanifold">supermanifold</a>, <a class="existingWikiWord" href="/nlab/show/SDiff">SDiff</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/NQ-supermanifold">NQ-supermanifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+vector+bundle">super vector bundle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/complex+supermanifold">complex supermanifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Euclidean+supermanifold">Euclidean supermanifold</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+spacetime">super spacetime</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Minkowski+spacetime">super Minkowski spacetime</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/integration+over+supermanifolds">integration over supermanifolds</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Berezin+integral">Berezin integral</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+Lie+group">super Lie group</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/super+Lie+group">super translation group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+Euclidean+group">super Euclidean group</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+%E2%88%9E-groupoid">super ∞-groupoid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+formal+smooth+%E2%88%9E-groupoid">super formal smooth ∞-groupoid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+line+2-bundle">super line 2-bundle</a> </li> </ul> <h2 id="supersymmetry">Supersymmetry</h2> <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/division+algebra+and+supersymmetry">division algebra and supersymmetry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">super Poincare Lie algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/supermultiplet">supermultiplet</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/BPS+state">BPS state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/M-theory+super+Lie+algebra">M-theory super Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/type+II+super+Lie+algebra">type II super Lie algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a>, <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a> </li> </ul> <h2 id="supersymmetric_field_theory">Supersymmetric field theory</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/superfield">superfield</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/supersymmetric+quantum+mechanics">supersymmetric quantum mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/adinkra">adinkra</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">super Yang-Mills theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/gauged+supergravity">gauged supergravity</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/superstring+theory">superstring theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a> </li> </ul> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometric+model+for+elliptic+cohomology">geometric model for elliptic cohomology</a></li> </ul> <div> <a href="/nlab/edit/supergeometry+-+contents">Edit this sidebar</a> </div></div></div> <h4 id="physics">Physics</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/physics">physics</a>, <a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a>, <a class="existingWikiWord" href="/nlab/show/philosophy+of+physics">philosophy of physics</a> <h2 id="surveys_textbooks_and_lecture_notes">Surveys, textbooks and lecture notes</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">(higher) category theory and physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a> </li> <li> <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> </li> </ul> <hr /> <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory (physics)</a>, <a class="existingWikiWord" href="/nlab/show/model+%28physics%29">model (physics)</a> <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> <ul> <li> <a class="existingWikiWord" href="/nlab/show/mechanics">mechanics</a> <ul> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/dynamics+on+Lie+groups">dynamics on Lie groups</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/rigid+body+dynamics">rigid body dynamics</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Lagrangian+mechanics">Lagrangian mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a>, <a class="existingWikiWord" href="/nlab/show/state">state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a> </li> <li> <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> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-symplectic+manifold">n-symplectic manifold</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/smooth+Lorentzian+manifold">smooth Lorentzian manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/special+relativity">special relativity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/dilaton+gravity">dilaton gravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a> </li> </ul> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/classical+field+theory">Classical field theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/classical+physics">classical physics</a> <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> <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">Quantum Mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+mechanics+in+terms+of+dagger-compact+categories">in terms of ∞-compact categories</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hamiltonian+operator">Hamiltonian operator</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantization">Quantization</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/path+integral">path integral quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/semiclassical+approximation">semiclassical approximation</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">Quantum Field Theory</a> <ul> <li> Axiomatizations <ul> <li> <a class="existingWikiWord" href="/nlab/show/AQFT">algebraic QFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+net">local net</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a> <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> <a class="existingWikiWord" href="/nlab/show/FQFT">functorial QFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem </li> <li> <a class="existingWikiWord" href="/nlab/show/extended+topological+quantum+field+theory">extended topological quantum field theory</a> </li> </ul> </li> </ul> </li> <li> Tools <ul> <li> <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-function+theory">geometric ∞-function theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+%28in+particle+phyiscs%29">models</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/fields+and+quanta+-+table">fields and quanta</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/GUT">Grand Unified Theories</a>, <a class="existingWikiWord" href="/nlab/show/MSSM">MSSM</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a> <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> Structural phenomena <ul> <li> <a class="existingWikiWord" href="/nlab/show/universality+class">universality class</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/instanton">instanton</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spontaneously+broken+symmetry">spontaneously broken symmetry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein mechanism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/integrable+systems">integrable systems</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/holonomic+quantum+fields">holonomic quantum fields</a> </li> </ul> </li> <li> Types of quantum field thories <ul> <li> <a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a> <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> <li> <a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory and elliptic cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/CFT">CFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a> </li> <li> <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> </li> <li> examples <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> <a class="existingWikiWord" href="/nlab/show/electric+charge">electric charge</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a> <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> <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spinors+in+Yang-Mills+theory">spinors in Yang-Mills theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a> <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> <a class="existingWikiWord" href="/nlab/show/general+covariance">general covariance</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gravity+as+a+BF-theory">gravity as a BF-theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> <ul> <li> <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> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/membrane">membrane</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/AKSZ+theory">AKSZ theory</a> </li> </ul> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/string+theory">String Theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+theory+results+applied+elsewhere">string theory results applied elsewhere</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/number+theory+and+physics">number theory and physics</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Riemann+hypothesis+and+physics">Riemann hypothesis and physics</a></li> </ul> </li> </ul> <div> <a href="/nlab/edit/physicscontents">Edit this sidebar</a> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <ul> <li><a href='#General'>General</a></li> <li><a href='#in_supergravity'>In supergravity</a></li> <li><a href='#in_superstring_theory'>In superstring theory</a></li> </ul> <li><a href='#InTermsOfHigherDifferentialGeometry'>Formalization in higher differential geometry</a></li> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#in_11d_supergravity'>In 11d Supergravity</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general_2'>General</a></li> <li><a href='#IntroductoryReferences'>Introductions, surveys and lectures</a></li> <li><a href='#in_supergravity_2'>In supergravity</a></li> <li><a href='#SpectralNetworksReferences'>Spectral networks</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <h3 id="General">General</h3> In <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetric</a> <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> with <a class="existingWikiWord" href="/nlab/show/extended+supersymmetry">extended supersymmetry</a>, certain extremal <a class="existingWikiWord" href="/nlab/show/supermultiplets">supermultiplets</a> have some of the <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetries</a> retained (have 0-<a class="existingWikiWord" href="/nlab/show/eigenvalue">eigenvalue</a> under some of the <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a> generators). These are called Bogomol’nyi–Prasad–Sommerfield saturated solutions. More in detail, where in a plain <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a> <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a> a suitable <a class="existingWikiWord" href="/nlab/show/basis">basis</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>Q</mi> <mi>A</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{Q_A\}</annotation></semantics></math> of supersymmetry generators has odd bracket proportional to the spacetime translation and hence to an <a class="existingWikiWord" href="/nlab/show/energy">energy</a>/<a class="existingWikiWord" href="/nlab/show/mass">mass</a> operator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> (with terminology as at <a class="existingWikiWord" href="/nlab/show/unitary+representation+of+the+Poincar%C3%A9+group">unitary representation of the Poincaré group</a>) <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>Q</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>Q</mi> <mi>B</mi></msub><mo stretchy="false">}</mo><mo>=</mo><mi>E</mi><msub><mi>δ</mi> <mrow><mi>A</mi><mi>B</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> \{Q_A, Q_B\} = E \delta_{A B} </annotation></semantics></math></div> for <a class="existingWikiWord" href="/nlab/show/extended+supersymmetry">extended supersymmetry</a> there are further bosonic super Lie algebra generators <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mrow><mi>A</mi><mi>B</mi></mrow></msub></mrow><annotation encoding="application/x-tex">K_{A B}</annotation></semantics></math> (charges) such that <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>Q</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>Q</mi> <mi>B</mi></msub><mo stretchy="false">}</mo><mo>=</mo><mi>E</mi><msub><mi>δ</mi> <mrow><mi>A</mi><mi>B</mi></mrow></msub><mo>−</mo><msub><mi>K</mi> <mrow><mi>A</mi><mi>B</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \{Q_A, Q_B\} = E \delta_{A B} - K_{A B} \,. </annotation></semantics></math></div> It follows from the supersymmetry algebra that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>E</mi><msub><mi>δ</mi> <mrow><mi>A</mi><mi>B</mi></mrow></msub><mo>−</mo><msub><mi>K</mi> <mrow><mi>A</mi><mi>B</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E \delta_{A B} - K_{A B})</annotation></semantics></math> is a positive definite <a class="existingWikiWord" href="/nlab/show/bilinear+form">bilinear form</a>, which puts a lower bound on the <a class="existingWikiWord" href="/nlab/show/energy">energy</a> given the values of these extra charges. This is called the BPS bound. See also at <a class="existingWikiWord" href="/nlab/show/Bridgeland+stability+condition">Bridgeland stability condition</a>. In particular when this bound is satisfied in that some of the <a class="existingWikiWord" href="/nlab/show/eigenvalues">eigenvalues</a> of the <a class="existingWikiWord" href="/nlab/show/matrix">matrix</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>K</mi> <mrow><mi>A</mi><mi>B</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(K_{A B})</annotation></semantics></math> are actually equal to the energy/mass, then the corresponding component of the right hand side in the above equation vanishes and hence then the corresponding supersymmetry generators may annihilate the given state, then called a BPS state. This way enhanced supersymmetry of states goes along with certain charges taken extremal values. States with similar behaviour are also considered also in some models of <a class="existingWikiWord" href="/nlab/show/soliton">soliton</a> theory (English Wikipedia: <a href="http://en.wikipedia.org/wiki/Bogomol'nyi%E2%80%93Prasad%E2%80%93Sommerfield_bound">Bogomol’nyi–Prasad–Sommerfield bound</a>). BPS states play a central role in the investigation of <a class="existingWikiWord" href="/nlab/show/moduli+spaces">moduli spaces</a> of classical <a class="existingWikiWord" href="/nlab/show/vacua">vacua</a> as they form part of the moduli problem which is often the most tractable. Several mathematical theories in <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a> are interpreted as counting BPS-states in the sense of integration on appropriate compactification of the <a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a> of BPS-states in a related physical model attached to the underlying geometry: most notably the <a class="existingWikiWord" href="/nlab/show/Gromov-Witten+invariants">Gromov-Witten invariants</a>, <a class="existingWikiWord" href="/nlab/show/Donaldson-Thomas+invariants">Donaldson-Thomas invariants</a> and the <a class="existingWikiWord" href="/nlab/show/Thomas-Pandharipande+invariants">Thomas-Pandharipande invariants</a>; all the three seem to be deeply interrelated though they are defined in rather very different terms. The compactification of the moduli space involves various <a class="existingWikiWord" href="/nlab/show/stability+conditions">stability conditions</a>. <h3 id="in_supergravity">In supergravity</h3> In the context of <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> BPS states correspond to <a class="existingWikiWord" href="/nlab/show/super+spacetimes">super spacetimes</a> admitting <a class="existingWikiWord" href="/nlab/show/Killing+vectors">Killing vectors</a>. These notably include extremal <a class="existingWikiWord" href="/nlab/show/black+brane">black brane</a> solutions. <h3 id="in_superstring_theory">In superstring theory</h3> Specifically in <a class="existingWikiWord" href="/nlab/show/superstring+theory">superstring theory</a> BPS states in <a class="existingWikiWord" href="/nlab/show/target+space">target space</a> correspond to string states on the worldsheet which are annihilated by the left-moving (say) half of the <a class="existingWikiWord" href="/nlab/show/Dirac-Ramond+operator">Dirac-Ramond operator</a>. These are counted by the <a class="existingWikiWord" href="/nlab/show/Witten+genus">Witten genus</a>, see at <a href="Witten+genus#RelationToBPSStateCounting">Witten genus – Relation to BPS states</a>. The degeneracy of BPS states in string theory has been used to provide a microscopic interpretation of <a class="existingWikiWord" href="/nlab/show/Bekenstein-Hawking+entropy">Bekenstein-Hawking entropy</a> of <a class="existingWikiWord" href="/nlab/show/black+holes">black holes</a>, see at <a class="existingWikiWord" href="/nlab/show/black+holes+in+string+theory">black holes in string theory</a>. <h2 id="InTermsOfHigherDifferentialGeometry">Formalization in higher differential geometry</h2> <blockquote> The following are some observations on the formalization of BPS states from the <a class="existingWikiWord" href="/nlab/show/nPOV">nPOV</a>, in <a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher differential geometry</a>, following (<a href="#SatiSchreiber15">Sati & Schreiber 2015</a>). </blockquote> Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mi>N</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d-1,1|N}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/super-Minkowski+spacetime">super-Minkowski spacetime</a>, let <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><mi>N</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(d,N,p)</annotation></semantics></math> be in the <a class="existingWikiWord" href="/nlab/show/brane+scan">brane scan</a> and write <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>≔</mo><mover><mi>ψ</mi><mo stretchy="false">¯</mo></mover><mo>∧</mo><msup><mi>E</mi> <mrow><mo>∧</mo><mi>p</mi></mrow></msup><mo>∧</mo><mi>ψ</mi><mo>∈</mo><msup><mi>Ω</mi> <mrow><mi>p</mi><mo>+</mo><mn>2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mi>N</mi></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \phi \coloneqq \bar{\psi} \wedge E^{\wedge p} \wedge \psi \in \Omega^{p+2}(\mathbb{R}^{d-1,1|N}) </annotation></semantics></math></div> for the correspoding <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a> <a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">cocycle</a>, as discussed at <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+action+functional">Green-Schwarz action functional</a>, see (<a href="#FSS13">FSS 13</a>) for the perspective invoked here. Consider then <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> a <a class="existingWikiWord" href="/nlab/show/super-spacetime">super-spacetime</a> locally modeled on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mi>N</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d-1,1|N}</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a>, solving the relevant <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> <a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a> (e.g. <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>11</mn></mrow><annotation encoding="application/x-tex">d= 11</annotation></semantics></math>, <a class="existingWikiWord" href="/nlab/show/heterotic+supergravity">heterotic supergravity</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">d = 10</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N = (1,0)</annotation></semantics></math>, <a class="existingWikiWord" href="/nlab/show/type+IIA+supergravity">type IIA supergravity</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">d = 10</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N= (1,1)</annotation></semantics></math> or <a class="existingWikiWord" href="/nlab/show/type+IIB+supergravity">type IIB supergravity</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">d = 10</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N= (2,0)</annotation></semantics></math>). This means in particular that <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> carries a <a class="existingWikiWord" href="/nlab/show/super+differential+form">super differential form</a> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>∈</mo><msup><mi>Ω</mi> <mrow><mi>p</mi><mo>+</mo><mn>2</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \omega \in \Omega^{p+2}(X) </annotation></semantics></math></div> which is <a class="existingWikiWord" href="/nlab/show/definite+form">definite</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math>. This is the <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> of the <a class="existingWikiWord" href="/nlab/show/WZW-term">WZW-term</a> which defines the relevant <a class="existingWikiWord" href="/nlab/show/super+p-brane+sigma-model">super p-brane sigma-model</a> with <a class="existingWikiWord" href="/nlab/show/target+space">target space</a> <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>. By (<a href="#AGIT89">AGIT 89</a>) <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 BPS state to the extent that it carries <a class="existingWikiWord" href="/nlab/show/Killing+spinors">Killing spinors</a> which form a central <a class="existingWikiWord" href="/nlab/show/Lie+algebra+extension">Lie algebra extension</a> of a sub-algebra of the <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a> algebra (i.e. of the <a class="existingWikiWord" href="/nlab/show/super+translation+Lie+algebra">super translation Lie algebra</a>) by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>dR</mi> <mi>p</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^p_{dR}(X)</annotation></semantics></math> which is classified by the <a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">cocycle</a> given by <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>ϵ</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>ϵ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>↦</mo><mi>ω</mi><mo stretchy="false">(</mo><msub><mi>ϵ</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>ϵ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>Ω</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mi>cl</mi></msub><mo stretchy="false">/</mo><mi>im</mi><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>d</mi></mstyle> <mi>dR</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\epsilon_1, \epsilon_2) \mapsto \omega(\epsilon_1,\epsilon_2) \in \Omega^p(X)_{cl}/im(\mathbf{d}_{dR}) \,. </annotation></semantics></math></div> Now we observe that by (<a href="#hgpII">hgpII, theorem 3.3.1</a>) this is precisely the <a class="existingWikiWord" href="/nlab/show/0-truncation">0-truncation</a> of the <a class="existingWikiWord" href="/nlab/show/super+L-infinity+algebra">super</a>-<a class="existingWikiWord" href="/nlab/show/Poisson+bracket+Lie+n-algebra">Poisson bracket Lie n-algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔓𝔬𝔦𝔰</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{Pois}(X,\omega)</annotation></semantics></math> induced by regarding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\omega)</annotation></semantics></math> as an <a class="existingWikiWord" href="/nlab/show/pre-n-plectic+manifold">pre-n-plectic</a> <a class="existingWikiWord" href="/nlab/show/supermanifold">supermanifold</a> and restricting along the inclusion of the <a class="existingWikiWord" href="/nlab/show/Killing+vectors">Killing vectors</a>/<a class="existingWikiWord" href="/nlab/show/Killing+spinors">Killing spinors</a> into all the <a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+fields">Hamiltonian vector fields</a>. <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>dR</mi> <mi>p</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>τ</mi> <mn>0</mn></msub><mi>𝔓𝔬𝔦𝔰</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>Vect</mi> <mi>Ham</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H^p_{dR}(X) \to \tau_0 \mathfrak{Pois}(X,\omega) \to Vect_{Ham}(X) </annotation></semantics></math></div> (Here we are using that if an <a class="existingWikiWord" href="/nlab/show/n-type">n-type</a> is an extension of a <a class="existingWikiWord" href="/nlab/show/0-type">0-type</a>, then its 0-truncation is still an extension by the 0-truncation of the original homotopy fiber.) The elements in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>dR</mi> <mi>p</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^p_{dR}(X)</annotation></semantics></math> here are precisely the <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 charges, as discussed in (<a href="#AGIT89">AGIT 89, p. 8</a>). Hence <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 the more BPS the more odd-graded elements there are in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mn>0</mn></msub><mi>𝔓𝔬𝔦𝔰</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tau_0 \mathfrak{Pois}(X,\omega)</annotation></semantics></math> (or its restriction to super-isometries). Hence <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 1/2 BPS state of supergravity if the odd dimension of this is half that of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>d</mi><mo stretchy="false">|</mo><mi>N</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d-1,d|N}</annotation></semantics></math>, it is 1/4 BPS if the odd dimension is one fourth of that of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>d</mi><mo stretchy="false">|</mo><mi>N</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d-1,d|N}</annotation></semantics></math>, etc. Notice that if <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>ℝ</mi><mo stretchy="false">/</mo><mi>Γ</mi><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mstyle mathvariant="bold"><mi>L</mi></mstyle> <mi>WZW</mi></msub></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>F</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>ω</mi></mover></mtd> <mtd><msubsup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>cl</mi> <mrow><mi>p</mi><mo>+</mo><mn>2</mn></mrow></msubsup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \mathbf{B}^{p+1} (\mathbb{R}/\Gamma)_{conn} \\ & {}^{\mathllap{\mathbf{L}_{WZW}}}\nearrow & \downarrow^{\mathrlap{F_{(-)}}} \\ X &\stackrel{\omega}{\longrightarrow}& \mathbf{\Omega}^{p+2}_{cl} } </annotation></semantics></math></div> is a <a class="existingWikiWord" href="/nlab/show/prequantum+line+n-bundle">prequantization</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>, i.e. an actual <a class="existingWikiWord" href="/nlab/show/WZW+term">WZW term</a> with <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔓𝔬𝔦𝔰</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{Pois}(X,\omega)</annotation></semantics></math> is supposed to be the <a class="existingWikiWord" href="/nlab/show/Lie+differentiation">Lie differentiation</a> of the <a class="existingWikiWord" href="/nlab/show/stabilizer+group">stabilizer group</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>L</mi></mstyle> <mi>WZW</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{L}_{WZW}</annotation></semantics></math>, which is the <a class="existingWikiWord" href="/nlab/show/quantomorphism+n-group">quantomorphism n-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>QuantMorph</mi><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>L</mi></mstyle> <mi>WZW</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">QuantMorph(\mathbf{L}_{WZW})</annotation></semantics></math>. (This Lie differentiation statement is strictly shown only for <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">p = 1</annotation></semantics></math> in <a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">dcct</a> but clearly should hold generally.) Hence we may regard <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>QuantMorph</mi></mstyle><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>L</mi></mstyle> <mi>WZW</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{QuantMorph}(\mathbf{L}_{WZW})</annotation></semantics></math> (or its restriction to super-<a class="existingWikiWord" href="/nlab/show/isometries">isometries</a>) as the <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> of the brane-charge extended supersymmetry algebra. By the discussion at <a href="conserved+current#InHigherPrequantumGeometry">conserved current – In higher differential geometry</a> this is indeed the <a class="existingWikiWord" href="/nlab/show/n-group">n-group</a> of conserved currents of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>L</mi></mstyle> <mi>WZW</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{L}_{WZW}</annotation></semantics></math> regraded as a <a class="existingWikiWord" href="/nlab/show/local+Lagrangian">local Lagrangian</a>, and so this conceptually connects back to the considerations in (<a href="#AGIT89">AGIT 89</a>). <h2 id="examples">Examples</h2> <h3 id="in_11d_supergravity">In 11d Supergravity</h3> In <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a> (<a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a>) there are four kinds of 1/2 BPS states (the <a class="existingWikiWord" href="/nlab/show/black+brane">black</a> <a class="existingWikiWord" href="/nlab/show/M-branes">M-branes</a>) (e.g. <a href="#Stelle98">Stelle 98, section 3</a> <a href="#EHKNT07">EHKNT 07</a>): <ul> <li> the <a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a>; </li> <li> the <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a>; </li> <li> the <a class="existingWikiWord" href="/nlab/show/M-wave">M-wave</a>; </li> <li> the <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+monopole">Kaluza-Klein monopole</a>. </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Bridgeland+stability+condition">Bridgeland stability condition</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Killing+spinor">Killing spinor</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/supermultiplet">supermultiplet</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/wall+crossing">wall crossing</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/protection+from+quantum+corrections">protection from quantum corrections</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/positive+energy+theorem">positive energy theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/enhanced+gauge+symmetry">enhanced gauge symmetry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+BPS+charges">geometry of physics – BPS charges</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/intersecting+branes">intersecting branes</a> <a class="existingWikiWord" href="/nlab/show/membrane+triple+junction">membrane triple junction</a> </li> </ul> <h2 id="references">References</h2> <h3 id="general_2">General</h3> The BPS bound derives its name from the discussion of <a class="existingWikiWord" href="/nlab/show/magnetic+monopoles">magnetic monopoles</a> in 4-dimensional <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a> in <ul> <li> <a class="existingWikiWord" href="/nlab/show/%D0%95.+%D0%91.+%D0%91%D0%BE%D0%B3%D0%BE%D0%BC%D0%BE%D0%BB%D1%8C%D0%BD%D1%8B%D0%B9">Е. Б. Богомольный</a>, Устойчивость классических решений, Яд. Физ 24 (1976) 449-454 Engl. tanslation: <a class="existingWikiWord" href="/nlab/show/Evgeny+B.+Bogomolnyj">Evgeny B. Bogomolnyj</a>, Stability of classical solutions, Sov. J. Nucl. Phys. 24 (1976) 449 and Yad. Fiz. 24 (1976) 861-870 [<a href="https://inspirehep.net/literature/101280">spire:101280</a>] reprinted in: Solitons and Particles, World Scientific (1984) 389-394 [<a href="https://doi.org/10.1142/0046">doi:10.1142/0046</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Manoj+K.+Prasad">Manoj K. Prasad</a>, <a class="existingWikiWord" href="/nlab/show/Charles+Sommerfield">Charles Sommerfield</a>, Exact classical solution for ‘t Hooft monopole and the Julia-Zee dyon, Phys. Rev. Lett. 35 (1975) 760-762 [<a href="https://doi.org/10.1103/PhysRevLett.35.760">doi:10.1103/PhysRevLett.35.760</a>] </li> </ul> The extension of the term “BPS-saturated state” from this case to situations in <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> seems to have happened in <ul> <li id="Witten95"><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, around (2.5) of <a class="existingWikiWord" href="/nlab/show/String+Theory+Dynamics+In+Various+Dimensions">String Theory Dynamics In Various Dimensions</a>, Nucl. Phys. B 443 (1995) 85-126 [<a href="https://arxiv.org/abs/hep-th/9503124">arXiv:hep-th/9503124</a>]</li> </ul> The original article identifying the role of BPS states in <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetric</a> <a class="existingWikiWord" href="/nlab/show/field+theory">field theory</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <a class="existingWikiWord" href="/nlab/show/David+Olive">David Olive</a>, Supersymmetry algebras that include topological charges (2002) (<a href="http://www.sciencedirect.com/science/article/pii/037026937890357X">journal</a>)</li> </ul> Exposition and review includes <ul> <li> <a class="existingWikiWord" href="/nlab/show/Andrew+Neitzke">Andrew Neitzke</a>, What is a BPS state?, 2012 (<a href="http://www.ma.utexas.edu/users/neitzke/expos/bps-expos.pdf">pdf</a>) </li> <li id="Dimofte10"> <a class="existingWikiWord" href="/nlab/show/Tudor+Dimofte">Tudor Dimofte</a>, Refined BPS invariants, Chern-Simons theory, and the quantum dilogarithm, 2010 (<a href="http://thesis.library.caltech.edu/5808/1/DimofteTDofficial.pdf">pdf</a>, <a href="http://thesis.library.caltech.edu/5808/">web</a>) </li> </ul> Further developments: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Jeffrey+Harvey">Jeffrey Harvey</a>, <a class="existingWikiWord" href="/nlab/show/Greg+Moore">Greg Moore</a>, Algebras, BPS states, and strings, Nucl.Phys. B 463 (1996) 315-368 [<a href="http://doi.org/10.1016/0550-3213%2895%2900605-2">doi:10.1016/0550-3213%2895%2900605-2</a>, <a href="https://arxiv.org/abs/hep-th/9510182">hep-th/9510182</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Jeffrey+Harvey">Jeffrey Harvey</a>, <a class="existingWikiWord" href="/nlab/show/Greg+Moore">Greg Moore</a>, On the algebras of BPS states, Comm. Math. Phys. 197 (1998), 489–-519, <a href="https://doi.org/10.1007/s002200050461">doi</a>, <a href="https://arxiv.org/abs/hep-th/9609017">hep-th/9609017</a>. </li> <li> <a class="existingWikiWord" href="/nlab/show/Ali+Chamseddine">Ali Chamseddine</a>, M. S. Volkov, Non-abelian BPS monopoles in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒩</mi><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">\mathcal{N}=4</annotation></semantics></math> gauged supergravity, Physical Review Letters 79 3343&-3346 (1997) [<a href="http://arxiv.org/abs/hep-th/9707176">hep-th/9707176</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Steven+Weinberg">Steven Weinberg</a>, The quantum theory of fields, vol. II </li> <li> <a class="existingWikiWord" href="/nlab/show/Tudor+Dimofte">Tudor Dimofte</a>, <a class="existingWikiWord" href="/nlab/show/Sergei+Gukov">Sergei Gukov</a>, Refined, Motivic, and Quantum, <a href="http://arXiv.org/abs/0904.1420">arXiv:0904.1420</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Davide+Gaiotto">Davide Gaiotto</a>, <a class="existingWikiWord" href="/nlab/show/Gregory+Moore">Gregory Moore</a>, <a class="existingWikiWord" href="/nlab/show/Andrew+Neitzke">Andrew Neitzke</a>, Wall-crossing, Hitchin systems, and the WKB approximation, <a href="https://arxiv.org/abs/0907.3987">arxiv:0907.3987</a> </li> <li> R. Pandharipande, R. P. Thomas, Stable pairs and BPS invariants, <a href="https://arxiv.org/abs/0711.3899">arXiv:0711.3899</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Markus+Reineke">Markus Reineke</a>, Cohomology of quiver moduli, functional equations, and integrality of Donaldson-Thomas type invariants, Compositio Mathematica 147:3 (2011) 943–964 <a href="https://doi.org/10.1112/S0010437X1000521X">doi</a> <a href="https://arXiv.org/abs/0903.0261">arXiv:0903.0261</a> </li> <li> Duiliu-Emanuel Diaconescu, Moduli of ADHM sheaves and local Donaldson-Thomas theory, J. Geom. & Physics 62:4 (2012) 763–799 <a href="https://arXiv.org/abs/0801.0820">arXiv:0801.0820</a> f<a href="https://doi.org/10.1016/j.geomphys.2011.12.018">doi</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Tom+Bridgeland">Tom Bridgeland</a>, Stability conditions on triangulated categories, Ann. of Math. 166 (2007) 317–345,<a href="https://arxiv.org/abs/math/0212237">math.AG/0212237</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Maxim+Kontsevich">Maxim Kontsevich</a>, <a class="existingWikiWord" href="/nlab/show/Yan+Soibelman">Yan Soibelman</a>, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, <a href="https://arxiv.org/abs/0811.2435">arXiv:0811.2435</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Maxim+Kontsevich">Maxim Kontsevich</a>, <a class="existingWikiWord" href="/nlab/show/Yan+Soibelman">Yan Soibelman</a>, Motivic Donaldson-Thomas invariants: summary of results, in: Mirror Symmetry and Tropical Geometry, Contemp. Math. 527 (2010) <a href="https://doi.org/10.1090/conm/527/10400">doi</a> <a href="https://arxiv.org/abs/0910.4315">arxiv/0910.4315</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Dominic+Joyce">Dominic Joyce</a>, Y. Song, A theory of generalized Donaldson-Thomas invariants, <a href="https://arxiv.org/abs/0810.5645">arxiv/0810.5645</a> </li> </ul> <h3 id="IntroductoryReferences">Introductions, surveys and lectures</h3> An introduction that starts at the beginning and then covers much of the ground in some detail is <ul> <li><a class="existingWikiWord" href="/nlab/show/Greg+Moore">Greg Moore</a>, PiTP Lectures on BPS states and wall-crossing in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">d = 4</annotation></semantics></math>, <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">\mathcal{N} = 2</annotation></semantics></math> theories (<a href="http://www.physics.rutgers.edu/~gmoore/PiTP_July26_2010.pdf">pdf</a>)</li> </ul> A survey of progress on the most general picture is in <ul> <li>Katzutoshi Ohta, BPS state counting and related physics (2005) (<a href="http://www2.yukawa.kyoto-u.ac.jp/~qft/2005/slides/ohta.pdf">pdf</a>)</li> </ul> <h3 id="in_supergravity_2">In supergravity</h3> Discussion of extremal/BPS <a class="existingWikiWord" href="/nlab/show/black+branes">black branes</a> in <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> (especially in <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a> and 10d <a class="existingWikiWord" href="/nlab/show/type+II+supergravity">type II supergravity</a>) includes <ul> <li id="Stelle98"> <a class="existingWikiWord" href="/nlab/show/Kellogg+Stelle">Kellogg Stelle</a>, BPS Branes in Supergravity, in: Quantum Field Theory: Perspective and Prospective, NATO Science Series 530, Springer (1999) [<a href="http://arxiv.org/abs/hep-th/9803116">arXiv:hep-th/9803116</a>, <a href="https://doi.org/10.1007/978-94-011-4542-8_12">doi:10.1007/978-94-011-4542-8_12</a>] </li> <li id="GauntlettHull99"> <a class="existingWikiWord" href="/nlab/show/Jerome+Gauntlett">Jerome Gauntlett</a>, <a class="existingWikiWord" href="/nlab/show/Chris+Hull">Chris Hull</a>, BPS states with extra supersymmetry, JHEP 0001 (2000) 004 [<a href="https://arxiv.org/abs/hep-th/9909098">arXiv:hep-th/9909098</a> <a href="https://doi.org/10.1088/1126-6708/2000/01/004">doi:10.1088/1126-6708/2000/01/004</a>] </li> <li id="EHKNT07"> <a class="existingWikiWord" href="/nlab/show/Francois+Englert">Francois Englert</a>, <a class="existingWikiWord" href="/nlab/show/Laurent+Houart">Laurent Houart</a>, <a class="existingWikiWord" href="/nlab/show/Axel+Kleinschmidt">Axel Kleinschmidt</a>, <a class="existingWikiWord" href="/nlab/show/Hermann+Nicolai">Hermann Nicolai</a>, Nassiba Tabti, An <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>9</mn></msub></mrow><annotation encoding="application/x-tex">E_9</annotation></semantics></math> multiplet of BPS states, JHEP 0705:065 (2007) [<a href="https://arxiv.org/abs/hep-th/0703285">arXiv:hep-th/0703285</a>] </li> <li> Andrew Callister, <a class="existingWikiWord" href="/nlab/show/Douglas+Smith">Douglas Smith</a>, Topological BPS charges in 10 and 11-dimensional supergravity, Phys. Rev. D 78 065042 (2008) [<a href="https://arxiv.org/abs/0712.3235">arXiv:0712.3235</a>] </li> <li> Andrew Callister, <a class="existingWikiWord" href="/nlab/show/Douglas+Smith">Douglas Smith</a>, Topological charges in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SL</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SL(2,\mathbb{R})</annotation></semantics></math> covariant massive 11-dimensional and Type IIB SUGRA, Phys.Rev.D80:125035,2009 (<a href="http://arxiv.org/abs/0907.3614">arXiv:0907.3614</a>) </li> <li> Andrew Callister, Topological BPS charges in 10- and 11-dimensional supergravity, thesis 2010 (<a href="http://inspirehep.net/record/1221591?ln=en">spire</a>) </li> <li> Cristine N. Ferreira, BPS solution for eleven-dimensional supergravity with a conical defect configuration (<a href="https://arxiv.org/abs/1312.0578">arXiv:1312.0578</a>) </li> </ul> Specifically for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo stretchy="false">/</mo><msup><mn>2</mn> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">1/2^n</annotation></semantics></math>-BPS states of intersecting <a class="existingWikiWord" href="/nlab/show/M-branes">M-branes</a> in 11d there is discussion in <ul> <li id="Tseytlin96"><a class="existingWikiWord" href="/nlab/show/Arkady+Tseytlin">Arkady Tseytlin</a>, Harmonic superpositions of M-branes, Nucl.Phys.B475:149-163,1996 (<a href="https://arxiv.org/abs/hep-th/9604035">arXiv:hep-th/9604035</a>) also in <a class="existingWikiWord" href="/nlab/show/Mike+Duff">Mike Duff</a> (ed.) chapter 5 of <a class="existingWikiWord" href="/nlab/show/The+World+in+Eleven+Dimensions">The World in Eleven Dimensions</a></li> </ul> see also <ul> <li> <a class="existingWikiWord" href="/nlab/show/Jerome+Gauntlett">Jerome Gauntlett</a>, Intersecting Branes (<a href="https://arxiv.org/abs/hep-th/9705011">hep-th/9705011</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Igor+Bandos">Igor Bandos</a>, <a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+de+Azc%C3%A1rraga">José de Azcárraga</a>, <a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+Izquierdo">José Izquierdo</a>, Jerzy Lukierski<a href="/nlab/new/Jerzy+Lukierski">?</a>, BPS states in M-theory and twistorial constituents, Phys. Rev. Lett. 86 (2001) 4451-4454 [<a href="https://arxiv.org/abs/hep-th/0101113">arXiv:hep-th/0101113</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Ulf+Gran">Ulf Gran</a>, <a class="existingWikiWord" href="/nlab/show/Jan+Gutowski">Jan Gutowski</a>, <a class="existingWikiWord" href="/nlab/show/George+Papadopoulos">George Papadopoulos</a>, Classification, geometry and applications of supersymmetric backgrounds [<a href="https://arxiv.org/abs/1808.07879">arXiv:1808.07879</a>] </li> </ul> Semiclassical approach: <ul> <li>T. Daniel Brennan, <a class="existingWikiWord" href="/nlab/show/Gregory+W.+Moore">Gregory W. Moore</a>, A note on the semiclassical formulation of BPS states in four-dimensional N = 2 theories, Prog. Theor. Exp. Phys. 2016, 12C110 (19 pages) <a href="https://doi.org/10.1093/ptep/ptw159">doi</a></li> </ul> Discussion in the context of multiple <a class="existingWikiWord" href="/nlab/show/M2-branes">M2-branes</a> in the <a class="existingWikiWord" href="/nlab/show/BLG+model">BLG model</a> is in <ul> <li id="BaggerLambert12"><a class="existingWikiWord" href="/nlab/show/Jonathan+Bagger">Jonathan Bagger</a>, <a class="existingWikiWord" href="/nlab/show/Neil+Lambert">Neil Lambert</a>, Sunil Mukhi, Constantinos Papageorgakis, section 1.6 of Multiple Membranes in M-theory (<a href="https://arxiv.org/abs/1203.3546">arXiv:1203.3546</a>)</li> </ul> Discussion for <a class="existingWikiWord" href="/nlab/show/4d+supergravity">4d supergravity</a>, hence in <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a> of <a class="existingWikiWord" href="/nlab/show/type+II+supergravity">type II supergravity</a> on a <a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifold">Calabi-Yau manifold</a> is due to <ul> <li id="Denef00"> <a class="existingWikiWord" href="/nlab/show/Frederik+Denef">Frederik Denef</a>, Supergravity flows and D-brane stability, JHEP 0008:050, 2000 (<a href="http://arxiv.org/abs/hep-th/0005049">arXiv:hep-th/0005049</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Frederik+Denef">Frederik Denef</a>, Quantum Quivers and Hall/Hole Halos, JHEP 0210:023, 2002 (<a href="http://arxiv.org/abs/hep-th/0206072">arXiv:hep-th/0206072</a>) </li> </ul> Discussion of more general classification of solutions to <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> preserving some <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a>, i.e. admitting some <a class="existingWikiWord" href="/nlab/show/Killing+spinors">Killing spinors</a> includes <ul> <li> <a class="existingWikiWord" href="/nlab/show/Jerome+Gauntlett">Jerome Gauntlett</a>, Stathis Pakis, The Geometry of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><mn>11</mn></mrow><annotation encoding="application/x-tex">D=11</annotation></semantics></math> Killing Spinors, JHEP 0304 (2003) 039 (<a href="http://arxiv.org/abs/hep-th/0212008">arXiv:hep-th/0212008</a>) </li> <li id="HEGKS08"> <a class="existingWikiWord" href="/nlab/show/Eric+D%27Hoker">Eric D'Hoker</a>, John Estes, Michael Gutperle, Darya Krym, Paul Sorba, Half-BPS supergravity solutions and superalgebras, JHEP 0812:047 (2008) [<a href="http://arxiv.org/abs/0810.1484">arXiv:0810.1484</a>, <a href="https://iopscience.iop.org/article/10.1088/1126-6708/2008/12/047">doi:10.1088/1126-6708/2008/12/047</a>] </li> </ul> The conceptual identification of the relevant brane-charge extension of the <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a> algebra as that of the <a class="existingWikiWord" href="/nlab/show/conserved+currents">conserved currents</a> of the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+super+p-brane+sigma+models">Green-Schwarz super p-brane sigma models</a> for branes is due to <ul> <li id="AGIT89"><a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+de+Azc%C3%A1rraga">José de Azcárraga</a>, <a class="existingWikiWord" href="/nlab/show/Jerome+Gauntlett">Jerome Gauntlett</a>, <a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+Izquierdo">J. M. Izquierdo</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Townsend">Paul Townsend</a>, Topological extensions of the supersymmetry algebra for extended objects, Phys. Rev. Lett. 63 (1989) 2443 [<a href="https://inspirehep.net/record/26393?ln=en">spire</a>]</li> </ul> reviewed in <ul> <li><a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+de+Azc%C3%A1rraga">José de Azcárraga</a>, <a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+Izquierdo">José M. Izquierdo</a>, section 8.8. of Lie groups, Lie algebras, cohomology and some applications in physics , Cambridge monographs of mathematical physics (1995)</li> </ul> This is for branes in the old <a class="existingWikiWord" href="/nlab/show/brane+scan">brane scan</a> (<a class="existingWikiWord" href="/nlab/show/strings">strings</a>, <a class="existingWikiWord" href="/nlab/show/membranes">membranes</a>, <a class="existingWikiWord" href="/nlab/show/NS5-branes">NS5-branes</a>), excluding <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> and <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a>. The generalization oft this perspective to the <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> is discussed in <ul> <li id="SorokinTownsend97"><a class="existingWikiWord" href="/nlab/show/Dmitri+Sorokin">Dmitri Sorokin</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Townsend">Paul Townsend</a>, M-theory superalgebra from the M-5-brane, Phys.Lett. B412 (1997) 265-273 (<a href="http://arxiv.org/abs/hep-th/9708003">arXiv:hep-th/9708003</a>)</li> </ul> and the generalization to <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> is discussed in <ul> <li id="Hammer97">Hanno Hammer, Topological Extensions of Noether Charge Algebras carried by D-p-branes, Nucl.Phys. B521 (1998) 503-546 (<a href="http://arxiv.org/abs/hep-th/9711009">arXiv:hep-th/9711009</a>)</li> </ul> Detailed discussion of examples for various backgrounds is in <ul> <li>Takeshi Sato, Superalgebras in Many Types of M-Brane Backgrounds and Various Supersymmetric Brane Configurations, Nucl.Phys. B548 (1999) 231-257 (<a href="https://arxiv.org/abs/hep-th/9812014">arXiv:hep-th/9812014</a>)</li> </ul> Discussion of this in <a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher differential geometry</a> via the <a class="existingWikiWord" href="/nlab/show/Poisson+bracket+Lie+n-algebra">Poisson bracket Lie n-algebra</a> is in <ul> <li id="SatiSchreiber15"><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/schreiber/show/Lie+n-algebras+of+BPS+charges">Lie n-algebras of BPS charges</a>, J. High Energ. Phys. 2017 87 (2017) [<a href="http://arxiv.org/abs/1507.08692">arXiv:1507.08692</a>, <a href="http://link.springer.com/article/10.1007/JHEP03(2017)087">doi:10.1007/JHEP03(2017)087</a>]</li> </ul> Discussion of relation of <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> BPS states to <a class="existingWikiWord" href="/nlab/show/knot+invariants">knot invariants</a> includes <ul> <li> <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, Fivebranes and Knots (<a href="http://arxiv.org/abs/1101.3216">arXiv:1101.3216</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Sergei+Gukov">Sergei Gukov</a>, Marko Stošić, Homological algebra of knots and BPS states, GTM 18 (2012) 309–367 (<a href="https://arxiv.org/abs/1112.0030">arXiv:1112.0030</a> <a href="https://doi.org/10.2140/gtm.2012.18.309">doi</a>) </li> <li> Ross Elliot, <a class="existingWikiWord" href="/nlab/show/Sergei+Gukov">Sergei Gukov</a>, Exceptional knot homology (<a href="https://arxiv.org/abs/1505.01635">arXiv:1505.01635</a>) </li> </ul> <h3 id="SpectralNetworksReferences">Spectral networks</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/Davide+Gaiotto">Davide Gaiotto</a>, <a class="existingWikiWord" href="/nlab/show/Greg+Moore">Greg Moore</a>, <a class="existingWikiWord" href="/nlab/show/Andrew+Neitzke">Andrew Neitzke</a>, Spectral networks (<a href="http://arxiv.org/abs/1204.4824">arXiv:1204.4824</a>, <a href="http://www.ma.utexas.edu/users/neitzke/spectral-network-movies/">illustrating animations</a>) </li> </ul> </body></html> </div> <div class="revisedby"> Last revised on September 12, 2024 at 09:02:01. 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