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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="wesszuminowitten_theory"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Wess-Zumino-Witten theory</h4> <div class="hide"><div> <ul> <li> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/infinity-Chern-Simons+theory+-+contents">∞-Chern-Simons theory</a> </li> <li> <a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Wess-Zumino-Witten+theory">∞-Wess-Zumino-Witten theory</a> </li> </ul> <h2 id="models">Models</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/1d+WZW+model">1d WZW model</a> </li> <li> 2d <a class="existingWikiWord" href="/nlab/show/Wess-Zumino-Witten+model">Wess-Zumino-Witten model</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/current+algebra">current algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+superstring">Green-Schwarz superstring</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/higher+dimensional+WZW+theory">higher dimensional WZW theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/4d+WZW+theory">4d WZW theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+action+functional">Green-Schwarz action functional</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/WZW-type+superstring+field+theory">WZW-type superstring field theory</a> </li> </ul> <div> <a href="/nlab/edit/infinity-Wess-Zumino-Witten+theory+-+contents">Edit this sidebar</a> </div></div></div> <h4 id="quantum_field_theory">Quantum field theory</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/FQFT">functorial quantum field theory</a> <h2 id="contents">Contents</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/cobordism+category">cobordism category</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/extended+cobordism">extended 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/bordism+categories+following+Stolz-Teichner">Riemannian bordism category</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/generalized+tangle+hypothesis">generalized tangle hypothesis</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/On+the+Classification+of+Topological+Field+Theories">classification of TQFTs</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/functorial+field+theory">functorial field theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/unitary+functorial+field+theory">unitary functorial field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/extended+functorial+field+theory">extended functorial field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+field+theory">CFT</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/vertex+operator+algebra">vertex operator algebra</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a> <ul> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/HQFT">HQFT</a> </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>, <a class="existingWikiWord" href="/nlab/show/Gromov-Witten+theory">Gromov-Witten theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a> </li> </ul> </li> </ul> </li> </ul> </li> <li> FQFT and <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> <ul> <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</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometric+models+for+tmf">geometric models for tmf</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/holographic+principle+of+higher+category+theory">holographic principle of higher category theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/holographic+principle">holographic principle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/AdS%2FCFT+correspondence">AdS/CFT correspondence</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantization+via+the+A-model">quantization via the A-model</a> </li> </ul> </li> </ul> </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" 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<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> 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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> <li><a href='#ActionFunctional'>Action functional</a></li> <ul> <li><a href='#KineticTerm'>Kinetic term</a></li> <li><a href='#TopologicalTerm'>Topological term – WZW term</a></li> <ul> <li><a href='#WZWTermFor2dModel'>For the 2d WZW model</a></li> <li><a href='#FormalizationGenerally'>Generally</a></li> </ul> </ul> <li><a href='#Properties'>Properties</a></li> <ul> <li><a href='#EquationsOfMotion'>Equations of motion</a></li> <li><a href='#HolographyAndRigorousConstruction'>Holography and Rigorous construction</a></li> <li><a href='#BraidRepresentationsViaTwisteddRCohomologyOfConfigurationSpaces'>Braid representations via twisted cohomology of configuration spaces</a></li> <li><a href='#DBranes'>D-branes for the WZW model</a></li> <li><a href='#quantization'>Quantization</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> <ul> <li><a href='#IntroductionsAndSurveys'>Introduction and survey</a></li> <li><a href='#WZWTermOfChiralPerturbationTheoryReferences'>The WZW term of QCD chiral perturbation theory</a></li> <ul> <li><a href='#WZWTermOfChiralPerturbationTheoryReferencesGeneral'>General</a></li> <li><a href='#WZWTermOfChiralPerturbationTheoryReferencesIncludingLightVectorMesons'>Including light vector mesons</a></li> <li><a href='#WZWTermOfChiralPerturbationTheoryReferencesIncludingHeavyScalarMesons'>Including heavy scalar mesons</a></li> <li><a href='#WZWTermOfChiralPerturbationTheoryReferencesIncludingHeavyVectorMesons'>Including heavy vector mesons</a></li> <li><a href='#WZWTermOfChiralPerturbationTheoryReferencesIncludingElectroweakInteractions'>Including electroweak interactions</a></li> </ul> <li><a href='#interpretation_via_cft_and_gerbes'>Interpretation via CFT and gerbes</a></li> <li><a href='#ReferencesRelationToGerbesAndCS'>Relation to gerbes and Chern-Simons theory</a></li> <li><a href='#partition_functions'>Partition functions</a></li> <li><a href='#ReferencesDBranes'>D-branes for the WZW model</a></li> <li><a href='#relation_to_dimensional_reduction_of_chernsimons'>Relation to dimensional reduction of Chern-Simons</a></li> <li><a href='#relation_to_extended_tqft'>Relation to extended TQFT</a></li> <li><a href='#in_solid_state_physics'>In solid state physics</a></li> <li><a href='#FractionalLevelWZWModelReferences'>On fractional-level WZW models as logarithmic CFTs</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> The Wess-Zumino-Witten model (or WZW model for short, also called Wess-Zumino-Novikov-Witten model, or short WZNW model) is a 2-dimensional <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> whose target space is a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>. This may be regarded as the boundary theory of <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> for Lie group <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>. The <a class="existingWikiWord" href="/nlab/show/vertex+operator+algebra">vertex operator algebra</a>s corresponding to the WZW model are <a class="existingWikiWord" href="/nlab/show/current+algebra">current algebra</a>s. <h2 id="ActionFunctional">Action functional</h2> For <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> a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, the <a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a> of the WZW over a 2-<a class="existingWikiWord" href="/nlab/show/dimension">dimension</a>al <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> <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> is the space of <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mi>Σ</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">g : \Sigma \to G</annotation></semantics></math>. The <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> of the WZW <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> is the sum of two terms, a kinetic term and a topological term <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>WZW</mi></msub><mo>=</mo><msub><mi>S</mi> <mi>kin</mi></msub><mo>+</mo><msub><mi>S</mi> <mi>top</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> S_{WZW} = S_{kin} + S_{top} \,. </annotation></semantics></math></div> <h3 id="KineticTerm">Kinetic term</h3> The Lie group canonically carries a <a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemannian metric</a> and the kinetic term is the standard one for <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a>s on Riemannian <a class="existingWikiWord" href="/nlab/show/target+space">target space</a>s. <h3 id="TopologicalTerm">Topological term – WZW term</h3> <h4 id="WZWTermFor2dModel">For the 2d WZW model</h4> In <a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher differential geometry</a>, then given any closed <a class="existingWikiWord" href="/nlab/show/differential+n-form">differential (p+2)-form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>∈</mo><msubsup><mi>Ω</mi> <mi>cl</mi> <mrow><mi>p</mi><mo>+</mo><mn>2</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\omega \in \Omega^{p+2}_{cl}(X)</annotation></semantics></math>, it is natural to ask for a <a class="existingWikiWord" href="/nlab/show/prequantization">prequantization</a> of it, namely for a <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle (p+1)-bundle with connection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math> (equivalently: <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> in degree-<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>-<a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a>) 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> whose <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mo>∇</mo></msub><mo>=</mo><mi>ω</mi></mrow><annotation encoding="application/x-tex">F_\nabla = \omega</annotation></semantics></math>. In terms of <a class="existingWikiWord" href="/nlab/show/moduli+stacks">moduli stacks</a> this means asking for lifts of the form <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><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mo>∇</mo></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}U(1)_{conn} \\ &{}^{\mathllap{\nabla}}\nearrow& \downarrow^{\mathrlap{F_{(-)}}} \\ X &\stackrel{\omega}{\longrightarrow}& \mathbf{\Omega}^{p+2}_{cl} } </annotation></semantics></math></div> in the <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> of <a class="existingWikiWord" href="/nlab/show/smooth+homotopy+types">smooth homotopy types</a>. This immediately raises the question for natural classes of examples of such prequantizations. One such class arises in <a class="existingWikiWord" href="/nlab/show/infinity-Lie+theory">infinity-Lie theory</a>, where <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> is a <a class="existingWikiWord" href="/nlab/show/left+invariant+form">left invariant form</a> on a <a class="existingWikiWord" href="/nlab/show/smooth+infinity-group">smooth infinity-group</a> given by a <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> in <a class="existingWikiWord" href="/nlab/show/L-infinity+algebra+cohomology">L-infinity algebra cohomology</a>. The <a class="existingWikiWord" href="/nlab/show/prequantum+n-bundles">prequantum n-bundles</a> arising this way are the higher <a class="existingWikiWord" href="/nlab/show/WZW+terms">WZW terms</a> discussed here. In low degree of traditional <a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a> this appears as follows: On <a class="existingWikiWord" href="/nlab/show/Lie+groups">Lie groups</a> <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>, those closed <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>-forms <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> which are <a class="existingWikiWord" href="/nlab/show/left+invariant+forms">left invariant forms</a> may be identified, via the general theory of <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebras">Chevalley-Eilenberg algebras</a>, with degree <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>-<a class="existingWikiWord" href="/nlab/show/cocycles">cocycles</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a> of the <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> <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">\mathfrak{g}</annotation></semantics></math> corresponding to <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 in turn may arise, via the <a class="existingWikiWord" href="/nlab/show/van+Est+map">van Est map</a>, as the <a class="existingWikiWord" href="/nlab/show/Lie+differentiation">Lie differentiation</a> of a degree-<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>-<a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo lspace="verythinmathspace">:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>p</mi><mo>+</mo><mn>2</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{c} \colon \mathbf{B}G \to \mathbf{B}^{p+2}U(1)</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/Lie+group+cohomology">Lie group cohomology</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> itself, with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in the <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>. This happens to be the case notably for <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> a <a class="existingWikiWord" href="/nlab/show/simply+connected+topological+space">simply connected</a> <a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact</a> <a class="existingWikiWord" href="/nlab/show/semisimple+Lie+group">semisimple Lie group</a> such as <a class="existingWikiWord" href="/nlab/show/special+unitary+group">SU</a> or <a class="existingWikiWord" href="/nlab/show/spin+group">Spin</a>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo>=</mo><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\mu = \langle -,[-,-]\rangle</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra 3-cocycle</a> in <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a> with the <a class="existingWikiWord" href="/nlab/show/Killing+form">Killing form</a> <a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle -,-\rangle</annotation></semantics></math>. This is, up to normalization, a representative of the de Rham image of a generator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>c</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{c}</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>3</mn></msup><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>H</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><mi>B</mi><mi>G</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">H^3(\mathbf{B}G, U(1)) \simeq H^4(B G, \mathbb{Z}) \simeq \mathbb{Z}</annotation></semantics></math>. Generally, by the discussion at <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+principal+bundles">geometry of physics – principal bundles</a>, the cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>c</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{c}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/modulating+morphism">modulates</a> an <a class="existingWikiWord" href="/nlab/show/infinity-group+extension">infinity-group extension</a> which is a <a class="existingWikiWord" href="/nlab/show/circle+n-group">circle p-group</a>-<a class="existingWikiWord" href="/nlab/show/principal+infinity-bundle">principal infinity-bundle</a> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>p</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>G</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>Ω</mi><mstyle mathvariant="bold"><mi>c</mi></mstyle></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}^p U(1) &\longrightarrow& \hat G \\ && \downarrow \\ && G &\stackrel{\Omega\mathbf{c}}{\longrightarrow}& \mathbf{B}^{p+1}U(1) } </annotation></semantics></math></div> whose higher <a class="existingWikiWord" href="/nlab/show/Dixmier-Douady+class">Dixmier-Douady class</a> class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∫</mo><mi>Ω</mi><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo>∈</mo><msup><mi>H</mi> <mrow><mi>p</mi><mo>+</mo><mn>2</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \int \Omega \mathbf{c} \in H^{p+2}(X,\mathbb{Z})</annotation></semantics></math> is an integral lift of the real cohomology class encoded by <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> under the <a class="existingWikiWord" href="/nlab/show/de+Rham+isomorphism">de Rham isomorphism</a>. In the example of <a class="existingWikiWord" href="/nlab/show/spin+group">Spin</a> 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> this extension is the <a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a>. Such a <a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theoretic</a> situation is concisely expressed by a diagram of <a class="existingWikiWord" href="/nlab/show/smooth+homotopy+types">smooth homotopy types</a> of the form <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></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>Ω</mi><mstyle mathvariant="bold"><mi>c</mi></mstyle></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>θ</mi> <mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>p</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>G</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> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><msub><mo>♭</mo> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>p</mi><mo>+</mo><mn>2</mn></mrow></msup><mi>ℝ</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ && &\longrightarrow& \mathbf{B}^{p+1}U(1) \\ &{}^{\mathllap{\Omega \mathbf{c}}}\nearrow& &\swArrow_\simeq& \downarrow^{\mathrlap{\theta_{\mathbf{B}^p U(1)}}} \\ G &\stackrel{\omega}{\longrightarrow}& \mathbf{\Omega}_{cl}^{p+2} &\stackrel{}{\longrightarrow}& \flat_{dR}\mathbf{B}^{p+2}\mathbb{R} } \,, </annotation></semantics></math></div> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>♭</mo> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>p</mi><mo>+</mo><mn>2</mn></mrow></msup><mi>ℝ</mi><mo>≃</mo><msub><mo>♭</mo> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>p</mi><mo>+</mo><mn>2</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\flat_{dR}\mathbf{B}^{p+2}\mathbb{R} \simeq \flat_{dR}\mathbf{B}^{p+2}U(1)</annotation></semantics></math> is the <a href="cohesive+infinity-topos+--+structures#deRhamCohomology">de Rham coefficients</a> (see also at <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+de+Rham+coefficients">geometry of physics – de Rham coefficients</a>) and where the homotopy filling the diagram is what exhibits <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> as a de Rham representative of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mstyle mathvariant="bold"><mi>c</mi></mstyle></mrow><annotation encoding="application/x-tex">\Omega \mathbf{c}</annotation></semantics></math>. Now, by the very <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a>-characterization of the <a class="existingWikiWord" href="/nlab/show/Deligne+complex">Deligne complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}^{p+1}U(1)_{conn}</annotation></semantics></math> (<a href="Deligne+cohomology#TheExactDifferentialCohomologyHexagon">here</a>), such a diagram is equivalently a <a class="existingWikiWord" href="/nlab/show/prequantization">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>: <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><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mo>∇</mo></mpadded></msup><mo>↗</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>θ</mi> <mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>p</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>G</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> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><msub><mo>♭</mo> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>p</mi><mo>+</mo><mn>2</mn></mrow></msup><mi>ℝ</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && \mathbf{B}^{p+1}U(1)_{conn} &\longrightarrow& \mathbf{B}^{p+1}U(1) \\ &{}^\mathllap{\nabla}\nearrow& \downarrow &\swArrow_\simeq& \downarrow^{\mathrlap{\theta_{\mathbf{B}^p U(1)}}} \\ G &\stackrel{\omega}{\longrightarrow}& \mathbf{\Omega}_{cl}^{p+2} &\stackrel{}{\longrightarrow}& \flat_{dR}\mathbf{B}^{p+2}\mathbb{R} } \,. </annotation></semantics></math></div> For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>=</mo><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\omega = \langle -,[-,-]\rangle</annotation></semantics></math> as above, we have <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> and so <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math> here is a <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 2-bundle with connection</a>, often referred to as a <a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe</a> <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle+gerbe">with connection</a>. As such, this is also known as the WZW gerbe or similar. This terminology arises as follows. In (<a href="Wess-Zumino-Witten+model#WessZumino71">Wess-Zumino 71</a>) the <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> for a <a class="existingWikiWord" href="/nlab/show/string">string</a> propagating on the <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>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> was considered, with only the standard <a class="existingWikiWord" href="/nlab/show/kinetic+action">kinetic action</a> term. Then in (<a href="Wess-Zumino-Witten+model#Witten84">Witten 84</a>) it was observed that for this <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> to give a <a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a> after <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a>, a certain <a class="existingWikiWord" href="/nlab/show/higher+gauge+theory">higher gauge</a> <a class="existingWikiWord" href="/nlab/show/interaction+term">interaction term</a> has to the added. The resulting <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> came to be known as the <a class="existingWikiWord" href="/nlab/show/Wess-Zumino-Witten+model">Wess-Zumino-Witten model</a> or WZW model for short, and the term that Witten added became the WZW term. In terms of <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> it describes the propagation of the <a class="existingWikiWord" href="/nlab/show/string">string</a> on the group <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> subject to a <a class="existingWikiWord" href="/nlab/show/force">force</a> of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> given by the <a class="existingWikiWord" href="/nlab/show/Killing+form">Killing form</a> <a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemannian metric</a> and subject to a <a class="existingWikiWord" href="/nlab/show/B-field">B-field</a> <a class="existingWikiWord" href="/nlab/show/higher+gauge+field">higher gauge force</a> whose <a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a> is <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>. In (<a href="Wess-Zumino-Witten+model#Gawedzki87">Gawedzki 87</a>) it was observed that when formulated properly and generally, this WZW term is the <a class="existingWikiWord" href="/nlab/show/surface+holonomy">surface holonomy</a> functional of a <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle+gerbe">connection on a bundle gerbe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math> 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>. This is equivalently the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math> that we just motivated above. Later WZW terms, or at least their curvature forms <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>, were recognized all over the place in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a>. For instance the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+sigma-models+for+super+p-branes">Green-Schwarz sigma-models for super p-branes</a> each have an <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> that is the sum of the standard <a class="existingWikiWord" href="/nlab/show/kinetic+action">kinetic action</a> plus a WZW term of degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>+</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">p+2</annotation></semantics></math>. In general WZW terms are “<a class="existingWikiWord" href="/nlab/show/gauged+WZW+model">gauged</a>” which means, as we will see, that they are not defined on the give <a class="existingWikiWord" href="/nlab/show/smooth+infinity-group">smooth infinity-group</a> <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> itself, but on a bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>G</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde G</annotation></semantics></math> of differential moduli stacks over that group, such that a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>→</mo><mover><mi>G</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\Sigma \to \tilde G</annotation></semantics></math> is a pair consisting of a map <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> and of a <a class="existingWikiWord" href="/nlab/show/higher+gauge+field">higher gauge field</a> on <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> (a “tensor multiplet” of fields). <h4 id="FormalizationGenerally">Generally</h4> The following (<a href="#FSS12">FSS 12</a>, <a href="#dcct">dcct</a>) is a general axiomatization of WZW terms in <a class="existingWikiWord" href="/nlab/show/cohesive+%28infinity%2C1%29-topos">cohesive homotopy theory</a>. In an ambient <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>, let <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">\mathbb{G}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/sylleptic+%E2%88%9E-group">sylleptic ∞-group</a>, equipped with a <a class="existingWikiWord" href="/nlab/show/Hodge+filtration">Hodge filtration</a>, hence in particular with a chosen morphism <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ι</mi><mo lspace="verythinmathspace">:</mo><msubsup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>cl</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>𝔾</mi><mo stretchy="false">)</mo><mo>⟶</mo><msub><mo>♭</mo> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>𝔾</mi></mrow><annotation encoding="application/x-tex"> \iota \colon \mathbf{\Omega}^2_{cl}(-,\mathbb{G}) \longrightarrow \flat_{dR} \mathbf{B}^2 \mathbb{G} </annotation></semantics></math></div> to its <a href="">de Rham coefficients</a> <div class="num_defn" id="RefinementOfHodgeFiltration"> <h6 id="definition">Definition</h6> Given an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> object <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> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> and given a <a class="existingWikiWord" href="/nlab/show/group+cohomology">group cocycle</a> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo lspace="verythinmathspace">:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>⟶</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>𝔾</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathbf{c} \colon \mathbf{B}G \longrightarrow \mathbf{B}^2 \mathbb{G} \,, </annotation></semantics></math></div> then a refinement of the <a class="existingWikiWord" href="/nlab/show/Hodge+filtration">Hodge filtration</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">\mathbb{G}</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>c</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{c}</annotation></semantics></math> is a completion of the <a class="existingWikiWord" href="/nlab/show/cospan">cospan</a> formed by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>♭</mo> <mi>dR</mi></msub><mstyle mathvariant="bold"><mi>c</mi></mstyle></mrow><annotation encoding="application/x-tex">\flat_{dR}\mathbf{c}</annotation></semantics></math> and by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ι</mi></mrow><annotation encoding="application/x-tex">\iota</annotation></semantics></math> above to a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> of the form <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msubsup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>flat</mi> <mn>1</mn></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mi>μ</mi></mover></mtd> <mtd><msubsup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>cl</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>𝔾</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>ι</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mo>♭</mo> <mi>dR</mi></msub><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mo>♭</mo> <mi>dR</mi></msub><mstyle mathvariant="bold"><mi>c</mi></mstyle></mrow></mover></mtd> <mtd><msub><mo>♭</mo> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>𝔾</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{\Omega}^1_{flat}(-,G) &\stackrel{\mu}{\longrightarrow}& \mathbf{\Omega}^2_{cl}(-,\mathbb{G}) \\ \downarrow && \downarrow^{\mathrlap{\iota}} \\ \flat_{dR}\mathbf{B}G &\stackrel{\flat_{dR}\mathbf{c}}{\longrightarrow}& \flat_{dR}\mathbf{B}^2 \mathbb{G} } \,. </annotation></semantics></math></div> We write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>G</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde G</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> of this refinement along the <a class="existingWikiWord" href="/nlab/show/Maurer-Cartan+form">Maurer-Cartan form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>θ</mi> <mi>G</mi></msub></mrow><annotation encoding="application/x-tex">\theta_G</annotation></semantics></math> 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> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mover><mi>G</mi><mo stretchy="false">˜</mo></mover></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>θ</mi> <mover><mi>G</mi><mo stretchy="false">˜</mo></mover></msub></mrow></mover></mtd> <mtd><msubsup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>flat</mi> <mn>1</mn></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>G</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>θ</mi> <mi>G</mi></msub></mrow></mover></mtd> <mtd><msub><mo>♭</mo> <mi>dR</mi></msub><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \tilde G &\stackrel{\theta_{\tilde G}}{\longrightarrow}& \mathbf{\Omega}^1_{flat}(-,G) \\ \downarrow && \downarrow \\ G &\stackrel{\theta_G}{\longrightarrow}& \flat_{dR}\mathbf{B}G } \,. </annotation></semantics></math></div></div> <div class="num_example"> <h6 id="example">Example</h6> Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔾</mi><mo>=</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>p</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{G} = \mathbf{B}^p U(1)</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/circle+n-group">circle (p+1)-group</a>. For <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> an ordinary <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math> may be taken to be the <a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cocycle</a> corresponding to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>c</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{c}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>G</mi><mo stretchy="false">˜</mo></mover><mo>≃</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">\tilde G \simeq G</annotation></semantics></math>. On the opposite extreme, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>p</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G = \mathbf{B}^p U(1)</annotation></semantics></math> itself with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>c</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{c}</annotation></semantics></math> the identity, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>G</mi><mo stretchy="false">˜</mo></mover><mo>=</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>p</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\tilde G = \mathbf{B}^p U (1)_{conn}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> for <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a> (the <a class="existingWikiWord" href="/nlab/show/Deligne+complex">Deligne complex</a> under <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> and <a class="existingWikiWord" href="/nlab/show/infinity-stackification">infinity-stackification</a>). Hence a more general case is a fibered product of these two, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>G</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde G</annotation></semantics></math> is such that a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>⟶</mo><mover><mi>G</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\Sigma \longrightarrow \tilde G</annotation></semantics></math> is equivalently a pair consisting of a map <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> and of differential <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>-form data on <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>. This is the case of relevance for WZW models of <a class="existingWikiWord" href="/nlab/show/super+p-branes">super p-branes</a> with “tensor multiplet” fields on them, such as the <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> and the <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a>. </div> <div class="num_prop"> <h6 id="proposition">Proposition</h6> In the situation of def. <a class="maruku-ref" href="#RefinementOfHodgeFiltration"></a> there is an essentially unique <a class="existingWikiWord" href="/nlab/show/prequantum+n-bundle">prequantization</a> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>L</mi></mstyle> <mi>WZW</mi></msub><mo lspace="verythinmathspace">:</mo><mover><mi>G</mi><mo stretchy="false">˜</mo></mover><mo>⟶</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>𝔾</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex"> \mathbf{L}_{WZW} \colon \tilde G \longrightarrow \mathbf{B}^2 \mathbb{G}_{conn} </annotation></semantics></math></div> of the closed differential form <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo stretchy="false">(</mo><msub><mi>θ</mi> <mover><mi>G</mi><mo stretchy="false">˜</mo></mover></msub><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mover><mi>G</mi><mo stretchy="false">˜</mo></mover><mover><mo>⟶</mo><mrow><msub><mi>θ</mi> <mover><mi>G</mi><mo stretchy="false">˜</mo></mover></msub></mrow></mover><msubsup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>flat</mi> <mn>1</mn></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mi>μ</mi></mover><msubsup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>cl</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>𝔾</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mu(\theta_{\tilde G}) \colon \tilde G \stackrel{\theta_{\tilde G}}{\longrightarrow} \mathbf{\Omega}^1_{flat}(-,G) \stackrel{\mu}{\longrightarrow} \mathbf{\Omega}^2_{cl}(-,\mathbb{G}) </annotation></semantics></math></div> whose underlying <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">\mathbb{G}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a> is <a class="existingWikiWord" href="/nlab/show/modulating+morphism">modulated</a> by the <a class="existingWikiWord" href="/nlab/show/looping+and+delooping">looping</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mstyle mathvariant="bold"><mi>c</mi></mstyle></mrow><annotation encoding="application/x-tex">\Omega \mathbf{c}</annotation></semantics></math> of the original cocycle. This we call the WZW term of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>c</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{c}</annotation></semantics></math> with respect to the chosen refinement of the Hodge structure. </div> <h2 id="Properties">Properties</h2> <h3 id="EquationsOfMotion">Equations of motion</h3> The <a class="existingWikiWord" href="/nlab/show/variational+calculus">variational derivative</a> of the WZW <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> is <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>δ</mi><msub><mi>S</mi> <mi>WZW</mi></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mfrac><mi>k</mi><mrow><mn>2</mn><mi>π</mi><mi>i</mi></mrow></mfrac><msub><mo>∫</mo> <mi>Σ</mi></msub><mo stretchy="false">⟨</mo><mo stretchy="false">(</mo><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>δ</mi><mi>g</mi><mo stretchy="false">)</mo><mo>,</mo><mo>∂</mo><mo stretchy="false">(</mo><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mover><mo>∂</mo><mo stretchy="false">¯</mo></mover><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">⟩</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \delta S_{WZW}(g) = -\frac{k}{2 \pi i } \int_\Sigma \langle (g^{-1}\delta g), \partial (g^{-1}\bar \partial g) \rangle \,. </annotation></semantics></math></div> Therefore the classical <a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a> for function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>Σ</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">g \colon \Sigma \to G</annotation></semantics></math> are <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mo stretchy="false">(</mo><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mover><mo>∂</mo><mo stretchy="false">¯</mo></mover><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⇔</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mover><mo>∂</mo><mo stretchy="false">¯</mo></mover><mo stretchy="false">(</mo><mi>g</mi><mo>∂</mo><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \partial(g^{-1}\bar \partial g) = 0 \;\;\;\;\;\; \Leftrightarrow \;\;\;\;\;\; \bar \partial(g \partial g^{-1}) = 0 \,. </annotation></semantics></math></div> The space of solutions to these equations is small. However, discussion of the <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> of the theory (<a href="#HolographyAndRigorousConstruction">below</a>) suggests to consider these equations with the real <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>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> replaced by its <a class="existingWikiWord" href="/nlab/show/complexification">complexification</a> to the <a class="existingWikiWord" href="/nlab/show/complex+Lie+group">complex Lie group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">(</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G({\mathbb{C}})</annotation></semantics></math>. Then the general solution to the equations of motion above has the form <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mover><mi>z</mi><mo stretchy="false">¯</mo></mover><mo stretchy="false">)</mo><mo>=</mo><msub><mi>g</mi> <mi>ℓ</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><msub><mi>g</mi> <mi>r</mi></msub><mo stretchy="false">(</mo><mover><mi>z</mi><mo stretchy="false">¯</mo></mover><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex"> g(z,\bar z) = g_{\ell}(z) g_r(\bar z)^{-1} </annotation></semantics></math></div> where hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mi>ℓ</mi></msub><mo lspace="verythinmathspace">:</mo><mi>Σ</mi><mo>→</mo><mi>G</mi><mo stretchy="false">(</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g_{\ell} \colon \Sigma \to G(\mathbb{C})</annotation></semantics></math> is any <a class="existingWikiWord" href="/nlab/show/holomorphic+function">holomorphic function</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mi>r</mi></msub></mrow><annotation encoding="application/x-tex">g_r</annotation></semantics></math> similarly any anti-holomorphic function. (e.g. <a href="#Gawedzki99">Gawedzki 99 (3.18), (3.19)</a>) <h3 id="HolographyAndRigorousConstruction">Holography and Rigorous construction</h3> By the <a class="existingWikiWord" href="/nlab/show/AdS3-CFT2+and+CS-WZW+correspondence">AdS3-CFT2 and CS-WZW correspondence</a> (see there for more details) the 2d WZW <a class="existingWikiWord" href="/nlab/show/CFT">CFT</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> is related to <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>-<a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>3</mn><mi>d</mi></mrow><annotation encoding="application/x-tex">3d</annotation></semantics></math>. In fact a rigorous constructions of the <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>-WZW model as a <a class="existingWikiWord" href="/nlab/show/rational+2d+CFT">rational 2d CFT</a> is via the <a class="existingWikiWord" href="/nlab/show/FRS-theorem+on+rational+2d+CFT">FRS-theorem on rational 2d CFT</a>, which constructs the model as a <a class="existingWikiWord" href="/nlab/show/boundary+field+theory">boundary field theory</a> of the <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>-<a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> as a <a class="existingWikiWord" href="/nlab/show/3d+TQFT">3d TQFT</a> incarnated via a <a class="existingWikiWord" href="/nlab/show/Reshetikhin-Turaev+construction">Reshetikhin-Turaev construction</a>. <div> <h3 id="BraidRepresentationsViaTwisteddRCohomologyOfConfigurationSpaces">Braid representations via twisted cohomology of configuration spaces</h3> The “<a class="existingWikiWord" href="/nlab/show/hypergeometric+construction+of+KZ+solutions">hypergeometric integral</a>” construction of <a class="existingWikiWord" href="/nlab/show/conformal+blocks">conformal blocks</a> for <a class="existingWikiWord" href="/nlab/show/affine+Lie+algebra">affine Lie algebra</a>/<a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a>-<a class="existingWikiWord" href="/nlab/show/2d+CFTs">2d CFTs</a> and of more general solutions to the <a class="existingWikiWord" href="/nlab/show/Knizhnik-Zamolodchikov+equation">Knizhnik-Zamolodchikov equation</a>, via <a class="existingWikiWord" href="/nlab/show/twisted+de+Rham+cohomology">twisted de Rham cohomology</a> of <a class="existingWikiWord" href="/nlab/show/configuration+spaces+of+points">configuration spaces of points</a>, originates with: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Vadim+Schechtman">Vadim Schechtman</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Varchenko">Alexander Varchenko</a>, Integral representations of N-point conformal correlators in the WZW model, Max-Planck-Institut für Mathematik, (1989) Preprint MPI/89- <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="http://cds.cern.ch/record/1044951">cds:1044951</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li id="DateJimboMatsuoMiwa90"> <a class="existingWikiWord" href="/nlab/show/Etsuro+Date">Etsuro Date</a>, <a class="existingWikiWord" href="/nlab/show/Michio+Jimbo">Michio Jimbo</a>, <a class="existingWikiWord" href="/nlab/show/Atsushi+Matsuo">Atsushi Matsuo</a>, <a class="existingWikiWord" href="/nlab/show/Tetsuji+Miwa">Tetsuji Miwa</a>, Hypergeometric-type integrals and the <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>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{sl}(2,\mathbb{C})</annotation></semantics></math>-Knizhnik-Zamolodchikov equation, International Journal of Modern Physics B 04 05 (1990) 1049-1057 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1142/S0217979290000528">doi:10.1142/S0217979290000528</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> <blockquote> (for <a class="existingWikiWord" href="/nlab/show/sl%282%29"><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>ℂ</mi> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">\mathfrak{sl}(2,\mathbb{C})</annotation> </semantics> </math></a>) </blockquote> </li> <li> <a class="existingWikiWord" href="/nlab/show/Atsushi+Matsuo">Atsushi Matsuo</a>, An application of Aomoto-Gelfand hypergeometric functions to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(n)</annotation></semantics></math> Knizhnik-Zamolodchikov equation, Communications in Mathematical Physics 134 (1990) 65–77 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1007/BF02102089">doi:10.1007/BF02102089</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> <a class="existingWikiWord" href="/nlab/show/Vadim+Schechtman">Vadim Schechtman</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Varchenko">Alexander Varchenko</a>, Hypergeometric solutions of Knizhnik-Zamolodchikov equations, Lett. Math. Phys. 20 (1990) 279–283 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1007/BF00626523">doi:10.1007/BF00626523</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> <a class="existingWikiWord" href="/nlab/show/Vadim+Schechtman">Vadim Schechtman</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Varchenko">Alexander Varchenko</a>, Arrangements of hyperplanes and Lie algebra homology, Inventiones mathematicae 106 1 (1991) 139-194 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://eudml.org/doc/143938">dml:143938</a>, <a class="existingWikiWord" href="/nlab/files/SchechtmanVarchenko_HyperplaneArrangements.pdf" title="pdf">pdf</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> </ul> following precursor observations due to: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Vladimir+S.+Dotsenko">Vladimir S. Dotsenko</a>, <a class="existingWikiWord" href="/nlab/show/Vladimir+A.+Fateev">Vladimir A. Fateev</a>, Conformal algebra and multipoint correlation functions in 2D statistical models, Nuclear Physics B 240 3 (1984) 312-348 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1016/0550-3213(84)90269-4">doi:10.1016/0550-3213(84)90269-4</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> <a class="existingWikiWord" href="/nlab/show/Philippe+Christe">Philippe Christe</a>, <a class="existingWikiWord" href="/nlab/show/Rainald+Flume">Rainald Flume</a>, The four-point correlations of all primary operators of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">d = 2</annotation></semantics></math> conformally invariant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(2)</annotation></semantics></math> <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 Wess-Zumino term, Nuclear Physics B 282 (1987) 466-494 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1016/0550-3213(87)90693-6">doi:10.1016/0550-3213(87)90693-6</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> </ul> The proof that for rational levels this construction indeed yields <a class="existingWikiWord" href="/nlab/show/conformal+blocks">conformal blocks</a> is due to: <ul> <li id="FeiginSchechtmanVarchenko90"> <a class="existingWikiWord" href="/nlab/show/Boris+Feigin">Boris Feigin</a>, <a class="existingWikiWord" href="/nlab/show/Vadim+Schechtman">Vadim Schechtman</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Varchenko">Alexander Varchenko</a>, On algebraic equations satisfied by correlators in Wess-Zumino-Witten models, Lett Math Phys 20 (1990) 291–297 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1007/BF00626525">doi:10.1007/BF00626525</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li id="FeiginSchechtmanVarchenko94"> <a class="existingWikiWord" href="/nlab/show/Boris+Feigin">Boris Feigin</a>, <a class="existingWikiWord" href="/nlab/show/Vadim+Schechtman">Vadim Schechtman</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Varchenko">Alexander Varchenko</a>, On algebraic equations satisfied by hypergeometric correlators in WZW models. I, Commun. Math. Phys. 163 (1994) 173–184 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1007/BF02101739">doi:10.1007/BF02101739</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> <blockquote> (for <a class="existingWikiWord" href="/nlab/show/sl%282%29"><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>ℂ</mi> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">\mathfrak{sl}(2, \mathbb{C})</annotation> </semantics> </math></a>) </blockquote> </li> <li id="FeiginSchechtmanVarchenko95"> <a class="existingWikiWord" href="/nlab/show/Boris+Feigin">Boris Feigin</a>, <a class="existingWikiWord" href="/nlab/show/Vadim+Schechtman">Vadim Schechtman</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Varchenko">Alexander Varchenko</a>, On algebraic equations satisfied by hypergeometric correlators in WZW models. II, Comm. Math. Phys. 170 1 (1995) 219-247 [<a href="https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-170/issue-1/On-algebraic-equations-satisfied-by-hypergeometric-correlators-in-WZW-models/cmp/1104272957.full">euclid:cmp/1104272957</a>] </li> </ul> Review: <ul> <li id="Varchenko95"> <a class="existingWikiWord" href="/nlab/show/Alexander+Varchenko">Alexander Varchenko</a>, Multidimensional Hypergeometric Functions and Representation Theory of Lie Algebras and Quantum Groups, Advanced Series in Mathematical Physics 21, World Scientific 1995 (<a href="https://doi.org/10.1142/2467">doi:10.1142/2467</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Ivan+Cherednik">Ivan Cherednik</a>, Section 8.2 of: Lectures on Knizhnik-Zamolodchikov equations and Hecke algebras, Mathematical Society of Japan Memoirs 1998 (1998) 1-96 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.2969/msjmemoirs/00101C010">doi:10.2969/msjmemoirs/00101C010</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li id="EtingofFrenkelKirillov98"> <a class="existingWikiWord" href="/nlab/show/Pavel+Etingof">Pavel Etingof</a>, <a class="existingWikiWord" href="/nlab/show/Igor+Frenkel">Igor Frenkel</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Kirillov">Alexander Kirillov</a>, Lecture 7 in: Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations, Mathematical surveys and monographs 58, American Mathematical Society (1998) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://bookstore.ams.org/surv-58">ISBN:978-1-4704-1285-2</a>, <a href="http://www.ams.org/journals/bull/2000-37-02/S0273-0979-00-00853-3/S0273-0979-00-00853-3.pdf">review pdf</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> <a class="existingWikiWord" href="/nlab/show/Toshitake+Kohno">Toshitake Kohno</a>, Homological representations of braid groups and KZ connections, Journal of Singularities 5 (2012) 94-108 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="http://dx.doi.org/10.5427/jsing.2012.5g">doi:10.5427/jsing.2012.5g</a>, <a href="https://www.journalofsing.org/volume5/kohno.pdf">pdf</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> <a class="existingWikiWord" href="/nlab/show/Toshitake+Kohno">Toshitake Kohno</a>, Local Systems on Configuration Spaces, KZ Connections and Conformal Blocks, Acta Math Vietnam 39 (2014) 575–598 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://link.springer.com/article/10.1007%2Fs40306-014-0088-6">doi:10.1007%2Fs40306-014-0088-6</a>, <a href="https://www.ms.u-tokyo.ac.jp/~kohno/papers/kohno_config.pdf">pdf</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> <a class="existingWikiWord" href="/nlab/show/Toshitake+Kohno">Toshitake Kohno</a>, Introduction to representation theory of braid groups, Peking 2018 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://www.math.pku.edu.cn/misc/puremath/summerschool/Peking_SummerSchool_kohno.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/KohnoBraidRepresentations.pdf" title="pdf">pdf</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> <blockquote> (motivation from <a class="existingWikiWord" href="/nlab/show/braid+representations">braid representations</a>) </blockquote> </li> </ul> See also: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Alexander+Varchenko">Alexander Varchenko</a>, Asymptotic solutions to the Knizhnik-Zamolodchikov equation and crystal base, Comm. Math. Phys. 171 1 (1995) 99-137 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/hep-th/9403102">arXiv:hep-th/9403102</a>, <a href="https://doi.org/10.1007/BF02103772">doi:10.1007/BF02103772</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> <a class="existingWikiWord" href="/nlab/show/Edward+Frenkel">Edward Frenkel</a>, <a class="existingWikiWord" href="/nlab/show/David+Ben-Zvi">David Ben-Zvi</a>, Section 14.3 in: Vertex Algebras and Algebraic Curves, Mathematical Surveys and Monographs 88, AMS 2004 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://bookstore.ams.org/surv-88-r">ISBN:978-1-4704-1315-6</a>, <a href="https://math.berkeley.edu/~frenkel/BOOK/">web</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> </ul> This “hypergeometric” construction uses results on the <a class="existingWikiWord" href="/nlab/show/twisted+de+Rham+cohomology">twisted de Rham cohomology</a> of <a class="existingWikiWord" href="/nlab/show/configuration+spaces+of+points">configuration spaces of points</a> due to: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Peter+Orlik">Peter Orlik</a>, <a class="existingWikiWord" href="/nlab/show/Louis+Solomon">Louis Solomon</a>, Combinatorics and topology of complements of hyperplanes, Invent Math 56 (1980) 167–189 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1007/BF01392549">doi:10.1007/BF01392549</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kazuhiko+Aomoto">Kazuhiko Aomoto</a>, Gauss-Manin connection of integral of difference products, J. Math. Soc. Japan 39 2 (1987) 191-208 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://projecteuclid.org/journals/journal-of-the-mathematical-society-of-japan/volume-39/issue-2/Gauss-Manin-connection-of-integral-of-difference-products/10.2969/jmsj/03920191.full">doi:10.2969/jmsj/03920191</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> <a class="existingWikiWord" href="/nlab/show/H%C3%A9l%C3%A8ne+Esnault">Hélène Esnault</a>, <a class="existingWikiWord" href="/nlab/show/Vadim+Schechtman">Vadim Schechtman</a>, <a class="existingWikiWord" href="/nlab/show/Eckart+Viehweg">Eckart Viehweg</a>, Cohomology of local systems on the complement of hyperplanes, Inventiones mathematicae 109.1 (1992) 557-561 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="http://page.mi.fu-berlin.de/esnault/preprints/helene/26-es_sch_vi.pdf">pdf</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> <a class="existingWikiWord" href="/nlab/show/Vadim+Schechtman">Vadim Schechtman</a>, H. Terao, <a class="existingWikiWord" href="/nlab/show/Alexander+Varchenko">Alexander Varchenko</a>, Local systems over complements of hyperplanes and the Kac-Kazhdan conditions for singular vectors, Journal of Pure and Applied Algebra 100 1–3 (1995) 93-102 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/hep-th/9411083">arXiv:hep-th/9411083</a>, <a href="https://doi.org/10.1016/0022-4049(95)00014-N">doi:10.1016/0022-4049(95)00014-N</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> </ul> also: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Peter+Orlik">Peter Orlik</a>, Hypergeometric integrals and arrangements, Journal of Computational and Applied Mathematics 105 (1999) 417–424 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1016/S0377-0427(99)00036-9">doi:10.1016/S0377-0427(99)00036-9</a>, <a href="https://core.ac.uk/download/pdf/82631708.pdf">pdf</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> <a class="existingWikiWord" href="/nlab/show/Daniel+C.+Cohen">Daniel C. Cohen</a>, <a class="existingWikiWord" href="/nlab/show/Peter+Orlik">Peter Orlik</a>, Arrangements and local systems, Math. Res. Lett. 7 (2000) 299-316 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/math/9907117">arXiv:math/9907117</a>, <a href="https://dx.doi.org/10.4310/MRL.2000.v7.n3.a5">doi:10.4310/MRL.2000.v7.n3.a5</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> </ul> reviewed in: <ul> <li>Yukihito Kawahara, The twisted de Rham cohomology for basic constructions of hyperplane arrangements and its applications, Hokkaido Math. J. 34 2 (2005) 489-505 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://projecteuclid.org/journals/hokkaido-mathematical-journal/volume-34/issue-2/The-twisted-de-Rham-cohomology-for-basic-constructionsof-hyperplane-arrangements/10.14492/hokmj/1285766233.full">doi:10.14492/hokmj/1285766233</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> Discussion for the special case of <a class="existingWikiWord" href="/nlab/show/level+%28Chern-Simons+theory%29">level</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">=0</annotation></semantics></math> (cf. at <a href="logarithmic+CFT#Examples">logarithmic CFT – Examples</a>): <ul> <li> <a class="existingWikiWord" href="/nlab/show/Fedor+A.+Smirnov">Fedor A. Smirnov</a>, Remarks on deformed and undeformed Knizhnik-Zamolodchikov equations, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/hep-th/9210051">arXiv:hep-th/9210051</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> <a class="existingWikiWord" href="/nlab/show/Fedor+A.+Smirnov">Fedor A. Smirnov</a>, Form factors, deformed Knizhnik-Zamolodchikov equations and finite-gap integration, Communications in Mathematical Physics 155 (1993) 459–487 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1007/BF02096723">doi:10.1007/BF02096723</a>, <a href="https://arxiv.org/abs/hep-th/9210052">arXiv:hep-th/9210052</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> S. Pakuliak, A. Perelomov, Relation Between Hyperelliptic Integrals, Mod. Phys. Lett. 9 19 (1994) 1791-1798 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1142/S0217732394001647">doi:10.1142/S0217732394001647</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> </ul> Interpretation of the <a class="existingWikiWord" href="/nlab/show/hypergeometric+construction+of+KZ+solutions">hypergeometric construction</a> as happening in <a class="existingWikiWord" href="/nlab/show/twisted+equivariant+differential+K-theory">twisted equivariant differential K-theory</a>, showing that the <a class="existingWikiWord" href="/nlab/show/K-theory+classification+of+D-brane+charge">K-theory classification of D-brane charge</a> and the <a class="existingWikiWord" href="/nlab/show/K-theory+classification+of+topological+phases+of+matter">K-theory classification of topological phases of matter</a> both reflect <a class="existingWikiWord" href="/nlab/show/braid+group+representations">braid group representations</a> as expected for <a class="existingWikiWord" href="/nlab/show/defect+branes">defect branes</a> and for <a class="existingWikiWord" href="/nlab/show/anyons">anyons</a>/<a class="existingWikiWord" href="/nlab/show/topological+order">topological order</a>, respectively: <ul> <li><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/Anyonic+defect+branes+in+TED-K-theory">Anyonic defect branes in TED-K-theory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/2203.11838">arXiv:2203.11838</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> </div> <h3 id="DBranes">D-branes for the WZW model</h3> The characterization of <a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a> <a class="existingWikiWord" href="/nlab/show/submanifolds">submanifolds</a> for the <a class="existingWikiWord" href="/nlab/show/open+string">open string</a> WZW model on a <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>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> comes from two consistency conditions: <ol> <li> geometrical condition: For the open string <a class="existingWikiWord" href="/nlab/show/CFT">CFT</a> to still have <a class="existingWikiWord" href="/nlab/show/current+algebra">current algebra</a> <a class="existingWikiWord" href="/nlab/show/worldsheet">worldsheet</a> symmetry, hence for half the current algebra symmetry of the closed WZW string to be preserved, the <a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a> <a class="existingWikiWord" href="/nlab/show/submanifolds">submanifolds</a> need to be <a class="existingWikiWord" href="/nlab/show/conjugacy+classes">conjugacy classes</a> of the group manifold (see e.g. <a href="#AlekseevSchomerus">Alekseev-Schomerus</a> for a brief review and further pointers). These conjugacy classes are therefore also called the symmetric D-branes. Notice that these conjugacy classes are equivalently the <a class="existingWikiWord" href="/nlab/show/leaves">leaves</a> of the <a class="existingWikiWord" href="/nlab/show/foliation">foliation</a> induced by the canonical <a class="existingWikiWord" href="/nlab/show/Cartan-Dirac+structure">Cartan-Dirac structure</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>, hence (by the discussion at <a class="existingWikiWord" href="/nlab/show/Dirac+structure">Dirac structure</a>), the leaves induced by the <a class="existingWikiWord" href="/nlab/show/Lagrangian+dg-submanifold">Lagrangian sub-Lie 2-algebroids</a> of the <a class="existingWikiWord" href="/nlab/show/Courant+Lie+2-algebroid">Courant Lie 2-algebroid</a> which is the <a class="existingWikiWord" href="/nlab/show/higher+gauge+groupoid">higher gauge groupoid</a> (see there) of the background <a class="existingWikiWord" href="/nlab/show/B-field">B-field</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>.(It has been suggested by <a class="existingWikiWord" href="/nlab/show/Chris+Rogers">Chris Rogers</a> that such a foliation be thought of as a higher real <a class="existingWikiWord" href="/nlab/show/polarization">polarization</a>.) </li> <li> cohomological condition: In order for the Kapustin-part of the <a class="existingWikiWord" href="/nlab/show/Freed-Witten-Kapustin+anomaly">Freed-Witten-Kapustin anomaly</a> of the <a class="existingWikiWord" href="/nlab/show/worldsheet">worldsheet</a> <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> of the open WZW string to vanish, the D-brane must be equipped with a <a class="existingWikiWord" href="/nlab/show/Chan-Paton+gauge+field">Chan-Paton gauge field</a>, hence a <a class="existingWikiWord" href="/nlab/show/twisted+unitary+bundle">twisted unitary bundle</a> (<a class="existingWikiWord" href="/nlab/show/bundle+gerbe+module">bundle gerbe module</a>) of some rank <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> for the restriction of the ambient <a class="existingWikiWord" href="/nlab/show/B-field">B-field</a> to the brane. For <a class="existingWikiWord" href="/nlab/show/simply+connected+topological+space">simply connected</a> Lie groups only the rank-1 Chan-Paton gauge fields and their direct sums play a role, and their existence corresponds to a trivialization of the underlying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}U(1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe</a>) of the restriction of the <a class="existingWikiWord" href="/nlab/show/B-field">B-field</a> to the brane. There is then a discrete finite collection of symmetric D-branes = <a class="existingWikiWord" href="/nlab/show/conjugacy+classes">conjugacy classes</a> satisfying this condition, and these are called the integral symmetric D-branes. (<a href="#AlekseevSchomerus">Alekseev-Schomerus</a>, <a href="#GW">Gawedzki-Reis</a>). As observed in <a href="#AlekseevSchomerus">Alekseev-Schomerus</a>, this may be thought of as identifying a D-brane as a variant kind of a <a class="existingWikiWord" href="/nlab/show/Bohr-Sommerfeld+leaf">Bohr-Sommerfeld leaf</a>. More generally, on non-simply connected group manifolds there are nontrivial higher rank <a class="existingWikiWord" href="/nlab/show/twisted+unitary+bundles">twisted unitary bundles</a>/<a class="existingWikiWord" href="/nlab/show/Chan-Paton+gauge+fields">Chan-Paton gauge fields</a> over conjugacy classes and hence the cohomological “integrality” or “Bohr-Sommerfeld”-condition imposed on symmetric D-branes becomes more refined (<a href="#Gawedzki04">Gawedzki 04</a>). </li> </ol> In summary, the <a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a> <a class="existingWikiWord" href="/nlab/show/submanifolds">submanifolds</a> in a Lie group which induce an <a class="existingWikiWord" href="/nlab/show/open+string">open string</a> WZW model that a) has one <a class="existingWikiWord" href="/nlab/show/current+algebra">current algebra</a> symmetry and b) is <a class="existingWikiWord" href="/nlab/show/Freed-Witten-Kapustin+anomaly">Kapustin-anomaly</a>-free are precisely the <a class="existingWikiWord" href="/nlab/show/conjugacy+class">conjugacy class</a>-submanifolds <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> equipped with a <a class="existingWikiWord" href="/nlab/show/twisted+unitary+bundle">twisted unitary bundle</a> for the restriction of the background <a class="existingWikiWord" href="/nlab/show/B-field">B-field</a> to the conjugacy class. <h3 id="quantization">Quantization</h3> on <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> of the WZW model, see at <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantization+of+loop+groups">quantization of loop groups</a>, </li> <li> <a class="existingWikiWord" href="/nlab/show/equivariant+elliptic+cohomology">equivariant elliptic cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantization+of+Chern-Simons+theory">quantization of Chern-Simons theory</a> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/exponentiated+pion+field">exponentiated pion field</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+WZW+terms">geometry of physics – WZW terms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/basic+bundle+gerbe">basic bundle gerbe</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/current+algebra">current algebra</a>, <a class="existingWikiWord" href="/nlab/show/affine+Lie+algebra">affine Lie algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Knizhnik-Zamolodchikov+equation">Knizhnik-Zamolodchikov equation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/coset+WZW+model">coset WZW model</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gauged+WZW+model">gauged WZW model</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/parameterized+WZW+model">parameterized WZW model</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/higher+dimensional+WZW+model">higher dimensional WZW model</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+action+functional">Green-Schwarz action functional</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/analytically+continued+Wess-Zumino-Witten+theory">analytically continued Wess-Zumino-Witten theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Gepner+model">Gepner model</a> </li> </ul> <h2 id="References">References</h2> <h3 id="IntroductionsAndSurveys">Introduction and survey</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/Peter+Goddard">Peter Goddard</a>, <a class="existingWikiWord" href="/nlab/show/David+Olive">David Olive</a>, Kac-Moody and Virasoro algebras in relation to quantum physics, International Journal of Modern Physics A 01 02 (1986) 303-414 [<a href="https://doi.org/10.1142/S0217751X86000149">doi:10.1142/S0217751X86000149</a>, <a href="https://inspirehep.net/literature/18583">spire:18583</a>]</li> </ul> Textbook accounts: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Philippe+Di+Francesco">Philippe Di Francesco</a>, Pierre Mathieu, David Sénéchal, Part C of: Conformal field theory, Springer (1997) [<a href="https://doi.org/10.1007/978-1-4612-2256-9">doi:10.1007/978-1-4612-2256-9</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Bojko+Bakalov">Bojko Bakalov</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Kirillov">Alexander Kirillov</a>, Wess-Zumino-Witten model, chapter 7 of: Lectures on tensor categories and modular functors, University Lecture Series 21, Amer. Math. Soc. (2001) [<a class="existingWikiWord" href="/nlab/files/BakalovKirillov-WZWModel-Ch7OfTensorCat.pdf" title="pdf">pdf</a>, <a href="http://www.math.stonybrook.edu/~kirillov/tensor/tensor.html">web</a>, <a href="https://bookstore.ams.org/view?ProductCode=ULECT/21">ams:ulect/21</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Ralph+Blumenhagen">Ralph Blumenhagen</a>, <a class="existingWikiWord" href="/nlab/show/Erik+Plauschinn">Erik Plauschinn</a>, Chapter 3 of: Introduction to Conformal Field Theory – With Applications to String Theory, Lecture Notes in Physics 779, Springer (2009) [<a href="https://doi.org/10.1007/978-3-642-00450-6">doi:10.1007/978-3-642-00450-6</a>] </li> </ul> Lecture notes: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Patrick+Meessen">Patrick Meessen</a>, Strings Moving on Group Manifolds: The WZW Model [<a href="http://www.unioviedo.es/hepth/people/Patrick/fysica/zooi/WZW_ClassMunoz.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Meessen-WZWModel.pdf" title="pdf">pdf</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Lorenz+Eberhardt">Lorenz Eberhardt</a>, Wess-Zumino-Witten models, lecture notes at <a href="https://conf.itp.phys.ethz.ch/esi-school/">YRISW 2019: A modern primer for 2D CFT</a>, Vienna (2019) [<a href="https://conf.itp.phys.ethz.ch/esi-school/Lecture_notes/WZW%20models.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Eberhardt-WZWModels.pdf" title="pdf">pdf</a>] </li> </ul> A basic introduction also to the super-WZW model (and with an eye towards the corresponding <a class="existingWikiWord" href="/nlab/show/2-spectral+triple">2-spectral triple</a>) is in <ul> <li id="FroehlichGawedzki93"><a class="existingWikiWord" href="/nlab/show/J%C3%BCrg+Fr%C3%B6hlich">Jürg Fröhlich</a>, <a class="existingWikiWord" href="/nlab/show/Krzysztof+Gawedzki">Krzysztof Gawedzki</a>, Conformal Field Theory and Geometry of Strings, extended lecture notes for lecture given at the Mathematical Quantum Theory Conference, Vancouver, Canada, August 4-8 (<a href="http://arxiv.org/abs/hep-th/9310187">arXiv:hep-th/9310187</a>)</li> </ul> A useful account of the WZW model that encompasses both its <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> and <a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a> quantization as well as the <a class="existingWikiWord" href="/nlab/show/current+algebra">current algebra</a> aspects of the QFT is in <ul> <li id="Gawedzki99"><a class="existingWikiWord" href="/nlab/show/Krzysztof+Gaw%C4%99dzki">Krzysztof Gawędzki</a>, Conformal field theory: a case study, in Y. Nutku, C. Saclioglu, T. Turgut (eds.) Conformal Field Theory – New Non-perturbative Methods In String And Field Theory, CRC Press (2000) [<a href="https://arxiv.org/abs/hep-th/9904145">arXiv:hep-th/9904145</a>, <a href="https://doi.org/10.1201/9780429502873">doi:10.1201/9780429502873</a>]</li> </ul> This starts in section 2 as a warmup with describing the 1d QFT of a particle propagating on a group manifold. The <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> of <a class="existingWikiWord" href="/nlab/show/states">states</a> is expressed in terms of the <a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie 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> and its <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> <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">\mathfrak{g}</annotation></semantics></math>. In section 4 the <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> of the 2d WZW model is discussed in analogy to that. In lack of a full formalization of the quantization procedure, the author uses the loop algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝓁</mi><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathcal{l} \mathfrak{g}</annotation></semantics></math> – the <a class="existingWikiWord" href="/nlab/show/affine+Lie+algebra">affine Lie algebra</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">\mathfrak{g}</annotation></semantics></math> as the evident analog that replaces <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">\mathfrak{g}</annotation></semantics></math> and discusses the <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> <a class="existingWikiWord" href="/nlab/show/space+of+states">of states</a> in terms of that. He also indicates how this may be understood as a space of <a class="existingWikiWord" href="/nlab/show/sections">sections</a> of a (<a class="existingWikiWord" href="/nlab/show/prequantum+line+bundle">prequantum</a>) <a class="existingWikiWord" href="/nlab/show/line+bundle">line bundle</a> over the <a class="existingWikiWord" href="/nlab/show/loop+group">loop group</a>. See also <ul> <li> L. Fehér, L. O’Raifeartaigh, P. Ruelle, I. Tsutsui, A. Wipf, On Hamiltonian reductions of the Wess-Zumino-Novikov-Witten theories, Phys. Rep. 222 (1992), no. 1, 64 pp. <a href="http://www.ams.org/mathscinet-getitem?mr=1192998">MR93i:81225</a>, <a href="http://dx.doi.org/10.1016/0370-1573(92)90026-V">doi</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Matthias+Blau">Matthias Blau</a>, <a class="existingWikiWord" href="/nlab/show/George+Thompson">George Thompson</a>, Equivariant Kähler Geometry and Localization in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">G/G</annotation></semantics></math> Model, Nucl. Phys. B 439 (1995) 367-394 [<a href="https://doi.org/10.1016/0550-3213(95)00058-Z">doi:10.1016/0550-3213(95)00058-Z</a>, <a href="https://arxiv.org/abs/hep-th/9407042">arXiv:hep-th/9407042</a>] <blockquote> (<a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetric</a> and using <a class="existingWikiWord" href="/nlab/show/group+algebra">group algebra</a>) </blockquote> </li> <li> <a class="existingWikiWord" href="/nlab/show/Krzysztof+Gawedzki">Krzysztof Gawedzki</a>, Rafal Suszek, <a class="existingWikiWord" href="/nlab/show/Konrad+Waldorf">Konrad Waldorf</a>, Global gauge anomalies in two-dimensional bosonic sigma models (<a href="http://arxiv.org/abs/1003.4154">arXiv:1003.4154</a>) </li> <li> Paul de Fromont, <a class="existingWikiWord" href="/nlab/show/Krzysztof+Gaw%C4%99dzki">Krzysztof Gawędzki</a>, Clément Tauber, Global gauge anomalies in coset models of conformal field theory (<a href="http://arxiv.org/abs/1301.2517">arXiv:1301.2517</a>) </li> </ul> <div> <h3 id="WZWTermOfChiralPerturbationTheoryReferences">The WZW term of QCD chiral perturbation theory</h3> The <a class="existingWikiWord" href="/nlab/show/gauged+WZW+model">gauged</a> <a class="existingWikiWord" href="/nlab/show/WZW+term">WZW term</a> of <a class="existingWikiWord" href="/nlab/show/chiral+perturbation+theory">chiral perturbation theory</a>/<a class="existingWikiWord" href="/nlab/show/quantum+hadrodynamics">quantum hadrodynamics</a> which reproduces the <a class="existingWikiWord" href="/nlab/show/chiral+anomaly">chiral anomaly</a> of <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a> in the <a class="existingWikiWord" href="/nlab/show/effective+field+theory">effective field theory</a> of <a class="existingWikiWord" href="/nlab/show/mesons">mesons</a> and <a class="existingWikiWord" href="/nlab/show/Skyrmions">Skyrmions</a>: <h4 id="WZWTermOfChiralPerturbationTheoryReferencesGeneral">General</h4> The original articles: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Julius+Wess">Julius Wess</a>, <a class="existingWikiWord" href="/nlab/show/Bruno+Zumino">Bruno Zumino</a>, Consequences of anomalous Ward identities, Phys. Lett. B 37 (1971) 95-97 (<a href="https://inspirehep.net/literature/67330">spire:67330</a>, <a href="https://doi.org/10.1016/0370-2693(71)90582-X">doi:10.1016/0370-2693(71)90582-X</a>) </li> <li id="Witten83a"> <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, Global aspects of current algebra, Nuclear Physics B Volume 223, Issue 2, 22 August 1983, Pages 422-432 (<a href="https://doi.org/10.1016/0550-3213(83)90063-9">doi:10.1016/0550-3213(83)90063-9</a>) </li> </ul> See also: <ul> <li>O. Kaymakcalan, S. Rajeev, J. Schechter, Nonabelian Anomaly and Vector Meson Decays, Phys. Rev. D 30 (1984) 594 (<a href="https://inspirehep.net/literature/194756">spire:194756</a>)</li> </ul> Corrections and streamlining of the computations: <ul> <li> Chou Kuang-chao, Guo Han-ying, Wu Ke, Song Xing-kang, On the gauge invariance and anomaly-free condition of the Wess-Zumino-Witten effective action, Physics Letters B Volume 134, Issues 1–2, 5 January 1984, Pages 67-69 (<a href="https://doi.org/10.1016/0370-2693(84)90986-9">doi:10.1016/0370-2693(84)90986-9</a>)) </li> <li> H. Kawai, S.-H. H. Tye, Chiral anomalies, effective lagrangians and differential geometry, Physics Letters B Volume 140, Issues 5–6, 14 June 1984, Pages 403-407 (<a href="https://doi.org/10.1016/0370-2693(84)90780-9">doi:10.1016/0370-2693(84)90780-9</a>) </li> <li> J. L. Mañes, Differential geometric construction of the gauged Wess-Zumino action, Nuclear Physics B Volume 250, Issues 1–4, 1985, Pages 369-384 (<a href="https://doi.org/10.1016/0550-3213(85)90487-0">doi:10.1016/0550-3213(85)90487-0</a>) </li> <li> Tomáš Brauner, Helena Kolešová, Gauged Wess-Zumino terms for a general coset space, Nuclear Physics B Volume 945, August 2019, 114676 (<a href="https://doi.org/10.1016/j.nuclphysb.2019.114676">doi:10.1016/j.nuclphysb.2019.114676</a>) </li> </ul> See also <ul> <li>Yasunori Lee, <a class="existingWikiWord" href="/nlab/show/Kantaro+Ohmori">Kantaro Ohmori</a>, <a class="existingWikiWord" href="/nlab/show/Yuji+Tachikawa">Yuji Tachikawa</a>, Revisiting Wess-Zumino-Witten terms (<a href="https://arxiv.org/abs/2009.00033">arXiv:2009.00033</a>)</li> </ul> Interpretation as <a class="existingWikiWord" href="/nlab/show/Skyrmion">Skyrmion</a>/<a class="existingWikiWord" href="/nlab/show/baryon+current">baryon current</a>: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Jeffrey+Goldstone">Jeffrey Goldstone</a>, <a class="existingWikiWord" href="/nlab/show/Frank+Wilczek">Frank Wilczek</a>, Fractional Quantum Numbers on Solitons, Phys. Rev. Lett. 47, 986 (1981) (<a href="https://doi.org/10.1103/PhysRevLett.47.986">doi:10.1103/PhysRevLett.47.986</a>) </li> <li id="Witten83b"> <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, Current algebra, baryons, and quark confinement, Nuclear Physics B Volume 223, Issue 2, 22 August 1983, Pages 433-444 (<a href="https://doi.org/10.1016/0550-3213(83)90064-0">doi:10.1016/0550-3213(83)90064-0</a>) </li> <li id="AdkinsNappi84"> <a class="existingWikiWord" href="/nlab/show/Gregory+Adkins">Gregory Adkins</a>, <a class="existingWikiWord" href="/nlab/show/Chiara+Nappi">Chiara Nappi</a>, Stabilization of Chiral Solitons via Vector Mesons, Phys. Lett. 137B (1984) 251-256 (<a href="http://inspirehep.net/record/194727">spire:194727</a>, <a href="https://doi.org/10.1016/0370-2693(84)90239-9">doi:10.1016/0370-2693(84)90239-9</a>) (beware that the two copies of the text at these two sources differ!) </li> <li id="RhoEtAl16"> <a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a> et al., Introduction, In: <a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a> et al. (eds.) <a class="existingWikiWord" href="/nlab/show/The+Multifaceted+Skyrmion">The Multifaceted Skyrmion</a>, World Scientific 2016 (<a href="https://doi.org/10.1142/9710">doi:10.1142/9710</a>) </li> </ul> Concrete form for <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>-<a class="existingWikiWord" href="/nlab/show/flavor+%28particle+physics%29">flavor</a> <a class="existingWikiWord" href="/nlab/show/quantum+hadrodynamics">quantum hadrodynamics</a> in 2d: <ul> <li>C. R. Lee, H. C. Yen, A Derivation of The Wess-Zumino-Witten Action from Chiral Anomaly Using Homotopy Operators, Chinese Journal of Physics, Vol 23 No. 1 (1985) (<a href="https://inspirehep.net/literature/16389">spire:16389</a>, <a class="existingWikiWord" href="/nlab/files/LeeYenWZW85.pdf" title="pdf">pdf</a>)</li> </ul> Concrete form for 2 <a class="existingWikiWord" href="/nlab/show/flavor+physics">flavors</a> in 4d: <ul> <li>Masashi Wakamatsu, On the electromagnetic hadron current derived from the gauged Wess-Zumino-Witten action, (<a href="https://arxiv.org/abs/1108.1236">arXiv:1108.1236</a>, <a href="https://inspirehep.net/literature/922302">spire:922302</a>)</li> </ul> <h4 id="WZWTermOfChiralPerturbationTheoryReferencesIncludingLightVectorMesons">Including light vector mesons</h4> Concrete form for 2-<a class="existingWikiWord" href="/nlab/show/flavor+%28particle+physics%29">flavor</a> <a class="existingWikiWord" href="/nlab/show/quantum+hadrodynamics">quantum hadrodynamics</a> in 4d with <a class="existingWikiWord" href="/nlab/show/light+meson">light</a> <a class="existingWikiWord" href="/nlab/show/vector+mesons">vector mesons</a> included (<a class="existingWikiWord" href="/nlab/show/omega-meson">omega-meson</a> and <a class="existingWikiWord" href="/nlab/show/rho-meson">rho-meson</a>): <ul> <li id="MeissnerZahed86"> <a class="existingWikiWord" href="/nlab/show/Ulf-G.+Meissner">Ulf-G. Meissner</a>, <a class="existingWikiWord" href="/nlab/show/Ismail+Zahed">Ismail Zahed</a>, equation (6) in: Skyrmions in the Presence of Vector Mesons, Phys. Rev. Lett. 56, 1035 (1986) (<a href="https://doi.org/10.1103/PhysRevLett.56.1035">doi:10.1103/PhysRevLett.56.1035</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Ulf-G.+Meissner">Ulf-G. Meissner</a>, <a class="existingWikiWord" href="/nlab/show/Norbert+Kaiser">Norbert Kaiser</a>, <a class="existingWikiWord" href="/nlab/show/Wolfram+Weise">Wolfram Weise</a>, equation (2.18) in: Nucleons as skyrme solitons with vector mesons: Electromagnetic and axial properties, Nuclear Physics A Volume 466, Issues 3–4, 11–18 May 1987, Pages 685-723 (<a href="https://doi.org/10.1016/0375-9474(87)90463-5">doi:10.1016/0375-9474(87)90463-5</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Ulf-G.+Meissner">Ulf-G. Meissner</a>, equation (2.45) in: Low-energy hadron physics from effective chiral Lagrangians with vector mesons, Physics Reports Volume 161, Issues 5–6, May 1988, Pages 213-361 (<a href="https://doi.org/10.1016/0370-1573(88)90090-7">doi:10.1016/0370-1573(88)90090-7</a>) </li> <li id="Kaiser00"> Roland Kaiser, equation (12) in: Anomalies and WZW-term of two-flavour QCD, Phys. Rev. D63:076010, 2001 (<a href="https://arxiv.org/abs/hep-ph/0011377">arXiv:hep-ph/0011377</a>, <a href="https://inspirehep.net/literature/537600">spire:537600</a>) </li> </ul> <h4 id="WZWTermOfChiralPerturbationTheoryReferencesIncludingHeavyScalarMesons">Including heavy scalar mesons</h4> Including <a class="existingWikiWord" href="/nlab/show/heavy+mesons">heavy</a> <a class="existingWikiWord" href="/nlab/show/scalar+mesons">scalar mesons</a>: specifically <a class="existingWikiWord" href="/nlab/show/kaons">kaons</a>: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Curtis+Callan">Curtis Callan</a>, <a class="existingWikiWord" href="/nlab/show/Igor+Klebanov">Igor Klebanov</a>, equation (4.1) in: Bound-state approach to strangeness in the Skyrme model, Nuclear Physics B Volume 262, Issue 2, 16 December 1985, Pages 365-382 (<a href="https://doi.org/10.1016/0550-3213(85)90292-5">doi10.1016/0550-3213(85)90292-5</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Igor+Klebanov">Igor Klebanov</a>, equation (99) of: Strangeness in the Skyrme model, in: D. Vauthrin, F. Lenz, J. W. Negele, Hadrons and Hadronic Matter, Plenum Press 1989 (<a href="https://link.springer.com/book/10.1007/978-1-4684-1336-6">doi:10.1007/978-1-4684-1336-6</a>) </li> <li> N. N. Scoccola, D. P. Min, H. Nadeau, <a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a>, equation (2.20) in: The strangeness problem: An <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(3)</annotation></semantics></math> skyrmion with vector mesons, Nuclear Physics A Volume 505, Issues 3–4, 25 December 1989, Pages 497-524 (<a href="https://doi.org/10.1016/0375-9474(89)90029-8">doi:10.1016/0375-9474(89)90029-8</a>) </li> </ul> specifically <a class="existingWikiWord" href="/nlab/show/D-mesons">D-mesons</a>: (…) specifically <a class="existingWikiWord" href="/nlab/show/B-mesons">B-mesons</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a>, D. O. Riska, N. N. Scoccola, above (2.1) in: The energy levels of the heavy flavour baryons in the topological soliton model, Zeitschrift für Physik A Hadrons and Nuclei volume 341, pages343–352 (1992) (<a href="https://doi.org/10.1007/BF01283544">doi:10.1007/BF01283544</a>)</li> </ul> <h4 id="WZWTermOfChiralPerturbationTheoryReferencesIncludingHeavyVectorMesons">Including heavy vector mesons</h4> Inclusion of <a class="existingWikiWord" href="/nlab/show/heavy+mesons">heavy</a> <a class="existingWikiWord" href="/nlab/show/vector+mesons">vector mesons</a>: specifically <a class="existingWikiWord" href="/nlab/show/K%2A-mesons">K*-mesons</a>: <ul> <li>S. Ozaki, H. Nagahiro, <a class="existingWikiWord" href="/nlab/show/Atsushi+Hosaka">Atsushi Hosaka</a>, Equations (3) and (9) in: Magnetic interaction induced by the anomaly in kaon-photoproductions, Physics Letters B Volume 665, Issue 4, 24 July 2008, Pages 178-181 (<a href="https://arxiv.org/abs/0710.5581">arXiv:0710.5581</a>, <a href="https://doi.org/10.1016/j.physletb.2008.06.020">doi:10.1016/j.physletb.2008.06.020</a>)</li> </ul> <h4 id="WZWTermOfChiralPerturbationTheoryReferencesIncludingElectroweakInteractions">Including electroweak interactions</h4> Including <a class="existingWikiWord" href="/nlab/show/electroweak+fields">electroweak fields</a>: <ul> <li> J. Bijnens, G. Ecker, A. Picha, The chiral anomaly in non-leptonic weak interactions, Physics Letters B Volume 286, Issues 3–4, 30 July 1992, Pages 341-347 (<a href="https://doi.org/10.1016/0370-2693(92)91785-8">doi:10.1016/0370-2693(92)91785-8</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Gerhard+Ecker">Gerhard Ecker</a>, <a class="existingWikiWord" href="/nlab/show/Helmut+Neufeld">Helmut Neufeld</a>, <a class="existingWikiWord" href="/nlab/show/Antonio+Pich">Antonio Pich</a>, Non-leptonic kaon decays and the chiral anomaly, Nuclear Physics B Volume 413, Issues 1–2, 31 January 1994, Pages 321-352 (<a href="https://doi.org/10.1016/0550-3213(94)90623-8">doi:10.1016/0550-3213(94)90623-8</a>) </li> </ul> Discussion for the full <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Jeffrey+Harvey">Jeffrey Harvey</a>, Christopher T. Hill, Richard J. Hill, Standard Model Gauging of the WZW Term: Anomalies, Global Currents and pseudo-Chern-Simons Interactions, Phys. Rev. D77:085017, 2008 (<a href="https://arxiv.org/abs/0712.1230">arXiv:0712.1230</a>)</li> </ul> </div> <h3 id="interpretation_via_cft_and_gerbes">Interpretation via CFT and gerbes</h3> Interpretation of the 3d WZW term as defining a <a class="existingWikiWord" href="/nlab/show/2d+CFT">2d CFT</a> <ul> <li id="Witten84"> <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, Non-Abelian bosonization in two dimensions Commun. Math. Phys. 92, 455 (1984) </li> <li id="KnizhnikZamolodchikov85"> <a class="existingWikiWord" href="/nlab/show/Vadim+Knizhnik">Vadim Knizhnik</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Zamolodchikov">Alexander Zamolodchikov</a>, Current algebra and Wess-Zumino model in two dimensions, Nucl. Phys. B 247, 83-103 (1984) </li> </ul> and hence as part of a <a class="existingWikiWord" href="/nlab/show/perturbative+string+theory+vacuum">perturbative string theory vacuum</a>/<a class="existingWikiWord" href="/nlab/show/target+space">target space</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Doron+Gepner">Doron Gepner</a>, <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, String theory on group manifolds, Nucl. Phys. B 278, 493-549 (1986) (<a href="http://inspirehep.net/record/230076">spire:230076</a>)</li> </ul> The WZ term on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\Sigma_2</annotation></semantics></math> was understood in terms of an integral of a 3-form over a cobounding manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\Sigma_3</annotation></semantics></math> in <ul> <li><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, Global aspects of current algebra, Nucl. Phys. B223, 422 (1983) (<a href="http://inspirehep.net/record/13234">spire:13234</a>, <a href="https://doi.org/10.1016/0550-3213(83)90063-9">doi:10.1016/0550-3213(83)90063-9</a>, <a href="https://www.phys.sinica.edu.tw/~spring8/users/jychen/pub/reference/NPB_v223_422.pdf">pdf</a>)</li> </ul> for the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\Sigma_2</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/closed+manifold">closed</a>, and generally, in terms of <a class="existingWikiWord" href="/nlab/show/surface+holonomy">surface holonomy</a> of <a class="existingWikiWord" href="/nlab/show/bundle+gerbes">bundle gerbes</a>/<a class="existingWikiWord" href="/nlab/show/circle+2-bundles+with+connection">circle 2-bundles with connection</a> in <ul> <li id="Gawedzki87"> <a class="existingWikiWord" href="/nlab/show/Krzysztof+Gawedzki">Krzysztof Gawedzki</a>, Topological Actions in two-dimensional Quantum Field Theories, in <a class="existingWikiWord" href="/nlab/show/Gerard+%27t+Hooft">Gerard 't Hooft</a> et. al (eds.) Nonperturbative quantum field theory Cargese 1987 proceedings, (<a href="http://inspirehep.net/record/257658?ln=de">web</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Giovanni+Felder">Giovanni Felder</a> , <a class="existingWikiWord" href="/nlab/show/Krzysztof+Gawedzki">Krzysztof Gawedzki</a>, A. Kupianen, Spectra of Wess-Zumino-Witten models with arbitrary simple groups. Commun. Math. Phys. 117, 127 (1988) </li> <li> <a class="existingWikiWord" href="/nlab/show/Krzysztof+Gawedzki">Krzysztof Gawedzki</a>, Topological actions in two-dimensional quantum field theories. In: Nonperturbative quantum field theory. ‘tHooft, G. et al. (eds.). London: Plenum Press 1988 </li> </ul> as the <a class="existingWikiWord" href="/nlab/show/surface+holonomy">surface holonomy</a> of a <a class="existingWikiWord" href="/nlab/show/circle+2-bundle+with+connection">circle 2-bundle with connection</a>. See also the references at <a class="existingWikiWord" href="/nlab/show/B-field">B-field</a> and at <a class="existingWikiWord" href="/nlab/show/Freed-Witten+anomaly+cancellation">Freed-Witten anomaly cancellation</a>. See also <ul> <li id="DeligneFreed99"> <a class="existingWikiWord" href="/nlab/show/Pierre+Deligne">Pierre Deligne</a>, <a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, chapter 6 of Classical field theory (1999) (<a href="https://publications.ias.edu/sites/default/files/79_ClassicalFieldTheory.pdf">pdf</a>) this is a chapter in <a class="existingWikiWord" href="/nlab/show/Pierre+Deligne">P. Deligne</a>, <a class="existingWikiWord" href="/nlab/show/Pavel+Etingof">P. Etingof</a>, <a class="existingWikiWord" href="/nlab/show/Dan+Freed">D.S. Freed</a>, L. Jeffrey, <a class="existingWikiWord" href="/nlab/show/David+Kazhdan">D. Kazhdan</a>, J. Morgan, D.R. Morrison, <a class="existingWikiWord" href="/nlab/show/Edward+Witten">E. Witten</a> (eds.) <a class="existingWikiWord" href="/nlab/show/Quantum+Fields+and+Strings">Quantum Fields and Strings</a>, A course for mathematicians, 2 vols. Amer. Math. Soc. Providence 1999. (<a href="http://www.math.ias.edu/qft">web version</a>) </li> </ul> For the fully general understanding as the <a class="existingWikiWord" href="/nlab/show/surface+holonomy">surface holonomy</a> of a <a class="existingWikiWord" href="/nlab/show/circle+2-bundle+with+connection">circle 2-bundle with connection</a> see the references <a href="#ReferencesRelationToGerbesAndCS">below</a>. See also <ul> <li><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, On holomorphic factorization of WZW and coset models, Comm. Math. Phys. Volume 144, Number 1 (1992), 189-212. (<a href="http://projecteuclid.org/euclid.cmp/1104249222">Euclid</a>)</li> </ul> <h3 id="ReferencesRelationToGerbesAndCS">Relation to gerbes and Chern-Simons theory</h3> Discussion of <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 2-bundles with connection</a> (expressed in terms of <a class="existingWikiWord" href="/nlab/show/bundle+gerbes">bundle gerbes</a>) and discussion of the WZW-background <a class="existingWikiWord" href="/nlab/show/B-field">B-field</a> (<a class="existingWikiWord" href="/nlab/show/WZW+term">WZW term</a>) in this language (cf. <a class="existingWikiWord" href="/nlab/show/basic+bundle+gerbe">basic bundle gerbe</a>) <ul> <li id="GW"> <a class="existingWikiWord" href="/nlab/show/Krzysztof+Gaw%C4%99dzki">Krzysztof Gawędzki</a>, <a class="existingWikiWord" href="/nlab/show/Nuno+Reis">Nuno Reis</a>, WZW branes and gerbes, Rev. Math. Phys. 14 (2002) 1281-1334 [<a href="https://arxiv.org/abs/hep-th/0205233">arXiv:hep-th/0205233</a>, <a href="https://doi.org/10.1142/S0129055X02001557">doi:10.1142/S0129055X02001557</a>] </li> <li id="SchweigertWaldorf07"> <a class="existingWikiWord" href="/nlab/show/Christoph+Schweigert">Christoph Schweigert</a>, <a class="existingWikiWord" href="/nlab/show/Konrad+Waldorf">Konrad Waldorf</a>, Gerbes and Lie Groups, in Trends and Developments in Lie Theory, Progress in Math., Birkhäuser (<a href="http://arxiv.org/abs/0710.5467">arXiv:0710.5467</a>) </li> </ul> Discussion of how this 2-bundle arises from the <a class="existingWikiWord" href="/nlab/show/Chern-Simons+circle+3-bundle">Chern-Simons circle 3-bundle</a> is in <ul> <li><a class="existingWikiWord" href="/nlab/show/Alan+Carey">Alan Carey</a>, Stuart Johnson, <a class="existingWikiWord" href="/nlab/show/Michael+Murray">Michael Murray</a>, <a class="existingWikiWord" href="/nlab/show/Danny+Stevenson">Danny Stevenson</a>, <a class="existingWikiWord" href="/nlab/show/Bai-Ling+Wang">Bai-Ling Wang</a>, Bundle gerbes for Chern-Simons and Wess-Zumino-Witten theories Commun.Math.Phys. 259 (2005) 577-613 (<a href="http://arxiv.org/abs/math/0410013">arXiv:math/0410013</a>)</li> </ul> and related discussion is in <ul> <li><a class="existingWikiWord" href="/nlab/show/Konrad+Waldorf">Konrad Waldorf</a>, Multiplicative Bundle Gerbes with Connection , Differential Geom. Appl. 28(3), 313-340 (2010) (<a href="http://arxiv.org/abs/0804.4835">arXiv:0804.4835</a>)</li> </ul> See also Section 2.3.18 and section 4.7 of <ul> <li><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a> (<a href="http://ncatlab.org/schreiber/files/Erlangen2011Schreiber.pdf">pdf slides</a>).</li> </ul> <h3 id="partition_functions">Partition functions</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/Terry+Gannon">Terry Gannon</a>, Partition Functions for Heterotic WZW Conformal Field Theories, Nucl.Phys. B402 (1993) 729-753 (<a href="http://arxiv.org/abs/hep-th/9209042">arXiv:hep-th/9209042</a>)</li> </ul> <h3 id="ReferencesDBranes">D-branes for the WZW model</h3> A characterization of <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> in the WZW model as those <a class="existingWikiWord" href="/nlab/show/conjugacy+classes">conjugacy classes</a> that in addition satisfy an integrality (<a class="existingWikiWord" href="/nlab/show/Bohr-Sommerfeld+quantization">Bohr-Sommerfeld</a>-type) condition missed in other parts of the literature is stated in <ul> <li id="AlekseevSchomerus"><a class="existingWikiWord" href="/nlab/show/Anton+Alekseev">Anton Alekseev</a>, <a class="existingWikiWord" href="/nlab/show/Volker+Schomerus">Volker Schomerus</a>, D-branes in the WZW model, Phys.Rev.D60:061901,1999 (<a href="http://arxiv.org/abs/hep-th/9812193v2">arXiv:hep-th/9812193v2</a>)</li> </ul> The refined interpretation of the integrality condition as a choice of trivialization of the underling <a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>/<a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe</a> of the <a class="existingWikiWord" href="/nlab/show/B-field">B-field</a> over the brane was then noticed in section 7 of <ul> <li><a class="existingWikiWord" href="/nlab/show/Krzysztof+Gawedzki">Krzysztof Gawedzki</a>, Nuno Reis, WZW branes and gerbes, Rev.Math.Phys. 14 (2002) 1281-1334 (<a href="http://arxiv.org/abs/hep-th/0205233">arXiv:hep-th/0205233</a>)</li> </ul> The observation that this is just the special rank-1 case of the more general structure provided by a <a class="existingWikiWord" href="/nlab/show/twisted+unitary+bundle">twisted unitary bundle</a> of some rank <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> on the D-brane (<a class="existingWikiWord" href="/nlab/show/gerbe+module">gerbe module</a>) which is twisted by the restriction of the <a class="existingWikiWord" href="/nlab/show/B-field">B-field</a> to the D-brane – the <a class="existingWikiWord" href="/nlab/show/Chan-Paton+gauge+field">Chan-Paton gauge field</a> – is due to <ul> <li id="Gawedzki04"><a class="existingWikiWord" href="/nlab/show/Krzysztof+Gawedzki">Krzysztof Gawedzki</a>, Abelian and non-Abelian branes in WZW models and gerbes, Commun.Math.Phys. 258 (2005) 23-73 (<a href="http://arxiv.org/abs/hep-th/0406072">arXiv:hep-th/0406072</a>).</li> </ul> The observation that the “multiplicative” structure of the WZW-<a class="existingWikiWord" href="/nlab/show/B-field">B-field</a> (induced from it being the <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a> of the <a class="existingWikiWord" href="/nlab/show/Chern-Simons+circle+3-bundle">Chern-Simons circle 3-connection</a> over the <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</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>-<a class="existingWikiWord" href="/nlab/show/principal+connections">principal connections</a>) induces the <a class="existingWikiWord" href="/nlab/show/Verlinde+ring">Verlinde ring</a> fusion product structure on symmetric D-branes equipped with <a class="existingWikiWord" href="/nlab/show/Chan-Paton+gauge+fields">Chan-Paton gauge fields</a> is discussed in <ul> <li><a class="existingWikiWord" href="/nlab/show/Alan+Carey">Alan Carey</a>, <a class="existingWikiWord" href="/nlab/show/Bai-Ling+Wang">Bai-Ling Wang</a>, Fusion of symmetric <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-branes and Verlinde rings, Commun. Math. Phys.277:577-625 (2008) (<a href="http://arxiv.org/abs/math-ph/0505040">arXiv:math-ph/0505040</a>)</li> </ul> The image in <a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a> of these <a class="existingWikiWord" href="/nlab/show/Chan-Paton+gauge+fields">Chan-Paton gauge fields</a> over conjugacy classes is shown to generate the <a class="existingWikiWord" href="/nlab/show/Verlinde+ring">Verlinde ring</a>/the <a class="existingWikiWord" href="/nlab/show/positive+energy+representations">positive energy representations</a> of the <a class="existingWikiWord" href="/nlab/show/loop+group">loop group</a> in <ul> <li><a class="existingWikiWord" href="/nlab/show/Eckhard+Meinrenken">Eckhard Meinrenken</a>, On the quantization of conjugacy classes, Enseign. Math. (2) 55 (2009), no. 1-2, 33-75 (<a href="http://arxiv.org/abs/0707.3963">arXiv:0707.3963</a>)</li> </ul> Formalization of WZW terms in <a class="existingWikiWord" href="/nlab/show/cohesive+%28infinity%2C1%29-topos">cohesive homotopy theory</a> is in <ul> <li id="dcct"><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></li> </ul> <h3 id="relation_to_dimensional_reduction_of_chernsimons">Relation to dimensional reduction of Chern-Simons</h3> One can also obtain the WZW-model by <a class="existingWikiWord" href="/nlab/show/KK-reduction">KK-reduction</a> from <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a>. E.g. <ul> <li><a class="existingWikiWord" href="/nlab/show/Matthias+Blau">Matthias Blau</a>, G. Thompson, Derivation of the Verlinde Formula from Chern-Simons Theory and the G/G model, Nucl.Phys. B408 (1993) 345-390 (<a href="http://arxiv.org/abs/hep-th/9305010">arXiv:hep-th/9305010</a>)</li> </ul> A discussion in <a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher differential geometry</a> via <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a> in <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a> is in <ul> <li> <a class="existingWikiWord" href="/schreiber/show/Extended+higher+cup-product+Chern-Simons+theories">Extended higher cup-product Chern-Simons theories</a> </li> <li> <a class="existingWikiWord" href="/schreiber/show/A+higher+stacky+perspective+on+Chern-Simons+theory">A higher stacky perspective on Chern-Simons theory</a> </li> </ul> <h3 id="relation_to_extended_tqft">Relation to extended TQFT</h3> Relation to <a class="existingWikiWord" href="/nlab/show/extended+TQFT">extended TQFT</a> is discussed in <ul> <li><a class="existingWikiWord" href="/nlab/show/Dan+Freed">Dan Freed</a>, <a class="existingWikiWord" href="/nlab/show/4-3-2+8-7-6">4-3-2 8-7-6</a></li> </ul> For a formulation of the WZW term in the presence of D-branes as an open-closed smooth <a class="existingWikiWord" href="/nlab/show/functorial+field+theory">functorial field theory</a>: <ul> <li id="BunkWaldorf21a"> <a class="existingWikiWord" href="/nlab/show/Severin+Bunk">Severin Bunk</a>, <a class="existingWikiWord" href="/nlab/show/Konrad+Waldorf">Konrad Waldorf</a>, Transgression of D-branes, Adv. Theor. Math. Phys. 25 5 (2021) 1095-1198 [<a href="https://arxiv.org/abs/arXiv:1808.04894">arXiv:1808.04894</a>, <a href="https://dx.doi.org/10.4310/ATMP.2021.v25.n5.a1">doi:10.4310/ATMP.2021.v25.n5.a1</a>] </li> <li id="BunkWaldorf21b"> <a class="existingWikiWord" href="/nlab/show/Severin+Bunk">Severin Bunk</a>, <a class="existingWikiWord" href="/nlab/show/Konrad+Waldorf">Konrad Waldorf</a>, Smooth functorial field theories from B-fields and D-branes, J. Homot. Rel. Struc. 16 1 (2021) 75-153 [<a href="https://doi.org/10.1007/s40062-020-00272-2">doi:10.1007/s40062-020-00272-2</a>, <a href="https://arxiv.org/abs/arXiv:1911.09990">arXiv:1911.09990</a>] </li> </ul> The formulation of the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+action+functional">Green-Schwarz action functional</a> for <a class="existingWikiWord" href="/nlab/show/superstrings">superstrings</a> (and other <a class="existingWikiWord" href="/nlab/show/branes">branes</a> of <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a>/<a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a>) as WZW-models (and <a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Wess-Zumino-Witten+theory">∞-WZW models</a>) on (<a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebra">super L-∞ algebra</a> <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+extensions">L-∞ extensions</a> of) the <a class="existingWikiWord" href="/nlab/show/super+translation+group">super translation group</a> is in <ul> <li id="FSS12"><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <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/The+brane+bouquet">Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields</a>, International Journal of Geometric Methods in Modern Physics, Volume 12, Issue 02 (2015) 1550018 (<a href="http://arxiv.org/abs/1308.5264">arXiv:1308.5264</a>)</li> </ul> <h3 id="in_solid_state_physics">In solid state physics</h3> The low-energy physics of a Heisenberg antiferromagnetic spin chain is argued to be described by a WZW model in <ul> <li>Zheng-Xin Liu, Guang-Ming Zhang, Classification of quantum critical states of integrable antiferromagnetic spin chains and their correspondent two-dimensional topological phases (<a href="http://arxiv.org/abs/1211.5421">arXiv:1211.5421</a>)</li> </ul> See also section 7.10 of Fradkin’s book. Discussion of <a class="existingWikiWord" href="/nlab/show/symmetry+protected+topological+order">symmetry protected topological order</a> phases of matter in <a class="existingWikiWord" href="/nlab/show/solid+state+physics">solid state physics</a> via <a class="existingWikiWord" href="/nlab/show/higher+dimensional+WZW+models">higher dimensional WZW models</a> is in <ul> <li id="CGLW11">Xie Chen, Zheng-Cheng Gu, Zheng-Xin Liu, <a class="existingWikiWord" href="/nlab/show/Xiao-Gang+Wen">Xiao-Gang Wen</a>, Symmetry protected topological orders and the group cohomology of their symmetry group, Phys. Rev. B 87, 155114 (2013) <a href="http://arxiv.org/abs/1106.4772">arXiv:1106.4772</a>; A short version in Science 338, 1604-1606 (2012) <a href="http://dao.mit.edu/~wen/pub/dDSPTsht.pdf">pdf</a></li> </ul> <div> <h3 id="FractionalLevelWZWModelReferences">On fractional-level WZW models as logarithmic CFTs</h3> On <a class="existingWikiWord" href="/nlab/show/WZW+models">WZW models</a> at <a class="existingWikiWord" href="/nlab/show/rational+number">fractional</a> <a class="existingWikiWord" href="/nlab/show/level+%28Chern-Simons+theory%29">level</a>: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Sunil+Mukhi">Sunil Mukhi</a>, Sudhakar Panda, Fractional-level current algebras and the classification of characters, Nuclear Physics B 338 1 (1990) 263-282 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1016/0550-3213(90)90632-N">doi:10.1016/0550-3213(90)90632-N</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li id="MooreRead91"> <a class="existingWikiWord" href="/nlab/show/Gregory+Moore">Gregory Moore</a>, <a class="existingWikiWord" href="/nlab/show/Nicholas+Read">Nicholas Read</a>, p. 389 of: Nonabelions in the fractional quantum hall effect, Nuclear Physics B 360 2–3 (1991) 362-396 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1016/0550-3213(91)90407-O">doi:10.1016/0550-3213(91)90407-O</a>, <a href="https://www.physics.rutgers.edu/~gmoore/MooreReadNonabelions.pdf">pdf</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> <blockquote> (suggesting the fractional level <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>5</mn><mo stretchy="false">/</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">k = -5/4</annotation></semantics></math> as related to <a class="existingWikiWord" href="/nlab/show/Laughlin+wavefunctions">Laughlin wavefunctions</a> of <a class="existingWikiWord" href="/nlab/show/anyons">anyons</a>) </blockquote> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hidetoshi+Awata">Hidetoshi Awata</a>, <a class="existingWikiWord" href="/nlab/show/Yasuhiko+Yamada">Yasuhiko Yamada</a>, Fusion rules for the fractional level <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mrow><mi>𝔰𝔩</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mo>^</mo></mover></mrow><annotation encoding="application/x-tex">\widehat{\mathfrak{sl}(2)}</annotation></semantics></math> algebra, Mod. Phys. Lett. A 7 (1992) 1185-1196 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://inspirehep.net/literature/332974">spire:332974</a>, <a href="https://doi.org/10.1142/S0217732392003645">doi:10.1142/S0217732392003645</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> P. Furlan, <a class="existingWikiWord" href="/nlab/show/A.+Ch.+Ganchev">A. Ch. Ganchev</a>, R. Paunov, <a class="existingWikiWord" href="/nlab/show/Valentina+B.+Petkova">Valentina B. Petkova</a>, Solutions of the Knizhnik-Zamolodchikov Equation with Rational Isospins and the Reduction to the Minimal Models, Nucl. Phys. B394 (1993) 665-706 (<a href="https://arxiv.org/abs/hep-th/9201080">arXiv:hep-th/9201080</a>, <a href="https://doi.org/10.1016/0550-3213(93)90227-G">doi:10.1016/0550-3213(93)90227-G</a>) </li> <li> J. L. Petersen, J. Rasmussen, M. Yu, Fusion, Crossing and Monodromy in Conformal Field Theory Based on <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 stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SL(2)</annotation></semantics></math> Current Algebra with Fractional Level, Nucl. Phys. B481 (1996) 577-624 (<a href="https://arxiv.org/abs/hep-th/9607129">arXiv:hep-th/9607129</a>, <a href="https://doi.org/10.1016/S0550-3213(96)00506-8">doi:10.1016/S0550-3213(96)00506-8</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Boris+Feigin">Boris Feigin</a>, <a class="existingWikiWord" href="/nlab/show/Feodor+Malikov">Feodor Malikov</a>, Modular functor and representation theory of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mrow><msub><mi>𝔰𝔩</mi> <mn>2</mn></msub></mrow><mo>^</mo></mover></mrow><annotation encoding="application/x-tex">\widehat{\mathfrak{sl}_2}</annotation></semantics></math> at a rational level, p. 357-405 in: Loday, Stasheff, Voronov (eds.) Operads: Proceedings of Renaissance Conferences, Contemporary Mathematics 202, AMS (1997) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/q-alg/9511011">arXiv:q-alg/9511011</a>, <a href="https://bookstore.ams.org/conm-202">ams:conm-202</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> <blockquote> (an <a class="existingWikiWord" href="/nlab/show/osp"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>𝔬𝔰𝔭</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">|</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">\mathfrak{osp}(1\vert2)</annotation> </semantics> </math></a>-factor appears) </blockquote> </li> </ul> with a good review in: <ul> <li><a class="existingWikiWord" href="/nlab/show/A.+Ch.+Ganchev">A. Ch. Ganchev</a>, <a class="existingWikiWord" href="/nlab/show/Valentina+B.+Petkova">Valentina B. Petkova</a>, G. M. T. Watts, A note on decoupling conditions for generic level <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mrow><mi>s</mi><mi>l</mi></mrow><mo>^</mo></mover><mo stretchy="false">(</mo><mn>3</mn><msub><mo stretchy="false">)</mo> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\widehat{s l}(3)_k</annotation></semantics></math> and fusion rules, Nucl. Phys. B 571 (2000) 457-478 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1016/S0550-3213(99)00745-2">doi:10.1016/S0550-3213(99)00745-2</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> On <a class="existingWikiWord" href="/nlab/show/braided+fusion+categories">braided fusion categories</a> formed by <a class="existingWikiWord" href="/nlab/show/affine+Lie+algebra">affine Lie algebra</a>-<a class="existingWikiWord" href="/nlab/show/representations">representations</a> at admissible fractional <a class="existingWikiWord" href="/nlab/show/level+%28Chern-Simons+theory%29">level</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Thomas+Creutzig">Thomas Creutzig</a>, <a class="existingWikiWord" href="/nlab/show/Yi-Zhi+Huang">Yi-Zhi Huang</a>, <a class="existingWikiWord" href="/nlab/show/Jinwei+Yang">Jinwei Yang</a>, Braided tensor categories of admissible modules for affine Lie algebras, Commun. Math. Phys. 362 (2018) 827–854 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/1709.01865">arXiv:1709.01865</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> On interpreting fractional level <a class="existingWikiWord" href="/nlab/show/WZW+models">WZW models</a> as <a class="existingWikiWord" href="/nlab/show/logarithmic+CFT">logarithmic</a> <a class="existingWikiWord" href="/nlab/show/2d+CFT">CFTs</a>: <ul> <li id="Gaberdiel01"> <a class="existingWikiWord" href="/nlab/show/Matthias+R.+Gaberdiel">Matthias R. Gaberdiel</a>, Fusion rules and logarithmic representations of a WZW model at fractional level, Nucl. Phys. B 618 (2001) 407-436 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/hep-th/0105046">arXiv:hep-th/0105046</a>, <a href="https://doi.org/10.1016/S0550-3213(01)00490-4">doi:10.1016/S0550-3213(01)00490-4</a>)<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li id="Gaberdiel01"> <a class="existingWikiWord" href="/nlab/show/Matthias+R.+Gaberdiel">Matthias R. Gaberdiel</a>, Section 5 of: An algebraic approach to logarithmic conformal field theory, Int. J. Mod. Phys. A 18 (2003) 4593-4638 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/hep-th/0111260">arXiv:hep-th/0111260</a>, <a href="https://doi.org/10.1142/S0217751X03016860">doi:10.1142/S0217751X03016860</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> <a class="existingWikiWord" href="/nlab/show/David+Ridout">David Ridout</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>𝔰𝔩</mi><mo>^</mo></mover><mo stretchy="false">(</mo><mn>2</mn><msub><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\widehat{\mathfrak{sl}}(2)_{-1/2}</annotation></semantics></math>: A Case Study, Nucl. Phys. B 814 (2009) 485-521 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/0810.3532">arXiv:0810.3532</a>, <a href="https://doi.org/10.1016/j.nuclphysb.2009.01.008">doi:10.1016/j.nuclphysb.2009.01.008</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> <a class="existingWikiWord" href="/nlab/show/Thomas+Creutzig">Thomas Creutzig</a>, <a class="existingWikiWord" href="/nlab/show/David+Ridout">David Ridout</a>, Modular Data and Verlinde Formulae for Fractional Level WZW Models I, Nuclear Physics B 865 1 (2012) 83-114 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/1205.6513">arXiv:1205.6513</a>, <a href="https://doi.org/10.1016/j.nuclphysb.2012.07.018">doi:10.1016/j.nuclphysb.2012.07.018</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> <a class="existingWikiWord" href="/nlab/show/Thomas+Creutzig">Thomas Creutzig</a>, <a class="existingWikiWord" href="/nlab/show/David+Ridout">David Ridout</a>, Modular Data and Verlinde Formulae for Fractional Level WZW Models II, Nuclear Physics B 875 2 (2013) 423-458 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/1306.4388">arXiv:1306.4388</a>, <a href="https://doi.org/10.1016/j.nuclphysb.2013.07.008">doi:10.1016/j.nuclphysb.2013.07.008</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> <a class="existingWikiWord" href="/nlab/show/Thomas+Creutzig">Thomas Creutzig</a>, <a class="existingWikiWord" href="/nlab/show/David+Ridout">David Ridout</a>, Section 4 of: Logarithmic conformal field theory: beyond an introduction, J. Phys. A: Math. Theor. 46 (2013) 494006 (<a href="https://iopscience.iop.org/article/10.1088/1751-8113/46/49/494006">doi:10.1088/1751-8113/46/49/494006</a>, <a href="https://arxiv.org/abs/1303.0847">arXiv:1303.0847</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Kazuya+Kawasetsu">Kazuya Kawasetsu</a>, <a class="existingWikiWord" href="/nlab/show/David+Ridout">David Ridout</a>, Relaxed highest-weight modules I: rank 1 cases, Commun. Math. Phys. 368 (2019) 627–663 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/1803.01989">arXiv:1803.01989</a>, <a href="https://doi.org/10.1007/s00220-019-03305-x">doi:10.1007/s00220-019-03305-x</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kazuya+Kawasetsu">Kazuya Kawasetsu</a>, <a class="existingWikiWord" href="/nlab/show/David+Ridout">David Ridout</a>, Relaxed highest-weight modules II: classifications for affine vertex algebras, Communications in Contemporary Mathematics, 24 05 (2022) 2150037 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/1906.02935">arXiv:1906.02935</a>, <a href="https://doi.org/10.1142/S0219199721500371">doi:10.1142/S0219199721500371</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> </ul> Reviewed in: <ul> <li id="Ridout10"> <a class="existingWikiWord" href="/nlab/show/David+Ridout">David Ridout</a>, Fractional Level WZW Models as Logarithmic CFTs (2010) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://researchers.ms.unimelb.edu.au/~dridout@unimelb/seminars/100225.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Ridout-FractionalLevelWZW2010.pdf" title="pdf">pdf</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li id="Ridout20"> <a class="existingWikiWord" href="/nlab/show/David+Ridout">David Ridout</a>, Fractional-level WZW models (2020) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://researchers.ms.unimelb.edu.au/~dridout@unimelb/seminars/200205.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Ridout-FractionalLevelWZW2020.pdf" title="pdf">pdf</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> </ul> In particular, the logarithmic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">c = -2</annotation></semantics></math> model is essentially an admissible-level WZW model (namely at level <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">k = 0</annotation></semantics></math>): <ul> <li id="Nichols02">Alexander Nichols, Extended chiral algebras in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><msub><mo stretchy="false">)</mo> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">SU(2)_0</annotation></semantics></math> WZNW model, JHEP 04 (2002) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://iopscience.iop.org/article/10.1088/1126-6708/2002/04/056">doi:10.1088/1126-6708/2002/04/056</a>, <a href="https://arxiv.org/abs/hep-th/0112094">arXiv:hep-th/0112094</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> with a comprehensive account in: <ul> <li id="NicholsThesis02">Alexander Nichols, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><msub><mo stretchy="false">)</mo> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">SU(2)_k</annotation></semantics></math> Logarithmic Conformal Field Theories, PhD thesis, Oxford (2002) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/hep-th/0210070">arXiv:hep-th/0210070</a>, <a href="https://inspirehep.net/literature/599081">spire:599081</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> On the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">c = -1</annotation></semantics></math> model as the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔰𝔲</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{su}(2)</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a> at fractional level <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">-1/2</annotation></semantics></math> and relation to the <a class="existingWikiWord" href="/nlab/show/beta-gamma+system">beta-gamma system</a>: <ul> <li id="LesageMathieuRasmussenSaleur02">F. Lesage, P. Mathieu, J. Rasmussen, H. Saleur, The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔰𝔲</mi><mo stretchy="false">(</mo><mn>2</mn><msub><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\mathfrak{su}(2)_{-1/2}</annotation></semantics></math> WZW model and the beta-gamma system, Nucl. Phys. B 647 (2002) 363-403 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/hep-th/0207201">arXiv:hep-th/0207201</a>, <a href="https://doi.org/10.1016/S0550-3213(02)00905-7">doi:10.1016/S0550-3213(02)00905-7</a>a<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> and its lift to a <a class="existingWikiWord" href="/nlab/show/logarithmic+CFT">logarithmic CFT</a>: <ul> <li id="LesageMathieuRasmussenSaleur04">F. Lesage, P. Mathieu, J. Rasmussen, H. Saleur, Logarithmic lift of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔰𝔲</mi><mo stretchy="false">(</mo><mn>2</mn><msub><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\mathfrak{su}(2)_{-1/2}</annotation></semantics></math> model, Nuclear Physics B 686 3 (2004) 313-346 &lbrack;<a href="https://doi.org/10.1016/j.nuclphysb.2004.02.039">doi:10.1016/j.nuclphysb.2004.02.039</a>&rbrack;</li> </ul> On quasi-characters at fractional level: <ul> <li><a class="existingWikiWord" href="/nlab/show/Sachin+Grover">Sachin Grover</a>, Quasi-Characters in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>𝔰𝔲</mi><mo>^</mo></mover><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\widehat{\mathfrak{su}}(2)</annotation></semantics></math> Current Algebra at Fractional Levels, SciPost Phys. Core 6 068 (2023) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/2208.09037">arXiv:2208.09037</a>, <a href="https://scipost.org/10.21468/SciPostPhysCore.6.4.068">doi:10.21468/SciPostPhysCore.6.4.068</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> Identification of would-be fractional level <a class="existingWikiWord" href="/nlab/show/su%282%29"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>𝔰𝔲</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">\mathfrak{su}(2)</annotation> </semantics> </math></a> <a class="existingWikiWord" href="/nlab/show/conformal+blocks">conformal blocks</a> in <a class="existingWikiWord" href="/nlab/show/twisted+equivariant+K-theory">twisted equivariant K-theory</a> of <a class="existingWikiWord" href="/nlab/show/configuration+spaces+of+points">configuration spaces of points</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, Rem. 2.3 in: <a class="existingWikiWord" href="/schreiber/show/Anyonic+defect+branes+in+TED-K-theory">Anyonic defect branes in TED-K-theory</a>, Rev. Math. Phys. 35 06 (2023) 2350009 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/2203.11838">arXiv:2203.11838</a>, <a href="https://doi.org/10.1142/S0129055X23500095">doi:10.1142/S0129055X23500095</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> </div></body></html> </div> <div class="revisedby"> Last revised on December 2, 2023 at 11:45:04. See the <a href="/nlab/history/Wess-Zumino-Witten+model" style="color: #005c19">history</a> of this page for a list of all contributions to it. </div> <div class="navigation navfoot"> <a href="/nlab/edit/Wess-Zumino-Witten+model" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/10035/#Item_9">Discuss</a><a href="/nlab/revision/Wess-Zumino-Witten+model/79" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a><a href="/nlab/show/diff/Wess-Zumino-Witten+model" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/Wess-Zumino-Witten+model" accesskey="S" class="navlink" id="history" rel="nofollow">History (79 revisions)</a> <a href="/nlab/show/Wess-Zumino-Witten+model/cite" style="color: black">Cite</a> <a href="/nlab/print/Wess-Zumino-Witten+model" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/Wess-Zumino-Witten+model" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>