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17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> nLab AGT correspondence </h1> <div class="navigation"> <a href='#navEnd'>Skip the Navigation Links</a> | <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/5942/#Item_6" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="duality_in_string_theory">Duality in string theory</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/duality+in+string+theory">duality in string theory</a> general mechanisms <ul> <li> <a class="existingWikiWord" href="/nlab/show/T-duality">T-duality</a> (<a class="existingWikiWord" href="/nlab/show/topological+T-duality">topological T-duality</a>, <a class="existingWikiWord" href="/nlab/show/non-abelian+T-duality">non-abelian T-duality</a>) <a class="existingWikiWord" href="/nlab/show/mirror+symmetry">mirror symmetry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/S-duality">S-duality</a> <a class="existingWikiWord" href="/nlab/show/electric-magnetic+duality">electric-magnetic duality</a>, <a class="existingWikiWord" href="/nlab/show/Montonen-Olive+duality">Montonen-Olive duality</a>, <a class="existingWikiWord" href="/nlab/show/geometric+Langlands+duality">geometric Langlands duality</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/U-duality">U-duality</a> <a class="existingWikiWord" href="/nlab/show/exceptional+generalized+geometry">exceptional generalized geometry</a> </li> </ul> string-fivebrane duality <ul> <li><a class="existingWikiWord" href="/nlab/show/dual+heterotic+string+theory">dual heterotic string theory</a></li> </ul> <a class="existingWikiWord" href="/nlab/show/string-string+dualities">string-string dualities</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/HET+-+I">HET - I</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/HET+-+II">HET - II</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/HET+-+M">HET - M</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/duality+between+M-theory+and+type+IIA+superstring+theory">M - IIA</a>. </li> <li> <a class="existingWikiWord" href="/nlab/show/duality+between+M-theory+on+Z2-orbifolds+and+type+IIB+string+theory+on+K3-fibrations">M- IIB</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/F-theory">F-theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/duality+between+M-theory+and+F-theory">M - F</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/duality+between+F-theory+and+heterotic+string+theory">F - HET</a> </li> </ul> string-QFT duality <ul> <li> <a class="existingWikiWord" href="/nlab/show/AdS-CFT+duality">AdS-CFT duality</a> (an <a class="existingWikiWord" href="/nlab/show/open%2Fclosed+string+duality">open/closed string duality</a>) <ul> <li> <a class="existingWikiWord" href="/nlab/show/AdS2%2FCFT1+duality">AdS2/CFT1 duality</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/AdS3%2FCFT2+duality">AdS3/CFT2 duality</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/p-adic+AdS%2FCFT+duality">p-adic AdS/CFT duality</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/AdS-QCD+correspondence">AdS-QCD correspondence</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory+as+topological+string+theory">Chern-Simons theory as topological string theory</a> </li> </ul> QFT-QFT duality: <ul> <li> <a class="existingWikiWord" href="/nlab/show/effective+QFT">effective QFT</a> incarnations of <a class="existingWikiWord" href="/nlab/show/open%2Fclosed+string+duality">open/closed string duality</a>, relating (<a class="existingWikiWord" href="/nlab/show/supergravity">super</a>-)<a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> to (<a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">super</a>-)<a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a>: <ul> <li> <a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/BCJ+relations">BCJ relations</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/classical+double+copy">classical double copy</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kramers-Wannier+duality">Kramers-Wannier duality</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Seiberg+duality">Seiberg duality</a> (swapping <a class="existingWikiWord" href="/nlab/show/NS5-branes">NS5-branes</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/AGT+correspondence">AGT correspondence</a> (<a class="existingWikiWord" href="/nlab/show/wrapped+brane">wrapped</a> <a class="existingWikiWord" href="/nlab/show/M5-branes">M5-branes</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/3d-3d+correspondence">3d-3d correspondence</a> (<a class="existingWikiWord" href="/nlab/show/wrapped+brane">wrapped</a> <a class="existingWikiWord" href="/nlab/show/M5-branes">M5-branes</a>) </li> </ul> </div></div> <h4 id="functorial_quantum_field_theory">Functorial 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="algebraic_quantum_field_theory">Algebraic Quantum Field Theory</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a> (<a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative</a>, <a class="existingWikiWord" href="/nlab/show/AQFT+on+curved+spacetime">on curved spacetimes</a>, <a class="existingWikiWord" href="/nlab/show/homotopical+algebraic+quantum+field+theory">homotopical</a>) <a class="existingWikiWord" href="/nlab/show/A+first+idea+of+quantum+field+theory">Introduction</a> <h2 id="concepts">Concepts</h2> <a class="existingWikiWord" href="/nlab/show/field+theory">field theory</a>: <ul> <li> <a class="existingWikiWord" href="/nlab/show/classical+field+theory">classical</a>, <a class="existingWikiWord" href="/nlab/show/prequantum+field+theory">pre-quantum</a>, <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum</a>, <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/relativistic+field+theory">relativistic</a>, <a class="existingWikiWord" href="/nlab/show/Euclidean+field+theory">Euclidean</a>, <a class="existingWikiWord" href="/nlab/show/thermal+quantum+field+theory">thermal</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/Lagrangian+field+theory">Lagrangian field theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/field+history">field history</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/space+of+field+histories">space of field histories</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Lagrangian+density">Lagrangian density</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+form">Euler-Lagrange form</a>, <a class="existingWikiWord" href="/nlab/show/presymplectic+current">presymplectic current</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange</a><a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/locally+variational+field+theory">locally variational field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Peierls-Poisson+bracket">Peierls-Poisson bracket</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/advanced+and+retarded+propagator">advanced and retarded propagator</a>, </li> <li> <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a> </li> </ul> </li> </ul> </li> </ul> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a><a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+symplectic+groupoids">of symplectic groupoids</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/algebraic+deformation+quantization">algebraic deformation quantization</a>, <a class="existingWikiWord" href="/nlab/show/star+algebra">star algebra</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/quantum+mechanical+system">quantum mechanical system</a>, <a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/subsystem">subsystem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/observables">observables</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/field+observables">field observables</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+observables">local observables</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/polynomial+observables">polynomial observables</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/microcausal+observables">microcausal observables</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a>, <a class="existingWikiWord" href="/nlab/show/C%2A-algebra">C*-algebra</a>, <a class="existingWikiWord" href="/nlab/show/von+Neumann+algebra">von Neumann algebra</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+net+of+observables">local net of observables</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/causal+locality">causal locality</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/field+net">field net</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">state on a star-algebra</a>, <a class="existingWikiWord" href="/nlab/show/expectation+value">expectation value</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a> <a class="existingWikiWord" href="/nlab/show/wave+function">wave function</a> <a class="existingWikiWord" href="/nlab/show/collapse+of+the+wave+function">collapse of the wave function</a>/<a class="existingWikiWord" href="/nlab/show/conditional+expectation+value">conditional expectation value</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/mixed+state">mixed state</a>, <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quasi-free+state">quasi-free state</a>, </li> <li> <a class="existingWikiWord" href="/nlab/show/Hadamard+state">Hadamard state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/picture+of+quantum+mechanics">picture of quantum mechanics</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/free+field">free field</a> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/star+algebra">star algebra</a>, <a class="existingWikiWord" href="/nlab/show/Moyal+deformation+quantization">Moyal deformation quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/canonical+commutation+relations">canonical commutation relations</a>, <a class="existingWikiWord" href="/nlab/show/Weyl+relations">Weyl relations</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/normal+ordered+product">normal ordered product</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Fock+space">Fock space</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/gauge+theories">gauge theories</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/gauge+symmetry">gauge symmetry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/BRST+complex">BRST complex</a>, <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+BV-BRST+complex">local BV-BRST complex</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/BV-operator">BV-operator</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+master+equation">quantum master equation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/master+Ward+identity">master Ward identity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gauge+anomaly">gauge anomaly</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/interacting+field+theory">interacting field</a> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a>, <a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative AQFT</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a>, <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/causal+additivity">causal additivity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a>, <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Feynman+diagram">Feynman diagram</a>, <a class="existingWikiWord" href="/nlab/show/Feynman+perturbation+series">Feynman perturbation series</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/effective+action">effective action</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+stability">vacuum stability</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/interacting+field+algebra">interacting field algebra</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Bogoliubov%27s+formula">Bogoliubov's formula</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+M%C3%B8ller+operator">quantum Møller operator</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/adiabatic+limit">adiabatic limit</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/infrared+divergence">infrared divergence</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/interacting+vacuum">interacting vacuum</a> </li> </ul> </li> </ul> <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/renormalization+scheme">("re-")normalization scheme</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/extension+of+distributions">extension of distributions</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/renormalization+condition">("re"-)normalization condition</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/renormalization+group">renormalization group</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/interaction+vertex+redefinition">interaction vertex redefinition</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/St%C3%BCckelberg-Petermann+renormalization+group">Stückelberg-Petermann renormalization group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/renormalization+group+flow">renormalization group flow</a>/<a class="existingWikiWord" href="/nlab/show/running+coupling+constants">running coupling constants</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/UV+cutoff">UV cutoff</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/counterterms">counterterms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/relative+effective+action">relative effective action</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wilsonian+RG">Wilsonian RG</a>, <a class="existingWikiWord" href="/nlab/show/Polchinski+flow+equation">Polchinski flow equation</a> </li> </ul> </li> </ul> <h2 id="Theorems">Theorems</h2> <h3 id="states_and_observables">States and observables</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure in quantum mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Alfsen-Shultz+theorem">Alfsen-Shultz theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Harding-D%C3%B6ring-Hamhalter+theorem">Harding-Döring-Hamhalter theorem</a> </li> </ul> </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/Fell%27s+theorem">Fell's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wigner+theorem">Wigner theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bub-Clifton+theorem">Bub-Clifton theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kadison-Singer+problem">Kadison-Singer problem</a> </li> </ul> <h3 id="operator_algebra">Operator algebra</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Wick%27s+theorem">Wick's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/cyclic+vector">cyclic vector</a>, <a class="existingWikiWord" href="/nlab/show/separating+vector">separating vector</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Stone-von+Neumann+theorem">Stone-von Neumann theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Haag%27s+theorem">Haag's theorem</a> </li> </ul> <h3 id="local_qft">Local QFT</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/DHR+superselection+theory">DHR superselection theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a> (<a class="existingWikiWord" href="/nlab/show/Wick+rotation">Wick rotation</a>) </li> </ul> <h3 id="perturbative_qft">Perturbative QFT</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Schwinger-Dyson+equation">Schwinger-Dyson equation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/main+theorem+of+perturbative+renormalization">main theorem of perturbative renormalization</a> </li> </ul> </div></div> <h4 id="string_theory">String theory</h4> <div class="hide"><div> <ul> <li> <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/books+about+string+theory">books about string theory</a> </li> </ul> <h3 id="ingredients">Ingredients</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a>, <ul> <li><a class="existingWikiWord" href="/nlab/show/CFT">CFT</a>, <a class="existingWikiWord" href="/nlab/show/perturbation+theory">perturbation theory</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/effective+QFT">effective background QFT</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>, <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a>, <a class="existingWikiWord" href="/nlab/show/quantum+gravity">quantum gravity</a></li> </ul> </li> </ul> <h3 id="critical_string_models">Critical string models</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a>, <a class="existingWikiWord" href="/nlab/show/differential+string+structure">differential string structure</a>, </li> <li> <a class="existingWikiWord" href="/nlab/show/dual+heterotic+string+theory">dual heterotic string theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+fivebrane+structure">differential fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/type+IIA+string+theory">type IIA string theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/type+IIB+string+theory">type IIB string theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/F-theory">F-theory</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/string+field+theory">string field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/duality+in+string+theory">duality in string theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/T-duality">T-duality</a>, <a class="existingWikiWord" href="/nlab/show/mirror+symmetry">mirror symmetry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/S-duality">S-duality</a>, <a class="existingWikiWord" href="/nlab/show/electric-magnetic+duality">electric-magnetic duality</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/U-duality">U-duality</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/open%2Fclosed+string+duality">open/closed string duality</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/AdS%2FCFT+correspondence">AdS/CFT correspondence</a>, <a class="existingWikiWord" href="/nlab/show/holographic+principle">holographic principle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a>, <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%C5%99ava-Witten+theory">Hořava-Witten theory</a></li> </ul> </li> </ul> <h3 id="extended_objects">Extended objects</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/brane">brane</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/D0-brane">D0-brane</a>, <a class="existingWikiWord" href="/nlab/show/D2-brane">D2-brane</a>, <a class="existingWikiWord" href="/nlab/show/D4-brane">D4-brane</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/D1-brane">D1-brane</a>, <a class="existingWikiWord" href="/nlab/show/D3-brane">D3-brane</a>, <a class="existingWikiWord" href="/nlab/show/D5-brane">D5-brane</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/RR-field">RR-field</a>, <a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/NS-brane">NS-brane</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> <ul> <li> <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/B2-field">B2-field</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/NS5-brane">NS5-brane</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/B6-field">B6-field</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/M-brane">M-brane</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/C3-field">C3-field</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/ABJM+theory">ABJM theory</a>, <a class="existingWikiWord" href="/nlab/show/BLG+model">BLG model</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/C6-field">C6-field</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a> </li> </ul> </li> </ul> </li> </ul> <h3 id="topological_strings">Topological strings</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a>, <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+M-theory">topological M-theory</a> </li> </ul> <h2 id="backgrounds">Backgrounds</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/target+space">target space</a>, <a class="existingWikiWord" href="/nlab/show/background+gauge+field">background gauge field</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/twisted+smooth+cohomology+in+string+theory">twisted smooth cohomology in string theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/landscape+of+string+theory+vacua">landscape of string theory vacua</a> </li> </ul> <h2 id="phenomenology">Phenomenology</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/string+phenomenology">string phenomenology</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/moduli+stabilization">moduli stabilization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/G%E2%82%82-MSSM">G₂-MSSM</a> </li> </ul> </li> </ul> <div> <a href="/nlab/edit/string+theory+-+contents">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='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> The AGT correspondence (<a href="#AGT09">AGT 09</a>) is a relation between <ol> <li> the <a class="existingWikiWord" href="/nlab/show/instanton">instanton</a>-<a class="existingWikiWord" href="/nlab/show/partition+function">partition function</a> of <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><msup><mo stretchy="false">)</mo> <mrow><mi>n</mi><mo>+</mo><mn>3</mn><mi>g</mi><mo>−</mo><mn>3</mn></mrow></msup></mrow><annotation encoding="application/x-tex">SU(2)^{n+3g-3}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/N%3D2+D%3D4+super+Yang-Mills+theory">N=2 D=4 super Yang-Mills theory</a> (Nekrasov’s partition function, e.g. <a href="#Szabo15">Szabo 15 (2.1)</a>) </li> <li> the <a class="existingWikiWord" href="/nlab/show/conformal+blocks">conformal blocks</a> of <a class="existingWikiWord" href="/nlab/show/Liouville+theory">Liouville theory</a> on an <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>-punctured <a class="existingWikiWord" href="/nlab/show/Riemann+surface">Riemann surface</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mrow><mi>g</mi><mo>,</mo><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">C_{g,n}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/genus">genus</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> </li> </ol> Here the idea is that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mrow><mi>g</mi><mo>,</mo><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">C_{g,n}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">super Yang-Mills theory</a> obtained by <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">compactifying</a> the worldvolume <a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a> of two <a class="existingWikiWord" href="/nlab/show/M5-branes">M5-branes</a>, see at <a class="existingWikiWord" href="/nlab/show/N%3D2+D%3D4+super+Yang-Mills+theory">N=2 D=4 super Yang-Mills theory</a>, the section <a href="N%3D2+D%3D4+super+Yang-Mills+theory#ConstructionByCompactificationOf5Branes">Construction by compactification</a>). This method generalizes a bit beyond super Yang-Mills, to <a class="existingWikiWord" href="/nlab/show/class+S">class S</a>. In particular, the <a class="existingWikiWord" href="/nlab/show/N%3D2+D%3D4+super+Yang-Mills+theory">N=2 D=4 super Yang-Mills theory</a> is a <a class="existingWikiWord" href="/nlab/show/quiver+gauge+theory">quiver gauge theory</a> and the correspondence matches the shape of its <a class="existingWikiWord" href="/nlab/show/quiver">quiver</a>-diagram to the <a class="existingWikiWord" href="/nlab/show/genus+of+a+surface">genus</a> and punctures of the <a class="existingWikiWord" href="/nlab/show/Riemann+surface">Riemann surface</a>: <center> <img src="https://ncatlab.org/nlab/files/AGTQuiver.jpg" width="400" /> </center> More generally, this construction yields something like a decomposition of the <a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-superconformal+QFT">6d (2,0)-superconformal QFT</a> into a <a class="existingWikiWord" href="/nlab/show/2d+SCFT">2d SCFT</a> “with values in <a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">4d SYM field theory</a>” (e.g. <a href="#Tachikawa10">Tachikawa 10, slide 25 (33 of 54)</a>). Hence composition with any kind of suitable invariant of the 4d field theories yields an actual <a class="existingWikiWord" href="/nlab/show/2d+SCFT">2d SCFT</a>, for instance taking the superconformal index in 4d yields a <a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a> (<a href="#GPRR10">GPRR 10</a>). In this picture of “4d-SYM field theory-valued <a class="existingWikiWord" href="/nlab/show/2d+SCFT">2d SCFT</a>” one has the following correspondences: <ul> <li> the <a class="existingWikiWord" href="/nlab/show/complex+structure">complex structure</a> in 2d is the <a class="existingWikiWord" href="/nlab/show/coupling+constants">coupling constants</a> and <a class="existingWikiWord" href="/nlab/show/theta+angles">theta angles</a> etc in the 4d <a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">super Yang-Mills theory</a>; </li> <li> the <a class="existingWikiWord" href="/nlab/show/mapping+class+group">mapping class group</a> (large <a class="existingWikiWord" href="/nlab/show/conformal+transformations">conformal transformations</a>) in 2d is the (generalized) <a class="existingWikiWord" href="/nlab/show/S-duality">S-duality</a> of the 4d theory. </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/M5-brane+elliptic+genus">M5-brane elliptic genus</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/duality+in+physics">duality in physics</a>, <a class="existingWikiWord" href="/nlab/show/duality+in+string+theory">duality in string theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Nekrasov+function">Nekrasov function</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/McKay+correspondence">McKay correspondence</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/3d-3d+correspondence">3d-3d correspondence</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/knots-quivers+correspondence">knots-quivers correspondence</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/class+S">class S</a> </li> </ul> Wrapping the <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> on a <a class="existingWikiWord" href="/nlab/show/3-manifold">3-manifold</a> instead yields: <a class="existingWikiWord" href="/nlab/show/3d-3d+correspondence">3d-3d correspondence</a>. <h2 id="references">References</h2> A <a class="existingWikiWord" href="/nlab/show/2d+SCFT">2d SCFT</a> argued to describe the <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a> of the <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> on a <a class="existingWikiWord" href="/nlab/show/4-manifold">4-manifold</a> (specifically: a <a class="existingWikiWord" href="/nlab/show/complex+surface">complex surface</a>) originates with <ul> <li><a class="existingWikiWord" href="/nlab/show/Juan+Maldacena">Juan Maldacena</a>, <a class="existingWikiWord" href="/nlab/show/Andrew+Strominger">Andrew Strominger</a>, <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, Black Hole Entropy in M-Theory, JHEP 9712:002, 1997 (<a href="https://arxiv.org/abs/hep-th/9711053">arXiv:hep-th/9711053</a>)</li> </ul> The origin of the AGT correspondence is: <ul> <li id="Gaiotto09"> <a class="existingWikiWord" href="/nlab/show/Davide+Gaiotto">Davide Gaiotto</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">N=2</annotation></semantics></math> dualities, JHEP 08 (2012) 034 [<a href="https://arxiv.org/abs/0904.2715">arXiv:0904.2715</a>] </li> <li id="AGT09"> <a class="existingWikiWord" href="/nlab/show/Luis+Alday">Luis Alday</a>, <a class="existingWikiWord" href="/nlab/show/Davide+Gaiotto">Davide Gaiotto</a>, <a class="existingWikiWord" href="/nlab/show/Yuji+Tachikawa">Yuji Tachikawa</a>, Liouville Correlation Functions from Four-dimensional Gauge Theories, Lett.Math.Phys.91:167-197, 2010 (<a href="https://arxiv.org/abs/0906.3219">arXiv:0906.3219</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Davide+Gaiotto">Davide Gaiotto</a>, <a class="existingWikiWord" href="/nlab/show/Gregory+Moore">Gregory Moore</a>, <a class="existingWikiWord" href="/nlab/show/Andrew+Neitzke">Andrew Neitzke</a>, Wall-crossing, Hitchin systems, and the WKB approximation, Advances in Mathematics 234 (2013) 239–403 (<a href="https://arxiv.org/abs/0907.3987">arXiv:0907.3987</a> <a href="https://doi.org/10.1016/j.aim.2012.09.027">doi</a>) </li> </ul> The <a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a> obtained from this by forming the 4d index is discussed in <ul> <li id="GPRR10">Abhijit Gadde, Elli Pomoni, <a class="existingWikiWord" href="/nlab/show/Leonardo+Rastelli">Leonardo Rastelli</a>, Shlomo S. Razamat, S-duality and 2d Topological QFT, JHEP 1003:032, 2010 (<a href="https://arxiv.org/abs/0910.2225">arXiv:0910.2225</a>)</li> </ul> Relation of the <a class="existingWikiWord" href="/nlab/show/AGT-correspondence">AGT-correspondence</a> to the <a class="existingWikiWord" href="/nlab/show/D%3D6+N%3D%282%2C0%29+SCFT">D=6 N=(2,0) SCFT</a> <ul> <li> Benjamin Assel, <a class="existingWikiWord" href="/nlab/show/Sakura+Schafer-Nameki">Sakura Schafer-Nameki</a>, Jin-Mann Wong, M5-branes on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>2</mn></msup><mo>×</mo><msub><mi>M</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">S^2 \times M_4</annotation></semantics></math>: Nahm’s Equations and 4d Topological Sigma-models, J. High Energ. Phys. (2016) 2016: 120 (<a href="https://arxiv.org/abs/1604.03606">arxiv:1604.03606</a>) (relating to the <a class="existingWikiWord" href="/nlab/show/moduli+space+of+monopoles">moduli space of monopoles</a>) </li> </ul> and to the <a class="existingWikiWord" href="/nlab/show/3d-3d+correspondence">3d-3d correspondence</a>: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Clay+Cordova">Clay Cordova</a>, <a class="existingWikiWord" href="/nlab/show/Daniel+Jafferis">Daniel Jafferis</a>, Toda Theory From Six Dimensions, J. High Energ. Phys. (2017) 2017: 106 (<a href="https://arxiv.org/abs/1605.03997">arxiv:1605.03997</a>) </li> <li> Sam van Leuven, Gerben Oling, Generalized Toda Theory from Six Dimensions and the Conifold, J. High Energ. Phys. (2017) 2017: 50 (<a href="https://arxiv.org/abs/1708.07840">arxiv:1708.07840</a>) </li> </ul> Brief surveys include <ul> <li id="Tachikawa10"> <a class="existingWikiWord" href="/nlab/show/Yuji+Tachikawa">Yuji Tachikawa</a>, M5-branes, 4d gauge theory and 2d CFT, 2010 (<a href="http://member.ipmu.jp/yuji.tachikawa/transp/4d-2d-caltech.pdf">pdf</a>) </li> <li> Abhijit Gadde, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒩</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\mathcal{N}= 2</annotation></semantics></math> Dualities and 2d TQFT 2012 (<a class="existingWikiWord" href="/nlab/files/Gadde2dTQFT.pdf" title="pdf">pdf</a>) </li> <li> Nikolay Bovev, New SCFTs from wrapped branes, 2013 (<a href="http://ipht.cea.fr/Meetings/Itzykson2013/Talks/bobev-Itzykson-july2013.pdf">pdf</a>) </li> <li> Giulio Bonelli, Variations on AGT Correspondence, 2013 (<a href="https://indico.cern.ch/event/217384/attachments/348854/486363/Bonelli.pdf">pdf</a>) </li> <li id="Taki16"> Masato Taki, String Theory as an Attempt of PolyMathematics, talk at 2016.4/28 iTHES-AIMR-IIS (<a href="https://indico2.riken.jp/event/2215/attachments/4106/4775/Taki_invited.pdf">pdf</a>) </li> </ul> More detailed review is in <ul> <li> Rober Rodger, A pedagogical introduction to the AGT conjecture, Master Thesis Utrecht (2013) (<a href="http://testweb.science.uu.nl/ITF/teaching/2013/R.J.Rodger.pdf">pdf</a>) </li> <li id="Szabo15"> <a class="existingWikiWord" href="/nlab/show/Richard+Szabo">Richard Szabo</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">N=2</annotation></semantics></math> gauge theories, instanton moduli spaces and geometric representation theory, Journal of Geometry and Physics Volume 109, November 2016, Pages 83-121 (<a href="https://arxiv.org/abs/1507.00685">arXiv:1507.00685</a>) </li> <li> Bruno Le Floch, A slow review of the AGT correspondence, J. Phys. A: Math. Theor. 55 (2022) 353002 (<a href="https://doi.org/10.1088/1751-8121/ac5945">doi</a><a href="https://arxiv.org/abs/2006.14025">arXiv:2006.14025</a>) </li> </ul> See also <ul> <li> <a class="existingWikiWord" href="/nlab/show/Alexander+Belavin">Alexander Belavin</a>, M. A. Bershtein, B. L. Feigin, A. V. Litvinov, G. M. Tarnopolsky, Instanton moduli spaces and bases in coset conformal field theory (<a href="http://arxiv.org/abs/1111.2803">arxiv/1111.2803</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Volker+Schomerus">Volker Schomerus</a>, Paulina Suchanek, Liouville’s imaginary shadow (<a href="http://arxiv.org/abs/1210.1856">arxiv/1210.1856</a>) </li> <li> A.Mironov, A.Morozov, The power of Nekrasov functions (<a href="http://arxiv.org/abs/0908.2190">arxiv/0908.2190</a>) </li> <li> D. Galakhov, A. Mironov, A. Morozov, S-duality as a beta-deformed Fourier transform (<a href="http://arxiv.org/abs/1205.4998">arxiv/1205.4998</a>) </li> <li> A. Mironov, Spectral duality in integrable systems from AGT conjecture (<a href="http://arxiv.org/abs/1204.0913">arxiv/1204.0913</a>) </li> <li> A. Belavin, V. Belavin, AGT conjecture and integrable structure of conformal field theory for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">c=1</annotation></semantics></math>, Nucl.Phys.B850:199-213 (2011) (<a href="http://arxiv.org/abs/1102.0343">arxiv/1102.0343</a>) </li> <li> A. Belavin, V. Belavin, M. Bershtein, Instantons and 2d Superconformal field theory (<a href="http://arxiv.org/abs/1106.4001">arxiv/1106.4001</a>) </li> <li> Kazunobu Maruyoshi, Quantum integrable systems, matrix models, and AGT correspondence, seminar (<a href="http://db.ipmu.jp/seminar/sysimg/seminar/428.pdf">slides pdf</a>) </li> <li> Giulio Bonelli, Alessandro Tanzini, Hitchin systems, N=2 gauge theories and W-gravity (<a href="http://arxiv.org/abs/0909.4031">arxiv/0909.4031</a>) </li> <li> Giulio Bonelli, Kazunobu Maruyoshi, Alessandro Tanzini, Quantum Hitchin systems via beta-deformed matrix models (<a href="http://arxiv.org/abs/1104.4016">arxiv/1104.4016</a>) </li> <li id="ChacaltanaDistler10"> <a class="existingWikiWord" href="/nlab/show/Oscar+Chacaltana">Oscar Chacaltana</a>, <a class="existingWikiWord" href="/nlab/show/Jacques+Distler">Jacques Distler</a>, Tinkertoys for Gaiotto Duality, JHEP 1011:099,2010, (<a href="http://arxiv.org/abs/arXiv:1008.5203">arXiv:1008.5203</a>) </li> <li> Satoshi Nawata, Givental J-functions, Quantum integrable systems, AGT relation with surface operator (<a href="http://arxiv.org/abs/1408.4132">arXiv/1408.4132</a>) </li> </ul> The AGT correspondence is treated with the help of a <a class="existingWikiWord" href="/nlab/show/Riemann-Hilbert+problem">Riemann-Hilbert problem</a> in <ul> <li> G. Vartanov, <a class="existingWikiWord" href="/nlab/show/J%C3%B6rg+Teschner">Jörg Teschner</a>, Supersymmetric gauge theories, quantization of moduli spaces of flat connections, and conformal field theory (<a href="http://arxiv.org/abs/1302.3778">arxiv/1302.3778</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Andrei+Mironov">Andrei Mironov</a>, A. Morozov, Superintegrability as the hidden origin of Nekrasov calculus [<a href="https://arxiv.org/abs/2207.08242">arXiv:2207.08242</a>] </li> </ul> Proof of the AGT conjecture in special cases: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Andrei+Mironov">Andrei Mironov</a>, <a class="existingWikiWord" href="/nlab/show/Andrey+Morozov">Andrey Morozov</a>, <a class="existingWikiWord" href="/nlab/show/Sh.+Shakirov">Sh. Shakirov</a>, A direct proof of AGT conjecture at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\beta = 1</annotation></semantics></math>, JHEP 1102:067 (2011) [<a href="https://arxiv.org/abs/1012.3137">arXiv:1012.3137</a>, <a href="https://doi.org/10.1007/JHEP02(2011)067">doi:10.1007/JHEP02(2011)067</a>] </li> <li> Qing-Jie Yuan, Shao-Ping Hu, Zi-Hao Huang, Kilar Zhang, A proof of An AGT conjecture at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\beta = 1</annotation></semantics></math> [<a href="https://arxiv.org/abs/2305.11839">arXiv:2305.11839</a>] </li> <li> Qian Shen, Zi-Hao Huang, Shao-Ping Hu, Qing-Jie Yuan, Kilar Zhang, Proof of 5D <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">A_n</annotation></semantics></math> AGT conjecture at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\beta=1</annotation></semantics></math> [<a href="https://arxiv.org/abs/2405.13676">arXiv:2405.13676</a>] </li> </ul> See also: <ul> <li>Leszek Hadasz, Błażej Ruba, Decomposition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>sl</mi><mo>^</mo></mover> <mrow><mn>2</mn><mo>,</mo><mi>k</mi></mrow></msub><mo>⊗</mo><msub><mover><mi>sl</mi><mo>^</mo></mover> <mrow><mn>2</mn><mo>,</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\widehat{sl}_{2,k} \otimes \widehat{sl}_{2,1}</annotation></semantics></math> highest weight representations for generic level <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> and equivalence between two dimensional CFT models [<a href="https://arxiv.org/abs/2312.14695">arXiv:2312.14695</a>]</li> </ul> <div class="property">category: <a class="category_link" href="/nlab/all_pages/physics">physics</a></div></body></html> </div> <div class="revisedby"> Last revised on May 24, 2024 at 05:42:57. 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