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1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> conformal field theory </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <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/1275/#Item_43" 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> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="algebraic_qft">Algebraic QFT</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a></strong> (<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>)</p> <p><a class="existingWikiWord" href="/nlab/show/A+first+idea+of+quantum+field+theory">Introduction</a></p> <h2 id="concepts">Concepts</h2> <p><strong><a class="existingWikiWord" href="/nlab/show/field+theory">field theory</a></strong>:</p> <ul> <li> <p><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></p> </li> <li> <p><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></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Lagrangian+field+theory">Lagrangian field theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field+history">field history</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/space+of+field+histories">space of field histories</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lagrangian+density">Lagrangian density</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+form">Euler-Lagrange form</a>, <a class="existingWikiWord" href="/nlab/show/presymplectic+current">presymplectic current</a></p> </li> <li> <p><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></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+variational+field+theory">locally variational field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Peierls-Poisson+bracket">Peierls-Poisson bracket</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/advanced+and+retarded+propagator">advanced and retarded propagator</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a><a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+symplectic+groupoids">of symplectic groupoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+deformation+quantization">algebraic deformation quantization</a>, <a class="existingWikiWord" href="/nlab/show/star+algebra">star algebra</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+mechanical+system">quantum mechanical system</a></strong>, <strong><a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/subsystem">subsystem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/observables">observables</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+observables">field observables</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+observables">local observables</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polynomial+observables">polynomial observables</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/microcausal+observables">microcausal observables</a></p> </li> <li> <p><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></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+net+of+observables">local net of observables</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/causal+locality">causal locality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field+net">field net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a></p> </li> </ul> </li> <li> <p><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></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a></p> <p><a class="existingWikiWord" href="/nlab/show/wave+function">wave function</a></p> <p><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></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mixed+state">mixed state</a>, <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-free+state">quasi-free state</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hadamard+state">Hadamard state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/picture+of+quantum+mechanics">picture of quantum mechanics</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/free+field">free field</a> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/star+algebra">star algebra</a>, <a class="existingWikiWord" href="/nlab/show/Moyal+deformation+quantization">Moyal deformation quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/canonical+commutation+relations">canonical commutation relations</a>, <a class="existingWikiWord" href="/nlab/show/Weyl+relations">Weyl relations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/normal+ordered+product">normal ordered product</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fock+space">Fock space</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/gauge+theories">gauge theories</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+symmetry">gauge symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BRST+complex">BRST complex</a>, <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+BV-BRST+complex">local BV-BRST complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BV-operator">BV-operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+master+equation">quantum master equation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/master+Ward+identity">master Ward identity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+anomaly">gauge anomaly</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/interacting+field+theory">interacting field</a> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a>, <a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative AQFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interaction">interaction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a>, <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/causal+additivity">causal additivity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a>, <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Feynman+diagram">Feynman diagram</a>, <a class="existingWikiWord" href="/nlab/show/Feynman+perturbation+series">Feynman perturbation series</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+action">effective action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+stability">vacuum stability</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interacting+field+algebra">interacting field algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Bogoliubov%27s+formula">Bogoliubov's formula</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+M%C3%B8ller+operator">quantum Møller operator</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adiabatic+limit">adiabatic limit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/infrared+divergence">infrared divergence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interacting+vacuum">interacting vacuum</a></p> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization+scheme">("re-")normalization scheme</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/extension+of+distributions">extension of distributions</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization+condition">("re"-)normalization condition</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization+group">renormalization group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/interaction+vertex+redefinition">interaction vertex redefinition</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/St%C3%BCckelberg-Petermann+renormalization+group">Stückelberg-Petermann renormalization group</a></p> </li> <li> <p><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></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/UV+cutoff">UV cutoff</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/counterterms">counterterms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relative+effective+action">relative effective action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wilsonian+RG">Wilsonian RG</a>, <a class="existingWikiWord" href="/nlab/show/Polchinski+flow+equation">Polchinski flow equation</a></p> </li> </ul> </li> </ul> <h2 id="Theorems">Theorems</h2> <h3 id="states_and_observables">States and observables</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure in quantum mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Alfsen-Shultz+theorem">Alfsen-Shultz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Harding-D%C3%B6ring-Hamhalter+theorem">Harding-Döring-Hamhalter theorem</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wigner+theorem">Wigner theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bub-Clifton+theorem">Bub-Clifton theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kadison-Singer+problem">Kadison-Singer problem</a></p> </li> </ul> <h3 id="operator_algebra">Operator algebra</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Wick%27s+theorem">Wick's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a></p> <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> <p><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Stone-von+Neumann+theorem">Stone-von Neumann theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Haag%27s+theorem">Haag's theorem</a></p> </li> </ul> <h3 id="local_qft">Local QFT</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/DHR+superselection+theory">DHR superselection theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a> (<a class="existingWikiWord" href="/nlab/show/Wick+rotation">Wick rotation</a>)</p> </li> </ul> <h3 id="perturbative_qft">Perturbative QFT</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Schwinger-Dyson+equation">Schwinger-Dyson equation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/main+theorem+of+perturbative+renormalization">main theorem of perturbative renormalization</a></p> </li> </ul> </div></div> <h4 id="functorial_qft">Functorial QFT</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/FQFT">functorial quantum field theory</a></strong></p> <h2 id="contents">Contents</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+category">cobordism category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+cobordism">extended cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bordism+categories+following+Stolz-Teichner">Riemannian bordism category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+tangle+hypothesis">generalized tangle hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/On+the+Classification+of+Topological+Field+Theories">classification of TQFTs</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functorial+field+theory">functorial field theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+functorial+field+theory">unitary functorial field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+functorial+field+theory">extended functorial field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+field+theory">CFT</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/vertex+operator+algebra">vertex operator algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Reshetikhin-Turaev+model">Reshetikhin-Turaev model</a> / <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/HQFT">HQFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a>, <a class="existingWikiWord" href="/nlab/show/Gromov-Witten+theory">Gromov-Witten theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p>FQFT and <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+models+for+tmf">geometric models for tmf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holographic+principle+of+higher+category+theory">holographic principle of higher category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/holographic+principle">holographic principle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AdS%2FCFT+correspondence">AdS/CFT correspondence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantization+via+the+A-model">quantization via the A-model</a></p> </li> </ul> </li> </ul> </div></div> <h4 id="physics">Physics</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/physics">physics</a></strong>, <a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a>, <a class="existingWikiWord" href="/nlab/show/philosophy+of+physics">philosophy of physics</a></p> <h2 id="surveys_textbooks_and_lecture_notes">Surveys, textbooks and lecture notes</h2> <ul> <li> <p><em><a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">(higher) category theory and physics</a></em></p> </li> <li> <p><em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/books+and+reviews+in+mathematical+physics">books and reviews</a>, <a class="existingWikiWord" href="/nlab/show/physics+resources">physics resources</a></p> </li> </ul> <hr /> <p><a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory (physics)</a>, <a class="existingWikiWord" href="/nlab/show/model+%28physics%29">model (physics)</a></p> <p><a class="existingWikiWord" href="/nlab/show/experiment">experiment</a>, <a class="existingWikiWord" href="/nlab/show/measurement">measurement</a>, <a class="existingWikiWord" href="/nlab/show/computable+physics">computable physics</a></p> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/mechanics">mechanics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/mass">mass</a>, <a class="existingWikiWord" href="/nlab/show/charge">charge</a>, <a class="existingWikiWord" href="/nlab/show/momentum">momentum</a>, <a class="existingWikiWord" href="/nlab/show/angular+momentum">angular momentum</a>, <a class="existingWikiWord" href="/nlab/show/moment+of+inertia">moment of inertia</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dynamics+on+Lie+groups">dynamics on Lie groups</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/rigid+body+dynamics">rigid body dynamics</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lagrangian+mechanics">Lagrangian mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a>, <a class="existingWikiWord" href="/nlab/show/state">state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange equations</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-symplectic+manifold">n-symplectic manifold</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+Lorentzian+manifold">smooth Lorentzian manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/special+relativity">special relativity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/dilaton+gravity">dilaton gravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/classical+field+theory">Classical field theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+physics">classical physics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+mechanics">classical mechanics</a></li> <li><a class="existingWikiWord" href="/nlab/show/waves">waves</a> and <a class="existingWikiWord" href="/nlab/show/optics">optics</a></li> <li><a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a></li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+mechanics">Quantum Mechanics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+mechanics+in+terms+of+dagger-compact+categories">in terms of ∞-compact categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+operator">Hamiltonian operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantization">Quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path+integral">path integral quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semiclassical+approximation">semiclassical approximation</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+field+theory">Quantum Field Theory</a></strong></p> <ul> <li> <p>Axiomatizations</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/AQFT">algebraic QFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+net">local net</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/boson">boson</a>, <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/FQFT">functorial QFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+topological+quantum+field+theory">extended topological quantum field theory</a></p> </li> </ul> </li> </ul> </li> <li> <p>Tools</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-function+theory">geometric ∞-function theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+%28in+particle+phyiscs%29">models</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fields+and+quanta+-+table">fields and quanta</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GUT">Grand Unified Theories</a>, <a class="existingWikiWord" href="/nlab/show/MSSM">MSSM</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/on-shell+recursion">on-shell recursion</a>, <a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></li> </ul> </li> </ul> </li> <li> <p>Structural phenomena</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universality+class">universality class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/instanton">instanton</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spontaneously+broken+symmetry">spontaneously broken symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein mechanism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integrable+systems">integrable systems</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holonomic+quantum+fields">holonomic quantum fields</a></p> </li> </ul> </li> <li> <p>Types of quantum field thories</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory and elliptic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CFT">CFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a>, <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>, <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixing</a></p> </li> <li> <p>examples</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a>, <a class="existingWikiWord" href="/nlab/show/QED">QED</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/electric+charge">electric charge</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>, <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spinors+in+Yang-Mills+theory">spinors in Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a></li> <li><a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a></li> <li><a class="existingWikiWord" href="/nlab/show/RR+field">RR field</a></li> <li><a class="existingWikiWord" href="/nlab/show/first-order+formulation+of+gravity">first-order formulation of gravity</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+covariance">general covariance</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity+as+a+BF-theory">gravity as a BF-theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/particle">particle</a>, <a class="existingWikiWord" href="/nlab/show/relativistic+particle">relativistic particle</a>, <a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental particle</a>, <a class="existingWikiWord" href="/nlab/show/spinning+particle">spinning particle</a>, <a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>, <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/membrane">membrane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AKSZ+theory">AKSZ theory</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+theory">String Theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+theory+results+applied+elsewhere">string theory results applied elsewhere</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/number+theory+and+physics">number theory and physics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Riemann+hypothesis+and+physics">Riemann hypothesis and physics</a></li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/physicscontents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#abstract'>Abstract</a></li> <li><a href='#2d_cft_on_the_plane'>2d CFT on the plane</a></li> <ul> <li><a href='#definition_in_terms_of_the_osterwalderschrader_axioms'>Definition in Terms of the Osterwalder-Schrader Axioms</a></li> <ul> <li><a href='#axiom_1'>Axiom 1</a></li> <li><a href='#axiom_2'>Axiom 2</a></li> <li><a href='#axiom_3'>Axiom 3</a></li> <li><a href='#axiom_4'>Axiom 4</a></li> <li><a href='#axiom_5'>Axiom 5</a></li> <li><a href='#pseudodefinition'>Pseudodefinition</a></li> <li><a href='#axiom_6'>Axiom 6</a></li> </ul> <li><a href='#properties'>Properties</a></li> </ul> <li><a href='#2d_cft_on_surfaces_or_arbitrary_genus'>2d CFT on surfaces or arbitrary genus</a></li> <li><a href='#conformal_anomaly'>Conformal anomaly</a></li> <li><a href='#cft_in_aqft_language'>CFT in AQFT language</a></li> <li><a href='#FullAndChiral'>Full versus chiral CFT</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#cft_on_complex_curvessurfaces_of_arbitrary_genus'>CFT on complex curves/surfaces of arbitrary genus</a></li> <li><a href='#formulation_by_conformal_nets'>Formulation by conformal nets</a></li> <li><a href='#formulation_in_full_aqft'>Formulation in full AQFT</a></li> <li><a href='#2dCFTAsFunctorialQFTReferences'><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> CFT as functorial field theory</a></li> <li><a href='#FRSFormalism'>Formulation by algebra in modular tensor categories</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>Recall that a <a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a> is an <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a> defined on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,n)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/%28infinity%2Cn%29-category+of+cobordisms">category of cobordisms</a> whose morphisms are plain cobordisms and diffeomorphisms between these.</p> <p>In a <em>conformal</em> quantum field theory the cobordisms are equipped with a <a class="existingWikiWord" href="/nlab/show/conformal+structure">conformal structure</a> (a <a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemannian metric</a> structure modulo pointwise rescaling): <em><a class="existingWikiWord" href="/nlab/show/conformal+cobordisms">conformal cobordisms</a></em>.</p> <p>A conformal field theory (CFT) is accordingly a functor on such a richer category of conformal <a class="existingWikiWord" href="/nlab/show/cobordisms">cobordisms</a>. See the discussion at <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a> for more details.</p> <p>The conformally invariant quantum field theories have fields for whom the <a class="existingWikiWord" href="/nlab/show/correlation+functions">correlation functions</a> have a specific behaviour accounting for the <em>conformal dimension</em> of the <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a>. This kind of constraints coming from conformal invariance, leads to constraints on the possible behaviour of correlation functions; this has being formulated in 1971 by Polyakov as a <strong><a class="existingWikiWord" href="/nlab/show/conformal+bootstrap">conformal bootstrap</a></strong> program: the conformal invariance should be sufficient to classify consistent conformal QFTs directly from the analysis of symmetries, rather than computing the <a class="existingWikiWord" href="/nlab/show/Feynman+diagram">Feynman diagram</a> <a class="existingWikiWord" href="/nlab/show/perturbation+series">perturbation series</a> from some <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>; this avoids the usual problems with regularization and <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a>.</p> <p>Precisely in 2-dimensions is the representation theory of the conformal group exceptionally interesting. In <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 \gt 2</annotation></semantics></math>, the global transformations, i.e. the elements of the <a class="existingWikiWord" href="/nlab/show/conformal+group">conformal group</a>, are given by the <a class="existingWikiWord" href="/nlab/show/Poincare+group">Poincare algebra</a>, dilations and special conformal transformations. In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math> dimensions, there are additionaly infinitely many local generators of conformal transformations whose commutation relations are given by the <a class="existingWikiWord" href="/nlab/show/Virasoro+algebra">Virasoro algebra</a>. This circumstance enables Belavin, Polyakov and Zamolodchikov to make a breakthrough in the bootstrap program in 1984 (what also helped a 1984-1985 revolution in string theory). There are well-developed tools for handling the theory locally (<a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a>s, <a class="existingWikiWord" href="/nlab/show/vertex+operator+algebra">vertex operator algebra</a>s) and at least in the <a class="existingWikiWord" href="/nlab/show/rational+conformal+field+theory">rational conformal field theory</a> case there are complete classification results for the full theories (defined on cobordisms of all genera).</p> <p>For this reason often in the literature the term “CFT” often implicitly refers to 2d CFT.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-dimensional conformal field theories have two major applications:</p> <ul> <li> <p>they describe critical phenomena on surfaces in condensed matter physics;</p> </li> <li> <p>they are the building blocks used in <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a></p> </li> </ul> <p>In the former application it is mostly the <em>local</em> behaviour of the CFT that is relevant. This is encoded in <a class="existingWikiWord" href="/nlab/show/vertex+operator+algebra">vertex operator algebra</a>s.</p> <p>In the <a class="existingWikiWord" href="/nlab/show/string+theory">string theoretic</a> applications the extension of the local theory to a full representation of the 2d <a class="existingWikiWord" href="/nlab/show/conformal+cobordism+category">conformal cobordism category</a> is crucial. This extension is called <em>solving the <a class="existingWikiWord" href="/nlab/show/sewing+constraints">sewing constraints</a></em> .</p> <h2 id="abstract">Abstract</h2> <p>In the definition paragraph we will show how to define a conformal field theory using the axiomatic approach of Wightman resp. Osterwalder-Schrader. There are several approaches to axiomatically define conformal field theory, the said approach is not the most “popular” or “elegant” one. There are two reasons to consider the Wightman approach however: If one is already familiar with the Wightman approach, it helps to put conformal field theories into context. The second reason is that several notions often used by physicists can be easily and rigorously defined and explained using this approach. Therefore, it may serve as a bridge between mathematicians and physics literature.</p> <h2 id="2d_cft_on_the_plane">2d CFT on the plane</h2> <h3 id="definition_in_terms_of_the_osterwalderschrader_axioms">Definition in Terms of the Osterwalder-Schrader Axioms</h3> <p>A definition of conformal field theories can be formulated using the appropriate version of the <a class="existingWikiWord" href="/nlab/show/Osterwalder--Schrader+axioms">Osterwalder–Schrader axioms</a>, that is by a system of axioms of the <a class="existingWikiWord" href="/nlab/show/correlation+functions">correlation functions</a>. From the correlation functions it is then possible to deduce the existence of field operators in the sense of the <a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a>, that is fields aka field operators are operator valued distributions. Both the Hilbert space and the field operators are therefore not defined in the axioms, but reconstructed from the correlations functions defined in the first three axioms.</p> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>M</mi> <mi>n</mi></msub><mo>:</mo><mo>=</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mi>z</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>ℂ</mi> <mi>n</mi></msup><mo>:</mo><msub><mi>z</mi> <mi>i</mi></msub><mo>≠</mo><msub><mi>z</mi> <mi>j</mi></msub><mspace width="thickmathspace"></mspace><mtext>for</mtext><mspace width="thickmathspace"></mspace><mi>i</mi><mo>≠</mo><mi>j</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> M_n := \{(z_1, ..., z_n) \in \mathbb{C}^n : z_i \neq z_j \; \text{for} \; i \neq j \} </annotation></semantics></math></div> <p>be the <strong>space of configuration points</strong>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">B_0</annotation></semantics></math> be a countable index set and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>:</mo><mo>=</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">⋃</mo> <mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow></msub><msubsup><mi>B</mi> <mn>0</mn> <mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">B := \bigcup_{n \in \mathbb{N}} B_0^{n}</annotation></semantics></math>.</p> <p>The correlation functions are a family <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>G</mi> <mi>i</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo>∈</mo><mi>B</mi></mrow></msub></mrow><annotation encoding="application/x-tex">(G_i)_{i \in B}</annotation></semantics></math> of continuous and polynomially bound functions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mi>i</mi> <mi>n</mi></msub></mrow></msub><mo>:</mo><msub><mi>M</mi> <mi>n</mi></msub><mo>→</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex"> G_{i_1, ..., i_n}: M_n \to \mathbb{C} </annotation></semantics></math></div> <div class="num_defn"> <h6 id="axiom_1">Axiom 1</h6> <p><strong>locality</strong></p> <p>For all indexes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>i</mi> <mn>1</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mi>i</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i_1, ..., i_n)</annotation></semantics></math>, configuration points <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mi>z</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(z_1, ..., z_n)</annotation></semantics></math> and permutations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo>:</mo><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo><mo>→</mo><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\pi: \{1,..., n \} \to \{1,..., n \}</annotation></semantics></math> one has</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mrow><mo stretchy="false">(</mo><msub><mi>i</mi> <mn>1</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mi>i</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mi>z</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>G</mi> <mrow><mo stretchy="false">(</mo><mi>π</mi><mo stretchy="false">(</mo><msub><mi>i</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mi>π</mi><mo stretchy="false">(</mo><msub><mi>i</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>π</mi><mo stretchy="false">(</mo><msub><mi>z</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mi>π</mi><mo stretchy="false">(</mo><msub><mi>z</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> G_{(i_1,..., i_n)}(z_1,..., z_n) = G_{(\pi(i_1),..., \pi(i_n))}(\pi(z_1),..., \pi(z_n)) </annotation></semantics></math></div></div> <p>In this context, covariance is meant with respect to the <a class="existingWikiWord" href="/nlab/show/Euclidean+group">Euclidean group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>=</mo><msub><mi>E</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">E = E_2</annotation></semantics></math>.</p> <div class="num_defn"> <h6 id="axiom_2">Axiom 2</h6> <p><strong>covariance</strong></p> <p>For every index <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><msub><mi>B</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">i \in B_0</annotation></semantics></math> there are <strong>conformal weights</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>h</mi> <mi>i</mi></msub><mo>,</mo><mover><mrow><msub><mi>h</mi> <mi>i</mi></msub></mrow><mo>¯</mo></mover><mo>∈</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">h_i, \overline{h_i} \in \mathbb{R}</annotation></semantics></math> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mrow><msub><mi>h</mi> <mi>i</mi></msub></mrow><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">\overline{h_i}</annotation></semantics></math> is completly independent from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>h</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">h_i</annotation></semantics></math>, it is <em>not</em> the complex conjugate, this notation is widly used in the physics literature so we use it here, too) such that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n \ge 1</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi><mo>∈</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">w \in E</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>w</mi> <mi>i</mi></msub><mo>:</mo><mo>=</mo><mi>w</mi><mo stretchy="false">(</mo><msub><mi>z</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">w_i := w(z_i)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>h</mi> <mi>j</mi></msub><mo>:</mo><mo>=</mo><msub><mi>h</mi> <mrow><msub><mi>i</mi> <mi>j</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">h_j := h_{i_j}</annotation></semantics></math> one has</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mrow><mo stretchy="false">(</mo><msub><mi>i</mi> <mn>1</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mi>i</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>,</mo><mover><mrow><msub><mi>z</mi> <mi>i</mi></msub></mrow><mo>¯</mo></mover><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mi>z</mi> <mi>n</mi></msub><mo>,</mo><mover><mrow><msub><mi>z</mi> <mi>n</mi></msub></mrow><mo>¯</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mo stretchy="false">(</mo><mfrac><mi>dw</mi><mi>dz</mi></mfrac><mo stretchy="false">(</mo><msub><mi>z</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><msub><mi>h</mi> <mi>j</mi></msub></mrow></msup><mo stretchy="false">(</mo><mover><mrow><mfrac><mi>dw</mi><mi>dz</mi></mfrac><mo stretchy="false">(</mo><msub><mi>z</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mrow><mo>¯</mo></mover><msup><mo stretchy="false">)</mo> <mover><mrow><msub><mi>h</mi> <mi>j</mi></msub></mrow><mo>¯</mo></mover></msup><msub><mi>G</mi> <mrow><mo stretchy="false">(</mo><msub><mi>i</mi> <mn>1</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mi>i</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>w</mi> <mn>1</mn></msub><mo>,</mo><mover><mrow><msub><mi>w</mi> <mn>1</mn></msub></mrow><mo>¯</mo></mover><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mi>w</mi> <mi>n</mi></msub><mo>,</mo><mover><mrow><msub><mi>w</mi> <mi>n</mi></msub></mrow><mo>¯</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> G_{(i_1,..., i_n)}(z_1, \overline{z_i}, ..., z_n, \overline{z_n}) = \prod (\frac{dw}{dz} (z_j))^{h_j} (\overline{\frac{dw}{dz}(z_j)})^{\overline{h_j}} G_{(i_1,..., i_n)}(w_1, \overline{w_1}..., w_n, \overline{w_n}) </annotation></semantics></math></div></div> <p>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mi>i</mi></msub><mo>:</mo><mo>=</mo><msub><mi>h</mi> <mi>i</mi></msub><mo>−</mo><mover><mrow><msub><mi>h</mi> <mi>i</mi></msub></mrow><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">s_i := h_i - \overline{h_i}</annotation></semantics></math> are called <strong>conformal spin</strong> (for the index <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>) and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>i</mi></msub><mo>:</mo><mo>=</mo><msub><mi>h</mi> <mi>i</mi></msub><mo>+</mo><mover><mrow><msub><mi>h</mi> <mi>i</mi></msub></mrow><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">d_i := h_i + \overline{h_i}</annotation></semantics></math> is the <strong>scaling dimension</strong>. As an axiom 2.2 we assume that all conformal spins and scaling dimensions are integers.</p> <p>For the next axiom, which is about reflection positivity, we need some notation:</p> <p>We write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>∈</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">z \in \mathbb{C}</annotation></semantics></math> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>=</mo><mi>t</mi><mo>+</mo><mi>iy</mi></mrow><annotation encoding="application/x-tex">z = t + iy</annotation></semantics></math> and identify <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> as the space coordinate and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math> as (imaginary) time. So, time reflection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math> is simply:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>θ</mi><mo>:</mo><mi>ℂ</mi><mo>→</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex"> \theta: \mathbb{C} \to \mathbb{C} </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>θ</mi><mo>:</mo><mi>z</mi><mo>=</mo><mi>t</mi><mo>+</mo><mi>iy</mi><mo>↦</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>t</mi><mo>+</mo><mi>iy</mi><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mover><mi>z</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex"> \theta: z = t + iy \mapsto -t + iy = - \overline{z} </annotation></semantics></math></div> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒮</mi><mo stretchy="false">(</mo><msup><mi>ℂ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{S}(\mathbb{C}^n)</annotation></semantics></math> be the space of <a class="existingWikiWord" href="/nlab/show/Schwartz+functions">Schwartz functions</a> and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>M</mi> <mi>n</mi> <mo>+</mo></msubsup><mo>:</mo><mo>=</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mi>z</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><msub><mi>M</mi> <mi>n</mi></msub><mo>:</mo><mo lspace="0em" rspace="thinmathspace">Re</mo><mo stretchy="false">(</mo><msub><mi>z</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>></mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> M_n^+ := \{ (z_1, ..., z_n) \in M_n: \operatorname{Re}(z_i) \gt 0 \} </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝒮</mi> <mn>0</mn> <mo>+</mo></msubsup><mo>:</mo><mo>=</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex"> \mathcal{S}_0^+ := \mathbb{C} </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝒮</mi> <mi>n</mi> <mo>+</mo></msubsup><mo>:</mo><mo>=</mo><mo stretchy="false">{</mo><mi>f</mi><mo>∈</mo><mi>𝒮</mi><mo stretchy="false">(</mo><msup><mi>ℂ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>:</mo><mo lspace="0em" rspace="thinmathspace">Supp</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>⊂</mo><msubsup><mi>M</mi> <mi>n</mi> <mo>+</mo></msubsup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> \mathcal{S}_n^+ := \{f \in \mathcal{S}(\mathbb{C}^n): \operatorname{Supp}(f) \subset M_n^+ \} </annotation></semantics></math></div> <p>Finally let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mrow><msup><mi>𝒮</mi> <mo>+</mo></msup></mrow><mo>̲</mo></munder></mrow><annotation encoding="application/x-tex">\underline{ \mathcal{S}^+ }</annotation></semantics></math> the space of all sequences <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>f</mi><mo>̲</mo></munder><mo>=</mo><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>i</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo>∈</mo><mi>B</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\underline{f} = (f_i)_{i \in B}</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub><mo>∈</mo><msubsup><mi>𝒮</mi> <mi>n</mi> <mo>+</mo></msubsup></mrow><annotation encoding="application/x-tex">f_i \in \mathcal{S}_n^+</annotation></semantics></math> fpr <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><msubsup><mi>B</mi> <mn>0</mn> <mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">i \in B_0^n</annotation></semantics></math> with only finitely many entries <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">\ne 0</annotation></semantics></math>.</p> <div class="num_defn"> <h6 id="axiom_3">Axiom 3</h6> <p><strong>reflection positivity</strong></p> <p>There is a map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>:</mo><msub><mi>B</mi> <mn>0</mn></msub><mo>→</mo><msub><mi>B</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex"> *: B_0 \to B_0 </annotation></semantics></math></div> <p>with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>*</mo> <mn>2</mn></msup><mo>=</mo><msub><mi>id</mi> <mrow><msub><mi>B</mi> <mn>0</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex">*^2 = id_{B_0}</annotation></semantics></math> that extends to a map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>:</mo><mi>B</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex"> *: B \to B </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>:</mo><mi>i</mi><mo>↦</mo><msup><mi>i</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex"> *: i \mapsto i^* </annotation></semantics></math></div> <p>so that</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>G</mi> <mrow><msup><mi>i</mi> <mo>*</mo></msup></mrow></msub><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>G</mi> <mrow><msup><mi>i</mi> <mo>*</mo></msup></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mover><mi>z</mi><mo>¯</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G_i(z) = G_{i^*}(\theta(z)) = G_{i^*}(- \overline{z})</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><munder><mi>f</mi><mo>̲</mo></munder><mo>,</mo><munder><mi>f</mi><mo>̲</mo></munder><mo stretchy="false">⟩</mo><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\langle \underline{f}, \underline{f} \rangle \ge 0</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>f</mi><mo>̲</mo></munder><mo>∈</mo><munder><mrow><msup><mi>𝒮</mi> <mo>+</mo></msup></mrow><mo>̲</mo></munder></mrow><annotation encoding="application/x-tex">\underline{f} \in \underline{\mathcal{S}^+}</annotation></semantics></math></p> </li> </ol> <p>The scalar product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><munder><mi>f</mi><mo>̲</mo></munder><mo>,</mo><munder><mi>f</mi><mo>̲</mo></munder><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle \underline{f}, \underline{f} \rangle</annotation></semantics></math> is defined as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi>B</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow></munder><msub><mo>∫</mo> <mrow><msub><mi>M</mi> <mrow><mi>n</mi><mo>+</mo><mi>m</mi></mrow></msub></mrow></msub><msub><mi>G</mi> <mrow><msup><mi>i</mi> <mo>*</mo></msup><mi>j</mi></mrow></msub><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">(</mo><msub><mi>z</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mi>θ</mi><mo stretchy="false">(</mo><msub><mi>z</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>,</mo><msub><mi>w</mi> <mn>1</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mi>w</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mover><mrow><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><mo>¯</mo></mover><msub><mi>f</mi> <mi>j</mi></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><msup><mi>d</mi> <mi>n</mi></msup><mi>z</mi><msup><mi>d</mi> <mi>m</mi></msup><mi>w</mi></mrow><annotation encoding="application/x-tex"> \sum_{i, j \in B, m, n \in \mathbb{N} } \int_{M_{n+m}} G_{i^* j} (\theta(z_1), ...,\theta(z_n), w_1, ..., w_n) \overline{f_i(z)} f_j(w) d^n z d^m w </annotation></semantics></math></div></div> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>Both a Hilbert space and the field operators of the theory can be (re-) constructed from axioms 1, 2 and 3.</p> <p>TODO: Details</p> </div> <div class="num_defn"> <h6 id="axiom_4">Axiom 4</h6> <p><strong>scaling covariance</strong></p> <p>The correlation functions satisfy the covariance condition also for dilatations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>e</mi> <mi>t</mi></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mi>t</mi><mo>∈</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">w(z) = e^t (t), t \in \mathbb{R}</annotation></semantics></math>.</p> </div> <p>Explicitly, the last condition says that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mi>z</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msup><mi>e</mi> <mi>t</mi></msup><msup><mo stretchy="false">)</mo> <mrow><msub><mi>h</mi> <mn>1</mn></msub><mo>+</mo><mover><mrow><msub><mi>h</mi> <mn>1</mn></msub></mrow><mo>¯</mo></mover><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>+</mo><msub><mi>h</mi> <mi>n</mi></msub><mo>+</mo><mover><mrow><msub><mi>h</mi> <mi>n</mi></msub></mrow><mo>¯</mo></mover></mrow></msup><msub><mi>G</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><msup><mi>e</mi> <mi>t</mi></msup><mo stretchy="false">(</mo><msub><mi>z</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msup><mi>e</mi> <mi>t</mi></msup><mo stretchy="false">(</mo><msub><mi>z</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> G_i(z_1, ..., z_n) = (e^t)^{h_1 + \overline{h_1} +...+ h_n + \overline{h_n}} G_i(e^t(z_1), ..., e^t(z_n)) </annotation></semantics></math></div> <p>Given axioms 1-4, the 2-point functions can be fully classified, see below.</p> <div class="num_defn"> <h6 id="axiom_5">Axiom 5</h6> <p><strong>existence of the energy-momentum tensor</strong></p> <p>There are four fields <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mrow><mi>μ</mi><mo>,</mo><mi>ν</mi></mrow></msub><mo>,</mo><mi>μ</mi><mo>.</mo><mi>ν</mi><mo>∈</mo><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">T_{\mu, \nu}, \mu. \nu \in \{0, 1\}</annotation></semantics></math> with the following properties:</p> <p>Symmetry:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mrow><mi>μ</mi><mo>,</mo><mi>ν</mi></mrow></msub><mo>=</mo><msub><mi>T</mi> <mrow><mi>ν</mi><mo>,</mo><mi>μ</mi></mrow></msub><mo>,</mo><mspace width="thickmathspace"></mspace><msub><mi>T</mi> <mrow><mi>μ</mi><mo>,</mo><mi>ν</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mo>=</mo><msub><mi>T</mi> <mrow><mi>ν</mi><mo>,</mo><mi>μ</mi></mrow></msub><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> T_{\mu, \nu} = T_{\nu, \mu}, \; T_{\mu, \nu}(z)^* = T_{\nu, \mu}(\theta (z)) </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∂</mo> <mi>t</mi></msub><msub><mi>T</mi> <mrow><mi>μ</mi><mo>,</mo><mn>0</mn></mrow></msub><mo>+</mo><msub><mo>∂</mo> <mi>y</mi></msub><msub><mi>T</mi> <mrow><mi>μ</mi><mo>,</mo><mn>1</mn></mrow></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> \partial_t T_{\mu, 0} + \partial_y T_{\mu, 1} = 0 </annotation></semantics></math></div> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mrow><mi>μ</mi><mo>,</mo><mi>ν</mi></mrow></msub></mrow><annotation encoding="application/x-tex">T_{\mu, \nu}</annotation></semantics></math> has scaling dimension 2 and the conformal spin <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math> is restricted by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo stretchy="false">(</mo><msub><mi>T</mi> <mn>00</mn></msub><mo>−</mo><msub><mi>T</mi> <mn>11</mn></msub><mo>±</mo><mn>2</mn><mi>i</mi><mspace width="thickmathspace"></mspace><msub><mi>T</mi> <mn>01</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo>±</mo><mn>2</mn></mrow><annotation encoding="application/x-tex"> s(T_{00} - T_{11} \pm 2i \; T_{01}) = \pm 2 </annotation></semantics></math></div></div> <p>The energy momentum tensor allows us to define two densly defined operators <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi></mrow></msub><mo>,</mo><mover><mrow><msub><mi>L</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi></mrow></msub></mrow><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">L_{-n}, \overline{L_{-n}}</annotation></semantics></math> that both define unitary representations of the <a class="existingWikiWord" href="/nlab/show/Virasoro+algebra">Virasoro algebra</a>.</p> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p><strong><a href="#LüscherMack75">Lüscher & Mack 1975</a></strong></p> <p>T is holomorphic. Therefore, the operators</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi></mrow></msub><mo>:</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>π</mi><mi>i</mi></mrow></mfrac><msub><mo>∮</mo> <mrow><mo stretchy="false">|</mo><mi>ζ</mi><mo stretchy="false">|</mo><mo>=</mo><mn>1</mn></mrow></msub><mfrac><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow><mrow><msup><mi>ζ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfrac><mi>d</mi><mi>ζ</mi></mrow><annotation encoding="application/x-tex"> L_{-n} := \frac{1}{2 \pi i} \oint_{| \zeta | = 1} \frac{T(\zeta)}{\zeta^{n+1}} d\zeta </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mrow><msub><mi>L</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi></mrow></msub></mrow><mo>¯</mo></mover><mo>:</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>π</mi><mi>i</mi></mrow></mfrac><msub><mo>∮</mo> <mrow><mo stretchy="false">|</mo><mi>ζ</mi><mo stretchy="false">|</mo><mo>=</mo><mn>1</mn></mrow></msub><mfrac><mrow><mover><mi>T</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow><mrow><msup><mi>ζ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfrac><mi>d</mi><mi>ζ</mi></mrow><annotation encoding="application/x-tex"> \overline{L_{-n}} := \frac{1}{2 \pi i} \oint_{| \zeta | = 1} \frac{\overline{T}(\zeta)}{\zeta^{n+1}} d\zeta </annotation></semantics></math></div> <p>are well defined and satisfy the commutation relations of two commuting <a class="existingWikiWord" href="/nlab/show/Virasoro+algebra">Virasoro algebras</a> with the same central charge.</p> </div> <div class="num_defn"> <h6 id="definition">Definition</h6> <p><strong>primary field</strong></p> <p>A conformal field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ψ</mi> <mi>i</mi></msub><mo>,</mo><mi>i</mi><mo>∈</mo><msub><mi>B</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\psi_i, i \in B_0</annotation></semantics></math>, is called a primary field if</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>L</mi> <mi>n</mi></msub><mo>,</mo><msub><mi>ψ</mi> <mi>i</mi></msub><mo stretchy="false">]</mo><mo>=</mo><msup><mi>z</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><msub><mo>∂</mo> <mi>z</mi></msub><msub><mi>ψ</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>h</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mi>z</mi> <mi>n</mi></msup><msub><mi>ψ</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> [L_n, \psi_i] = z^{n+1} \partial_z \psi_i(z) + h_i (n+1) z^n \psi_i(z) </annotation></semantics></math></div> <p>and likewise for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>z</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">\overline{z}</annotation></semantics></math>.</p> </div> <p>So, primary fields are the fields whose correlation functions are covariant “infinitesimally” with respect to all holomorphic functions, the “infinitesimal symmetry” expressed in the definition.</p> <p>The fields are operator valued distributions and cannot be multiplied in general. The possibility of multiplication of fields evaluated at different points and some control of the singularities of this product are part of the axioms of conformal field theory:</p> <div class="num_defn"> <h6 id="pseudodefinition">Pseudodefinition</h6> <p><strong>operator product expansion</strong></p> <p>An operator product expansion (<strong>OPE</strong>) for a family of fields means that there is for all fields and all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>z</mi> <mn>1</mn></msub><mo>≠</mo><msub><mo lspace="0em" rspace="thinmathspace">z</mo> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">z_1 \neq \z_2</annotation></semantics></math> a relation of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ψ</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi>z</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><msub><mi>ψ</mi> <mi>j</mi></msub><mo stretchy="false">(</mo><msub><mi>z</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>∼</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>k</mi><mo>∈</mo><msub><mi>B</mi> <mn>0</mn></msub></mrow></munder><msub><mi>C</mi> <mi>ijk</mi></msub><mo stretchy="false">(</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>−</mo><msub><mi>z</mi> <mn>2</mn></msub><msup><mo stretchy="false">)</mo> <mrow><msub><mi>h</mi> <mi>k</mi></msub><mo>−</mo><msub><mi>h</mi> <mi>i</mi></msub><mo>−</mo><msub><mi>h</mi> <mi>j</mi></msub></mrow></msup><msub><mi>ψ</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><msub><mi>z</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \psi_i(z_1) \psi_j(z_2) \sim \sum_{k \in B_0} C_{ijk} (z_1 - z_2)^{h_k - h_i - h_j} \psi_k(z_2) </annotation></semantics></math></div> <p>Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∼</mo></mrow><annotation encoding="application/x-tex">\sim</annotation></semantics></math> means modulo regular functions.</p> </div> <p>A rigorous interpretation of an OPE would interpret the given relation as a relation of e.g. matrix elements or vacuum expectation values.</p> <p>An OPE is called <strong>associative</strong> if the expansion of a product of more than two fields does not depend on the order of the expansion of the products of two factors. Since the OPE has no interpretation as defining products of operators, or more generally the product in a ring, the notion of associativity does not refer to the associativity of a product in a ring, as the term may suggest.</p> <div class="num_defn"> <h6 id="axiom_6">Axiom 6</h6> <p><strong>existence of associative operator product expansion</strong></p> <p>The primary fields have an associative OPE.</p> </div> <p>Many <em>formal</em> calculations of physicists in CFT involving OPE can be justified by using <a class="existingWikiWord" href="/nlab/show/vertex+operator+algebra">vertex operator algebra</a>s, so that <a class="existingWikiWord" href="/nlab/show/vertex+operator+algebra">vertex operator algebra</a>s have become a standard way to formulate CFT.</p> <h3 id="properties">Properties</h3> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>Any 2-point function satisfying axioms 1-4 has the following form:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mi>ij</mi></msub><mo stretchy="false">(</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>z</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>C</mi> <mi>ij</mi></msub><mo stretchy="false">(</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>−</mo><msub><mi>z</mi> <mn>2</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">(</mo><msub><mi>h</mi> <mi>i</mi></msub><mo>+</mo><msub><mi>h</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><mover><mrow><msub><mi>z</mi> <mn>1</mn></msub></mrow><mo>¯</mo></mover><mo>−</mo><mover><mrow><msub><mi>z</mi> <mn>2</mn></msub></mrow><mo>¯</mo></mover><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">(</mo><mover><mrow><msub><mi>h</mi> <mi>i</mi></msub></mrow><mo>¯</mo></mover><mo>+</mo><mover><mrow><msub><mi>h</mi> <mi>j</mi></msub></mrow><mo>¯</mo></mover><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex"> G_{ij}(z_1, z_2) = C_{ij} (z_1 - z_2)^{-(h_i + h_j)} (\overline{z_1} - \overline{z_2})^{-(\overline{h_i} + \overline{h_j})} </annotation></semantics></math></div> <p>with some constant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>ij</mi></msub><mo>∈</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">C_{ij} \in \mathbb{C}</annotation></semantics></math>.</p> </div> <h2 id="2d_cft_on_surfaces_or_arbitrary_genus">2d CFT on surfaces or arbitrary genus</h2> <p>…</p> <h2 id="conformal_anomaly">Conformal anomaly</h2> <p>The <a class="existingWikiWord" href="/nlab/show/Liouville+cocycle">Liouville cocycle</a> appears when one moves from genuine <a class="existingWikiWord" href="/nlab/show/representation">representation</a>s of 2-dimensional <a class="existingWikiWord" href="/nlab/show/conformal+cobordisms">conformal cobordisms</a> to <a class="existingWikiWord" href="/nlab/show/projective+representation">projective representations</a>. The obstruction for such a projective representation to be a genuine representation is precisely given by the central charge <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>; when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">c\neq 0</annotation></semantics></math>, one says that the conformal field theory has a <em>conformal <a class="existingWikiWord" href="/nlab/show/quantum+anomaly">anomaly</a></em>.</p> <p>…</p> <h2 id="cft_in_aqft_language">CFT in AQFT language</h2> <p>In <a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a> conformal field theory is modeled in terms of <a class="existingWikiWord" href="/nlab/show/local+net">local net</a>s that are <a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a>s.</p> <h2 id="FullAndChiral">Full versus chiral CFT</h2> <blockquote> <p>some discussion of full vs. chiral CFT goes here… then:</p> </blockquote> <p>The following result establishes which pairs of <a class="existingWikiWord" href="/nlab/show/vertex+operator+algebras">vertex operator algebras</a> can appear as the left and right chiral parts of a <em><a class="existingWikiWord" href="/nlab/show/full+field+algebra">full field algebra</a></em> in the sense of (<a href="#HuangKong05">Huang-Kong 05</a>), etc. (see <a class="existingWikiWord" href="/nlab/show/vertex+operator+algebra">vertex operator algebra</a>).</p> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>Two <a class="existingWikiWord" href="/nlab/show/modular+tensor+categories">modular tensor categories</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> and <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> are said to have the same <strong>Witt class</strong> if there exist two <a class="existingWikiWord" href="/nlab/show/spherical+categories">spherical categories</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> such that we have an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence</a> of <a class="existingWikiWord" href="/nlab/show/ribbon+categories">ribbon categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>⊠</mo><mi>𝒵</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>D</mi><mo>⊠</mo><mi>𝒵</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> C \boxtimes \mathcal{Z}(S) \simeq D \boxtimes \mathcal{Z}(T) </annotation></semantics></math></div></div> <p>Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒵</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{Z}(S)</annotation></semantics></math> is the category whose objects are pairs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Z, z)</annotation></semantics></math> consisting of an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">Z \in S</annotation></semantics></math> and a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>z</mi> <mi>X</mi></msub><mo>:</mo><mi>Z</mi><mo>⊗</mo><mi>X</mi><mover><mo>→</mo><mo>≃</mo></mover><mi>X</mi><mo>⊗</mo><mi>Z</mi></mrow><annotation encoding="application/x-tex">z_X : Z \otimes X \stackrel{\simeq}{\to} X \otimes Z</annotation></semantics></math>, such that for al objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">X, Y \in S</annotation></semantics></math> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Z</mi><mo>⊗</mo><mi>X</mi><mo>⊗</mo><mi>Y</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>z</mi> <mrow><mi>X</mi><mo>⊗</mo><mi>Y</mi></mrow></msub></mrow></mover></mtd> <mtd></mtd> <mtd><mi>X</mi><mo>⊗</mo><mi>Y</mi><mo>⊗</mo><mi>Z</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>z</mi> <mi>x</mi></msub><mo>⊗</mo><mi>Id</mi></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↗</mo> <mrow><mi>Id</mi><mo>⊗</mo><mpadded width="0"><mrow><msubsup><mi>z</mi> <mi>y</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow></mpadded></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi><mo>⊗</mo><mi>Z</mi><mo>⊗</mo><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Z \otimes X \otimes Y &&\stackrel{z_{X \otimes Y}}{\to}&& X \otimes Y \otimes Z \\ & {}_{\mathllap{z_x \otimes Id}}\searrow && \nearrow_{Id \otimes \mathrlap{z_y^{-1}}} \\ && X \otimes Z \otimes Y } </annotation></semantics></math></div> <p>This becomes a monoidal category itself by setting</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo><mo>⊗</mo><mo stretchy="false">(</mo><mi>W</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mo stretchy="false">(</mo><mi>Z</mi><mo>⊗</mo><mi>W</mi><mo>,</mo><mi>z</mi><mo>⊗</mo><mi>w</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (Z,z) \otimes (W,w) := (Z \otimes W, z \otimes w) \,. </annotation></semantics></math></div> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Michael+M%C3%BCger">Michael Müger</a>)</strong></p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/spherical+category">spherical category</a> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒵</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{Z}(S)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/modular+tensor+category">modular tensor category</a>.</p> </div> <div class="num_theorem"> <h6 id="theorem_2">Theorem</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Michael+M%C3%BCger">Michael Müger</a>)</strong></p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/modular+tensor+category">modular tensor category</a> then there is an equivalence of <a class="existingWikiWord" href="/nlab/show/ribbon+categories">ribbon categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒵</mi><mover><mo>←</mo><mo>≃</mo></mover><mi>C</mi><mo>⊠</mo><mover><mi>C</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex"> \mathcal{Z} \stackrel{\simeq}{\leftarrow} C \boxtimes \bar C </annotation></semantics></math></div> <p>An inverse functor is explicitly constructed by <strong>(<a class="existingWikiWord" href="/nlab/show/Jin-Cheng+Guu">Jin-Cheng Guu</a>)</strong> and <strong>(<span class="newWikiWord">Ying Hong Tham<a href="/nlab/new/Ying+Hong+Tham">?</a></span>)</strong> in (<a href="#GuuTham21">Guu, Tham 21</a>) using graphical calculus. The functor exists for the non-modular case, but it is then just an ambidextrous adjoint.</p> </div> <h2 id="examples">Examples</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ising+model">Ising model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/parafermion">parafermion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/logarithmic+CFT">logarithmic CFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brownian+loop+soup">Brownian loop soup</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a></p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sewing+constraints">sewing constraints</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/modular+invariance">modular invariance</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cardy+condition">Cardy condition</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+structure">conformal structure</a>, <a class="existingWikiWord" href="/nlab/show/moduli+space+of+conformal+structures">moduli space of conformal structures</a>, <a class="existingWikiWord" href="/nlab/show/Teichm%C3%BCller+theory">Teichmüller theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+transformation">conformal transformation</a>, <a class="existingWikiWord" href="/nlab/show/M%C3%B6bius+transformation">Möbius transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vertex+operator+algebra">vertex operator algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+anomaly">conformal anomaly</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rational+conformal+field+theory">rational conformal field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/minimal+model+CFT">minimal model CFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gepner+model">Gepner model</a>, <a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Liouville+cocycle">Liouville cocycle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yangian">Yangian</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/current+algebra">current algebra</a>, <a class="existingWikiWord" href="/nlab/show/affine+Lie+algebra">affine Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Knizhnik-Zamolodchikov+connection">Knizhnik-Zamolodchikov connection</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Liouville+theory">Liouville theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/SCFT">SCFT</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/2d+SCFT">2d SCFT</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/boundary+conformal+field+theory">boundary conformal field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/automorphism+of+a+2d+conformal+field+theory">automorphism of a 2d conformal field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TT+deformation">TT deformation</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="general">General</h3> <p>The mathematical axioms of CFT, as well as its relevance for surface phenomena goes back to</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Alexander+Belavin">Alexander Belavin</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Polyakov">Alexander Polyakov</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Zamolodchikov">Alexander Zamolodchikov</a>, <em>Infinite conformal symmetry in two–dimensional quantum field theory</em>, Nuclear Physics B Volume 241, Issue 2, 23 July 1984, Pages 333-380 (<a href="https://doi.org/10.1016/0550-3213(84)90052-X">doi:10.1016/0550-3213(84)90052-X</a>)</li> </ul> <p>Monographs:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Philippe+Di+Francesco">Philippe Di Francesco</a>, Pierre Mathieu, David Sénéchal: <em>Conformal field theory</em>, Springer (1997) [<a href="https://doi.org/10.1007/978-1-4612-2256-9">doi:10.1007/978-1-4612-2256-9</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Martin+Schottenloher">Martin Schottenloher</a>, <em>A Mathematical Introduction to Conformal Field Theory</em>, Lecture Notes in Physics <strong>759</strong>, Springer 2008 (<a href="https://link.springer.com/book/10.1007/978-3-540-68628-6">doi:10.1007/978-3-540-68628-6</a>, <a href="https://www.mathematik.uni-muenchen.de/~schotten/LNP-cft-pdf/">web</a>)</p> </li> </ul> <p>See also:</p> <ul> <li id="LüscherMack75"><a class="existingWikiWord" href="/nlab/show/Martin+L%C3%BCscher">Martin Lüscher</a>, <a class="existingWikiWord" href="/nlab/show/Gerhard+Mack">Gerhard Mack</a>: <em>Global Conformal Invariance in Quantum Field Theory</em>, Comm. Math. Phys. <strong>41</strong> 3 (1975) 203-234 [<a href="https://doi.org/10.1007/BF01608988">doi:10.1007/BF01608988</a>, <a href="https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-41/issue-3/Global-conformal-invariance-in-quantum-field-theory/cmp/1103898909.full">jstor:cmp/1103898909</a>, <a href="https://inspirehep.net/literature/90687">inspire:90687</a>]</li> </ul> <p>With emphases on <a class="existingWikiWord" href="/nlab/show/braid+group+representations">braid group representations</a> constituted by <a class="existingWikiWord" href="/nlab/show/conformal+blocks">conformal blocks</a> via the <a class="existingWikiWord" href="/nlab/show/Knizhnik-Zamolodchikov+equation">Knizhnik-Zamolodchikov equation</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Toshitake+Kohno">Toshitake Kohno</a>, <em>Conformal field theory and topology</em>, transl. from the 1998 Japanese original by the author. Translations of Mathematical Monographs <strong>210</strong>. Iwanami Series in Modern Mathematics. Amer. Math. Soc. (2002) [<a href="https://bookstore.ams.org/mmono-210">AMS:mmono-210</a>]</li> </ul> <p>Introduction and surveys:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Paul+Ginsparg">Paul Ginsparg</a>, <em>Applied Conformal Field Theory</em>, lectures at: <em><a href="https://inis.iaea.org/search/search.aspx?orig_q=RN:21047524">Fields, strings, critical phenomena, Les Houche Summer School 1988</a>(<a href="https://arxiv.org/abs/hep-th/9108028">arXiv:hep-th/9108028</a>)</em></p> </li> <li id="Gawedzki99"> <p><a class="existingWikiWord" href="/nlab/show/Krzysztof+Gawedzki">Krzysztof Gawedzki</a>, <em>Conformal field theory: a case study</em> (<a href="http://arxiv.org/abs/hep-th/9904145">arXiv:hep-th/9904145</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ingo+Runkel">Ingo Runkel</a>, <em>Boundary problems in conformal field theory</em> (<a href="http://www.math.uni-hamburg.de/home/runkel/PDF/phd.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ralph+Blumenhagen">Ralph Blumenhagen</a>, <a class="existingWikiWord" href="/nlab/show/Erik+Plauschinn">Erik Plauschinn</a>, <em>Introduction to Conformal Field Theory – With Applications to String Theory</em>, Lecture Notes in Physics <strong>779</strong>, Springer (2009) [<a href="https://doi.org/10.1007/978-3-642-00450-6">doi:10.1007/978-3-642-00450-6</a>]</p> </li> <li> <p>Yu Nakayama, <em>A lecture note on scale invariance vs conformal invariance</em>, <a href="http://arxiv.org/abs/1302.0884">arXiv:1302.0884</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Katrin+Wendland">Katrin Wendland</a>, <em>Snapshots of Conformal Field Theory</em>, in:</p> <p>Mathematical Aspects of Quantum Field Theories. Mathematical Physics Studies. Springer 2015 (<a href="http://de.arxiv.org/abs/1404.3108">arXiv:1404.3108</a>, <a href="https://doi.org/10.1007/978-3-319-09949-1_4">doi:10.1007/978-3-319-09949-1_4</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/J%C3%B6rg+Teschner">Jörg Teschner</a>, <em>A guide to two-dimensional conformal field theory</em> (<a href="http://arxiv.org/abs/1708.00680">arXiv:1708.00680</a>)</p> </li> <li> <p>Joaquin Liniado, <em>Two Dimensional Conformal Field Theory and a Primer to Chiral Algebras</em> (<a href="https://arxiv.org/abs/2110.15164">arXiv:2110.15164</a>)</p> </li> <li> <p>Marc Gillioz, <em>Conformal field theory for particle physicists</em> [<a href="https://arxiv.org/abs/2207.09474">arXiv:2207.09474</a>]</p> </li> <li> <p>Satoshi Nawata, Runkai Tao, Daisuke Yokoyama, <em>Fudan lectures on 2d conformal field theory</em> [<a href="https://arxiv.org/abs/2208.05180">arXiv2208.05180</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/J%C3%BCrgen+Fuchs">Jürgen Fuchs</a>, <a class="existingWikiWord" href="/nlab/show/Christoph+Schweigert">Christoph Schweigert</a>, <a class="existingWikiWord" href="/nlab/show/Simon+Wood">Simon Wood</a>, <a class="existingWikiWord" href="/nlab/show/Yang+Yang">Yang Yang</a>, <em>Algebraic structures in two-dimensional conformal field theory</em>, in: <em><a class="existingWikiWord" href="/nlab/show/Encyclopedia+of+Mathematical+Physics+2nd+ed">Encyclopedia of Mathematical Physics 2nd ed</a></em>, Elsevier (2024) [<a href="https://arxiv.org/abs/2305.02773">arXiv:2305.02773</a>]</p> </li> <li> <p>Andrew M. Evans, Alexandra Miller, Aaron Russell, <em>A Conformal Field Theory Primer in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>≥</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">D \geq 3</annotation></semantics></math></em> [<a href="https://arxiv.org/abs/2309.10107">arXiv:2309.10107</a>]</p> </li> </ul> <p>For a discussion of mathematical formalization (<a class="existingWikiWord" href="/nlab/show/vertex+operator+algebras">vertex operator algebras</a>, <a class="existingWikiWord" href="/nlab/show/conformal+nets">conformal nets</a>, <a class="existingWikiWord" href="/nlab/show/functorial+QFT">functorial QFT</a>) see</p> <ul> <li id="Tener18"> <p>James E. Tener, <em>Representation theory in chiral conformal field theory: from fields to observables</em> (<a href="https://arxiv.org/abs/1810.08168">arXiv:1810.08168</a>)</p> </li> <li id="HuangKong05"> <p><a class="existingWikiWord" href="/nlab/show/Yi-Zhi+Huang">Yi-Zhi Huang</a>, <a class="existingWikiWord" href="/nlab/show/Liang+Kong">Liang Kong</a>, <em>Full field algebras</em>, Commun.Math.Phys.272:345-396,2007 (<a href="http://arxiv.org/abs/math/0511328">arXiv:0511328</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Liang+Kong">Liang Kong</a>, <em>Full field algebras, operads and tensor categories</em>, Adv. Math.213:271-340, 2007 (<a href="http://arxiv.org/abs/math/0603065">arXiv:0603065</a>)</p> </li> </ul> <p>For a survey of perspectives in CFT with an eye towards <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> see various contributions in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a> (eds.), <em><a class="existingWikiWord" href="/schreiber/show/Mathematical+Foundations+of+Quantum+Field+and+Perturbative+String+Theory">Mathematical Foundations of Quantum Field and Perturbative String Theory</a></em></li> </ul> <p>Discussion in relation to the <a class="existingWikiWord" href="/nlab/show/AdS-CFT+correspondence">AdS-CFT correspondence</a>:</p> <ul> <li>Matteo Broccoli, <em>Aspects of Conformal Field Theory</em> [<a href="https://arxiv.org/abs/2212.11829">arXiv:2212.11829</a>]</li> </ul> <h3 id="cft_on_complex_curvessurfaces_of_arbitrary_genus">CFT on complex curves/surfaces of arbitrary genus</h3> <p>For chiral 2d CFT:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Edward+Frenkel">Edward Frenkel</a>, <a class="existingWikiWord" href="/nlab/show/David+Ben-Zvi">David Ben-Zvi</a>, <em>Vertex algebras and algebraic curves</em>, Math. Surveys and Monographs <strong>88</strong>, AMS 2001,</p> <p>xii+348 pp. (Bull. AMS. <a href="http://www.ams.org/journals/bull/2002-39-04/S0273-0979-02-00955-2/S0273-0979-02-00955-2.pdf">review</a>, <a href="http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1106.17035&format=complete">ZMATH entry</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alexander+Beilinson">Alexander Beilinson</a>, <a class="existingWikiWord" href="/nlab/show/Vladimir+Drinfeld">Vladimir Drinfeld</a>, <em><a class="existingWikiWord" href="/nlab/show/Chiral+Algebras">Chiral Algebras</a></em>, Colloqium Publications <strong>51</strong>, Amer. Math. Soc. 2004, <a href="http://books.google.hr/books?id=yHZh3p-kFqQC&lpg=PP1&vq=%22Two-dimensional%20conformal%20geometry%20and%20vertex%20operator%20algebras%22&dq=isbn%3A0817638296&pg=PP1#v=onepage&q=%22Two-dimensional%20conformal%20geometry%20and%20vertex%20operator%20algebras%22&f=false">gbooks</a></p> </li> </ul> <h3 id="formulation_by_conformal_nets">Formulation by conformal nets</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/Chris+Douglas">Chris Douglas</a>, <a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Henriques">André Henriques</a>, <em>Topological modular forms and conformal nets</em>, in <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a> (eds.), <em><a class="existingWikiWord" href="/schreiber/show/Mathematical+Foundations+of+Quantum+Field+and+Perturbative+String+Theory">Mathematical Foundations of Quantum Field and Perturbative String Theory</a></em></li> </ul> <p>For further references see <a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a>.</p> <h3 id="formulation_in_full_aqft">Formulation in full AQFT</h3> <p>Formulation in full <a class="existingWikiWord" href="/nlab/show/AQFT+on+curved+spacetimes">AQFT on curved spacetimes</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Marco+Benini">Marco Benini</a>, <a class="existingWikiWord" href="/nlab/show/Luca+Giorgetti">Luca Giorgetti</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Schenkel">Alexander Schenkel</a>, <em>A skeletal model for 2d conformal AQFTs</em> (<a href="https://arxiv.org/abs/2111.01837">arXiv:2111.01837</a>)</li> </ul> <div> <h3 id="2dCFTAsFunctorialQFTReferences"><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> CFT as functorial field theory</h3> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/D%3D2+conformal+field+theory">D=2 conformal field theory</a> as a <a class="existingWikiWord" href="/nlab/show/functorial+field+theory">functorial field theory</a>, namely as a <a class="existingWikiWord" href="/nlab/show/monoidal+functor">monoidal</a> <a class="existingWikiWord" href="/nlab/show/functor">functor</a> from a 2d <a class="existingWikiWord" href="/nlab/show/conformal+cobordism+category">conformal cobordism category</a> to <a class="existingWikiWord" href="/nlab/show/Hilbert+spaces">Hilbert spaces</a>:</p> <ul> <li id="Segal88"><a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a>, <em>The definition of conformal field theory</em>, in: K. Bleuler, M. Werner (eds.), <em>Differential geometrical methods in theoretical physics</em> (Proceedings of Research Workshop, Como 1987), NATO Adv. Sci. Inst., Ser. C: Math. Phys. Sci. <strong>250</strong> Kluwer Acad. Publ., Dordrecht (1988) 165-171 <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/10.1007/978-94-015-7809-7">doi:10.1007/978-94-015-7809-7</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> <p>and including discussion of <a class="existingWikiWord" href="/nlab/show/modular+functors">modular functors</a>:</p> <ul> <li id="Segal89"> <p><a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a>, <em>Two-dimensional conformal field theories and modular functors</em>, in: <em>Proceedings of the IXth International Congress on Mathematical Physics</em>, Swansea, 1988, Hilger, Bristol (1989) 22-37.</p> </li> <li id="Segal04"> <p><a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a>, <em>The definition of conformal field theory</em>, in: <a class="existingWikiWord" href="/nlab/show/Ulrike+Tillmann">Ulrike Tillmann</a> (ed.), <em>Topology, geometry and quantum field theory</em> , London Math. Soc. Lect. Note Ser. <strong>308</strong>, Cambridge University Press (2004) 421-577 <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/10.1017/CBO9780511526398.019">doi:10.1017/CBO9780511526398.019</a>, <a href="https://people.maths.ox.ac.uk/segalg/0521540496txt.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/SegalDefinitionCFT.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></p> </li> </ul> <p>General construction for the case of <a class="existingWikiWord" href="/nlab/show/rational+2d+conformal+field+theory">rational 2d conformal field theory</a> is given by the</p> <ul> <li><em><a class="existingWikiWord" href="/nlab/show/FRS-theorem+on+rational+2d+CFT">FRS-theorem on rational 2d CFT</a></em></li> </ul> <p>See also:</p> <ul> <li id="MooreSegal06"> <p><a class="existingWikiWord" href="/nlab/show/Greg+Moore">Greg Moore</a>, <a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a>, <em>D-branes and K-theory in 2D topological field theory</em> (<a href="http://arxiv.org/abs/hep-th/0609042">arXiv:hep-th/0609042</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Richard+Blute">Richard Blute</a>, <a class="existingWikiWord" href="/nlab/show/Prakash+Panangaden">Prakash Panangaden</a>, <a class="existingWikiWord" href="/nlab/show/Dorette+Pronk">Dorette Pronk</a>, <em>Conformal field theory as a nuclear functor</em>, Electronic Notes in Theoretical Computer Science Volume 172, 1 April 2007, Pages 101-132 GDP Festschrift (<a href="http://aix1.uottawa.ca/~rblute/conf.pdf">pdf</a>, <a href="https://doi.org/10.1016/j.entcs.2007.02.005">doi:10.1016/j.entcs.2007.02.005</a>)</p> </li> </ul> <p>A different but closely analogous development for chiral 2d CFT (<a class="existingWikiWord" href="/nlab/show/vertex+operator+algebras">vertex operator algebras</a>, see <a href="vertex+operator+algebra#AsOperadAlgebras">there</a> for more):</p> <ul> <li id="Huang91"><a class="existingWikiWord" href="/nlab/show/Yi-Zhi+Huang">Yi-Zhi Huang</a>, <em>Geometric interpretation of vertex operator algebras</em>, Proc. Natl. Acad. Sci. USA <strong>88</strong> (1991) pp. 9964-9968 (<a href="https://doi.org/10.1073/pnas.88.22.9964">doi:10.1073/pnas.88.22.9964</a>)</li> </ul> <p>Discussion of the case of <a class="existingWikiWord" href="/nlab/show/Liouville+theory">Liouville theory</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Colin+Guillarmou">Colin Guillarmou</a>, <a class="existingWikiWord" href="/nlab/show/Antti+Kupiainen">Antti Kupiainen</a>, <a class="existingWikiWord" href="/nlab/show/R%C3%A9mi+Rhodes">Rémi Rhodes</a>, <a class="existingWikiWord" href="/nlab/show/Vincent+Vargas">Vincent Vargas</a>, <em>Segal’s axioms and bootstrap for Liouville Theory</em> &lbrack;<a href="https://arxiv.org/abs/2112.14859">arXiv:2112.14859</a>&rbrack;</li> </ul> <p>Early suggestions to refine this to an <a class="existingWikiWord" href="/nlab/show/extended+TQFT">extended</a> <a class="existingWikiWord" href="/nlab/show/2-functor">2-functorial</a> construction:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Stefan+Stolz">Stefan Stolz</a>, <a class="existingWikiWord" href="/nlab/show/Peter+Teichner">Peter Teichner</a>, <em><a class="existingWikiWord" href="/nlab/show/What+is+an+elliptic+object%3F">What is an elliptic object?</a></em></li> </ul> <p>A step towards generalization to <a class="existingWikiWord" href="/nlab/show/2d+super-conformal+field+theory">2d super-conformal field theory</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Stephan+Stolz">Stephan Stolz</a>, <a class="existingWikiWord" href="/nlab/show/Peter+Teichner">Peter Teichner</a>, <em>Supersymmetric field theories and generalized cohomology</em>, in: <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">H. Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">U. Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Mathematical+Foundations+of+Quantum+Field+and+Perturbative+String+Theory">Mathematical Foundations of Quantum Field and Perturbative String Theory</a></em>, Proceedings of Symposia in Pure Mathematics 83 (2011), 279–340 (<a href="https://arxiv.org/abs/1108.0189">arXiv:1108.0189</a>, <a href="https://doi.org/10.1090/pspum/083">doi:10.1090/pspum/083/2742432</a>)</li> </ul> <p>Discussion of 2-functorial chiral 2d CFT:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Henriques">André Henriques</a>, <em>The complex cobordism 2-category</em>, 2021 (<a href="http://andreghenriques.com/ComplexCob2CatandCentralExt.mp4">video</a>)</li> </ul> </div> <h3 id="FRSFormalism">Formulation by algebra in modular tensor categories</h3> <p>For full CFT, The special case of <em>rational</em> CFT has been essentially entirely formalized and classified. The classification result for full rational 2d CFT was given by Fjelstad–Fuchs–<a class="existingWikiWord" href="/nlab/show/Ingo+Runkel">Runkel</a>–<a class="existingWikiWord" href="/nlab/show/Christoph+Schweigert">Schweigert</a></p> <ul> <li> <p><a href="http://golem.ph.utexas.edu/string/archives/000747.html">FRS reviews</a></p> </li> <li> <p><a href="http://golem.ph.utexas.edu/string/archives/000813.html">The FRS theorem of RCFT</a></p> </li> <li id="KapustinSaulina"> <p><a class="existingWikiWord" href="/nlab/show/Anton+Kapustin">Anton Kapustin</a>, <a class="existingWikiWord" href="/nlab/show/Natalia+Saulina">Natalia Saulina</a> <em>Surface operators in 3d TFT and 2d Rational CFT</em>, in <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a> (eds.), <em><a class="existingWikiWord" href="/schreiber/show/Mathematical+Foundations+of+Quantum+Field+and+Perturbative+String+Theory">Mathematical Foundations of Quantum Field and Perturbative String Theory</a></em> (<a href="http://arxiv.org/abs/1012.0911">arXiv:1012.0911</a>)</p> </li> <li id="Kong"> <p><a class="existingWikiWord" href="/nlab/show/Liang+Kong">Liang Kong</a>, <em>Conformal field theory and a new geometry</em> , in <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a> (eds.), <em><a class="existingWikiWord" href="/schreiber/show/Mathematical+Foundations+of+Quantum+Field+and+Perturbative+String+Theory">Mathematical Foundations of Quantum Field and Perturbative String Theory</a></em> (<a href="http://arxiv.org/abs/1107.3649">arXiv:1107.3649</a>)</p> </li> </ul> <p>…</p> </body></html> </div> <div class="revisedby"> <p> Last revised on July 31, 2024 at 19:51:02. 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