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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> D=6 N=(2,0) SCFT </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/3066/#Item_20" 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" 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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="supergeometry">Super-Geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a></strong> and (<a class="existingWikiWord" href="/nlab/show/synthetic+differential+supergeometry">synthetic</a> ) <strong><a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a></strong></p> <h2 id="background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/graded+object">graded object</a></p> </li> </ul> <h2 id="introductions">Introductions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+superalgebra">geometry of physics – superalgebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+supergeometry">geometry of physics – supergeometry</a></p> </li> </ul> <h2 id="superalgebra">Superalgebra</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/super+commutative+monoid">super commutative monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+abelian+group">super abelian group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+ring">super ring</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supercommutative+ring">supercommutative ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exterior+ring">exterior ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Clifford+ring">Clifford ring</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+module">super module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+vector+space">super vector space</a>, <a class="existingWikiWord" href="/nlab/show/SVect">SVect</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+algebra">super algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supercommutative+algebra">supercommutative algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exterior+algebra">exterior algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Clifford+algebra">Clifford algebra</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/superdeterminant">superdeterminant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">super Poincare Lie algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+complex+of+super+vector+spaces">chain complex of super vector spaces</a> (<a class="existingWikiWord" href="/nlab/show/model+structure+on+chain+complexes+of+super+vector+spaces">model structure</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+graded-commutative+superalgebra">differential graded-commutative superalgebra</a> (<a class="existingWikiWord" href="/nlab/show/model+structure+on+differential+graded-commutative+superalgebras">model structure</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+L-infinity+algebra">super L-infinity algebra</a></p> </li> </ul> <h2 id="supergeometry">Supergeometry</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/superpoint">superpoint</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Cartesian+space">super Cartesian space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supermanifold">supermanifold</a>, <a class="existingWikiWord" href="/nlab/show/SDiff">SDiff</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/NQ-supermanifold">NQ-supermanifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+vector+bundle">super vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+supermanifold">complex supermanifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+supermanifold">Euclidean supermanifold</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+spacetime">super spacetime</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Minkowski+spacetime">super Minkowski spacetime</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integration+over+supermanifolds">integration over supermanifolds</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Berezin+integral">Berezin integral</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Lie+group">super Lie group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Lie+group">super translation group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Euclidean+group">super Euclidean group</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+%E2%88%9E-groupoid">super ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+formal+smooth+%E2%88%9E-groupoid">super formal smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+line+2-bundle">super line 2-bundle</a></p> </li> </ul> <h2 id="supersymmetry">Supersymmetry</h2> <p><a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/division+algebra+and+supersymmetry">division algebra and supersymmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">super Poincare Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supermultiplet">supermultiplet</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BPS+state">BPS state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/M-theory+super+Lie+algebra">M-theory super Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/type+II+super+Lie+algebra">type II super Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a>, <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a></p> </li> </ul> <h2 id="supersymmetric_field_theory">Supersymmetric field theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/superfield">superfield</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supersymmetric+quantum+mechanics">supersymmetric quantum mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adinkra">adinkra</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">super Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/gauged+supergravity">gauged supergravity</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/superstring+theory">superstring theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a></p> </li> </ul> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometric+model+for+elliptic+cohomology">geometric model for elliptic cohomology</a></li> </ul> <div> <p> <a href="/nlab/edit/supergeometry+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="string_theory">String theory</h4> <div class="hide"><div> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a></strong></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/books+about+string+theory">books about string theory</a></p> </li> </ul> <h3 id="ingredients">Ingredients</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a>,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/CFT">CFT</a>, <a class="existingWikiWord" href="/nlab/show/perturbation+theory">perturbation theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+QFT">effective background QFT</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>, <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a>, <a class="existingWikiWord" href="/nlab/show/quantum+gravity">quantum gravity</a></li> </ul> </li> </ul> <h3 id="critical_string_models">Critical string models</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a>, <a class="existingWikiWord" href="/nlab/show/differential+string+structure">differential string structure</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dual+heterotic+string+theory">dual heterotic string theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+fivebrane+structure">differential fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/type+IIA+string+theory">type IIA string theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/type+IIB+string+theory">type IIB string theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/F-theory">F-theory</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+field+theory">string field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/duality+in+string+theory">duality in string theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/T-duality">T-duality</a>, <a class="existingWikiWord" href="/nlab/show/mirror+symmetry">mirror symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/S-duality">S-duality</a>, <a class="existingWikiWord" href="/nlab/show/electric-magnetic+duality">electric-magnetic duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/U-duality">U-duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open%2Fclosed+string+duality">open/closed string duality</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/AdS%2FCFT+correspondence">AdS/CFT correspondence</a>, <a class="existingWikiWord" href="/nlab/show/holographic+principle">holographic principle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a>, <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%C5%99ava-Witten+theory">Hořava-Witten theory</a></li> </ul> </li> </ul> <h3 id="extended_objects">Extended objects</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/brane">brane</a></p> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/D0-brane">D0-brane</a>, <a class="existingWikiWord" href="/nlab/show/D2-brane">D2-brane</a>, <a class="existingWikiWord" href="/nlab/show/D4-brane">D4-brane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D1-brane">D1-brane</a>, <a class="existingWikiWord" href="/nlab/show/D3-brane">D3-brane</a>, <a class="existingWikiWord" href="/nlab/show/D5-brane">D5-brane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/RR-field">RR-field</a>, <a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/NS-brane">NS-brane</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>, <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/B2-field">B2-field</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/NS5-brane">NS5-brane</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/B6-field">B6-field</a></li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/M-brane">M-brane</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/C3-field">C3-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ABJM+theory">ABJM theory</a>, <a class="existingWikiWord" href="/nlab/show/BLG+model">BLG model</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/C6-field">C6-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a></p> </li> </ul> </li> </ul> </li> </ul> <h3 id="topological_strings">Topological strings</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a>, <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+M-theory">topological M-theory</a></p> </li> </ul> <h2 id="backgrounds">Backgrounds</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/target+space">target space</a>, <a class="existingWikiWord" href="/nlab/show/background+gauge+field">background gauge field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+smooth+cohomology+in+string+theory">twisted smooth cohomology in string theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/landscape+of+string+theory+vacua">landscape of string theory vacua</a></p> </li> </ul> <h2 id="phenomenology">Phenomenology</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/string+phenomenology">string phenomenology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/moduli+stabilization">moduli stabilization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/G%E2%82%82-MSSM">G₂-MSSM</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/string+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#GeometricEngineering'>Geometric engineering</a></li> <li><a href='#holographic_dual'>Holographic dual</a></li> <li><a href='#realization_of_quantum_chromodynamics'>Realization of quantum chromodynamics</a></li> <li><a href='#solitonic_1branes'>Solitonic 1-branes</a></li> <li><a href='#CompactificationOnARiemannSurface'>Compactification on a Riemann surface and AGT correspondence</a></li> <li><a href='#twistor_space_description'>Twistor space description</a></li> </ul> <li><a href='#related_entries'>Related entries</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#compactification_to_5d_superyangmills'>Compactification to 5d super-Yang-Mills</a></li> <li><a href='#CompactificationTo4DSYM'>Compactification to 4d super-Yang-Mills</a></li> <li><a href='#CompactificationTo2dCFT'>Compactification to 2d CFT</a></li> <li><a href='#ade_classification'>ADE classification</a></li> <li><a href='#ModelsAndSpecialProperties'>Models and special properties</a></li> <li><a href='#ReferencesOnTheHolographicDual'>On the holographic dual</a></li> <li><a href='#solitonic_1brane_excitations'>Solitonic 1-brane excitations</a></li> <li><a href='#TheoryXAsAnFQFTReferences'>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><mn>6</mn></mrow><annotation encoding="application/x-tex">D=6</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒩</mi><mo>=</mo><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{N}=(2,0)</annotation></semantics></math> SCFT as an extended functorial field theory</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>According to the <a href="supersymmetry#ClassificationSuperconformal">classification of superconformal symmetry</a>, there should exist <a class="existingWikiWord" href="/nlab/show/superconformal+field+theories">superconformal field theories</a> in 6 dimensions…</p> <div> <table><thead><tr><th><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></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></th><th><a class="existingWikiWord" href="/nlab/show/superconformal+super+Lie+algebra">superconformal super Lie algebra</a></th><th><a class="existingWikiWord" href="/nlab/show/R-symmetry">R-symmetry</a></th><th><a class="existingWikiWord" href="/nlab/show/black+brane">black brane</a> <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> <br /> <a class="existingWikiWord" href="/nlab/show/superconformal+field+theory">superconformal field theory</a> <br /> via <a class="existingWikiWord" href="/nlab/show/AdS-CFT">AdS-CFT</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mn>3</mn><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}3\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}2k+1\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mi>B</mi><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo><mo>≃</mo></mrow><annotation encoding="application/x-tex">\phantom{A}B(k,2) \simeq </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/osp">osp</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">|</mo><mn>4</mn><mo stretchy="false">)</mo><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">(2k+1 \vert 4)\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mi>SO</mi><mo stretchy="false">(</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}SO(2k+1)\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mn>3</mn><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}3\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mn>2</mn><mi>k</mi><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}2k\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mi>D</mi><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo><mo>≃</mo></mrow><annotation encoding="application/x-tex">\phantom{A}D(k,2)\simeq </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/osp">osp</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>k</mi><mo stretchy="false">|</mo><mn>4</mn><mo stretchy="false">)</mo><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">(2k \vert 4)\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mi>SO</mi><mo stretchy="false">(</mo><mn>2</mn><mi>k</mi><mo stretchy="false">)</mo><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}SO(2k)\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a> <br /> <a class="existingWikiWord" href="/nlab/show/D%3D3+SYM">D=3 SYM</a> <br /> <a class="existingWikiWord" href="/nlab/show/BLG+model">BLG model</a> <br /> <a class="existingWikiWord" href="/nlab/show/ABJM+model">ABJM model</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mn>4</mn><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}4\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mi>k</mi><mo>+</mo><mn>1</mn><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}k+1\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mi>A</mi><mo stretchy="false">(</mo><mn>3</mn><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>𝔰𝔩</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">|</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}A(3,k)\simeq \mathfrak{sl}(4 \vert k+1)\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mi>U</mi><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}U(k+1)\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D3-brane">D3-brane</a> <br /> <a class="existingWikiWord" href="/nlab/show/D%3D4+N%3D4+SYM">D=4 N=4 SYM</a> <br /> <a class="existingWikiWord" href="/nlab/show/D%3D4+N%3D2+SYM">D=4 N=2 SYM</a> <br /> <a class="existingWikiWord" href="/nlab/show/D%3D4+N%3D1+SYM">D=4 N=1 SYM</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mn>5</mn><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}5\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mn>1</mn><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}1\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mi>F</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}F(4)\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mi>SO</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}SO(3)\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D4-brane">D4-brane</a> <br /> <a class="existingWikiWord" href="/nlab/show/D%3D5+SYM">D=5 SYM</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mn>6</mn><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}6\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mi>k</mi><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}k\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mi>D</mi><mo stretchy="false">(</mo><mn>4</mn><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo>≃</mo></mrow><annotation encoding="application/x-tex">\phantom{A}D(4,k) \simeq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/osp">osp</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>8</mn><mo stretchy="false">|</mo><mn>2</mn><mi>k</mi><mo stretchy="false">)</mo><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">(8 \vert 2k)\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mi>Sp</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}Sp(k)\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> <br /> <a class="existingWikiWord" href="/nlab/show/D%3D6+N%3D%282%2C0%29+SCFT">D=6 N=(2,0) SCFT</a> <br /> <a class="existingWikiWord" href="/nlab/show/D%3D6+N%3D%281%2C0%29+SCFT">D=6 N=(1,0) SCFT</a></td></tr> </tbody></table> <p>(<a href="supersymmetry#Shnider88">Shnider 88</a>, also <a href="supersymmetry#Nahm78">Nahm 78</a>, see <a href="supersymmetry#Minwalla98">Minwalla 98, section 4.2</a>)</p> </div> <p>…with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,0)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a>, that contain a <a class="existingWikiWord" href="/nlab/show/self-dual+higher+gauge+theory">self-dual higher gauge theory</a> whose field configurations are <a class="existingWikiWord" href="/nlab/show/connections+on+a+2-bundle">connections on a 2-bundle</a> (a <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 2-bundle with connection</a> in the abelian case).</p> <p>This was derived by <a href="#ClausKalloshProeyen97">Claus, Kallosh & van Proeyen 1997</a>, in the abelian case and to low order, as the small fluctuations of the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+sigma-model">Green-Schwarz sigma-model</a> of the <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> around the embedding in the <a class="existingWikiWord" href="/nlab/show/asymptotic+boundary">asymptotic boundary</a> of the <a class="existingWikiWord" href="/nlab/show/AdS-spacetime">AdS-spacetime</a> that is the <a class="existingWikiWord" href="/nlab/show/near-horizon+geometry">near-horizon geometry</a> of the <a class="existingWikiWord" href="/nlab/show/black+brane">black</a> M5-brane.</p> <p>In accord with this the <a href="AdS-CFT#AdS7CFT6">AdS7-CFT6</a> correspondence predicts that the nonabelian 6d theory is the corresponding theory for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math> coincident M5-branes.</p> <p>In the non-abelian case it is expected (<a href="#Witten07">Witten 07</a>) that the compactification of such theories are at the heart of the phenomenon that leads to <a class="existingWikiWord" href="/nlab/show/S-duality">S-duality</a> in <a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">super Yang-Mills theory</a> and further to <a class="existingWikiWord" href="/nlab/show/geometric+Langlands+duality">geometric Langlands duality</a> (<a href="#Witten09">Witten 09</a>).</p> <p>Due to the <a class="existingWikiWord" href="/nlab/show/self-dual+higher+gauge+theory">self-duality</a> a characterization of these theories by an <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> is subtle. Therefore more direct descriptions are still under investigation (for instance <a href="#SSW11">SSW11</a>). Review includes (<a href="#Moore11">Moore11</a>, <a href="#Moore12">Moore 12</a>).</p> <h2 id="properties">Properties</h2> <h3 id="GeometricEngineering">Geometric engineering</h3> <p>For <a class="existingWikiWord" href="/nlab/show/geometric+engineering">geometric engineering</a> of the <a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-superconformal+QFT">6d (2,0)-superconformal QFT</a>, see at <em><a href="duality+between+M-theory+on+Z2-orbifolds+and+type+IIB+string+theory+on+K3-fibrations#GeometricEngineeringOfThe6d2SuperconformalQFT">duality between M-theory on Z2-orbifolds and type IIB string theory on K3-fibrations – Geometric engineering of 6d (2,0)-SCFT</a></em>.</p> <h3 id="holographic_dual">Holographic dual</h3> <p>Under <a class="existingWikiWord" href="/nlab/show/AdS-CFT">AdS7/CFT6</a> the 6d <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,0)</annotation></semantics></math>-superconformal QFT is supposed to be dual to <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">M-theory</a> on <a class="existingWikiWord" href="/nlab/show/anti+de+Sitter+spacetime">anti de Sitter spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>AdS</mi> <mn>7</mn></msub><mo>×</mo><msup><mi>S</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">AdS_7 \times S^4</annotation></semantics></math>.</p> <p>See <a class="existingWikiWord" href="/nlab/show/AdS%2FCFT+correspondence">AdS/CFT correspondence</a> for more on this.</p> <h3 id="realization_of_quantum_chromodynamics">Realization of quantum chromodynamics</h3> <p>See at <em><a class="existingWikiWord" href="/nlab/show/AdS-QCD+correspondence">AdS-QCD correspondence</a></em>.</p> <h3 id="solitonic_1branes">Solitonic 1-branes</h3> <p>The 5d <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,0)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/SCFT">SCFT</a> has tensionless 1-<a class="existingWikiWord" href="/nlab/show/brane">brane</a> configurations. From the point of view of the ambient <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a> these are the boundaries of <a class="existingWikiWord" href="/nlab/show/M2-branes">M2-branes</a> ending on the <a class="existingWikiWord" href="/nlab/show/M5-branes">M5-branes</a>. (<a href="#GGT">GGT</a>)</p> <h3 id="CompactificationOnARiemannSurface">Compactification on a Riemann surface and AGT correspondence</h3> <div class="float_right_image" style="margin: 0px 20px 10px 20px"> <img src="/nlab/files/6d_qft_graph.gif" width="600px" alt="Compactification diagram" /> </div> <blockquote> <p>(graphics taken from (<a href="#Workshop14">Workshop 14</a>))</p> </blockquote> <p>The <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">compactification</a> of the 5-brane on a <a class="existingWikiWord" href="/nlab/show/Riemann+surface">Riemann surface</a> yields as <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory</a> <a class="existingWikiWord" href="/nlab/show/N%3D2+D%3D4+super+Yang-Mills+theory">N=2 D=4 super Yang-Mills theory</a>. See at <em><a href="N%3D2+D%3D4+super+Yang-Mills+theory#ConstructionByCompactificationOf5Branes">N=2 D=4 SYM – Construction by compactification of 5-branes</a></em>.</p> <p>The <em><a class="existingWikiWord" href="/nlab/show/AGT+correspondence">AGT correspondence</a></em> relates the <a class="existingWikiWord" href="/nlab/show/partition+function">partition function</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><msup><mo stretchy="false">)</mo> <mrow><mi>n</mi><mo>+</mo><mn>3</mn><mi>g</mi><mo>−</mo><mn>3</mn></mrow></msup></mrow><annotation encoding="application/x-tex">SU(2)^{n+3g-3}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/N%3D2+D%3D4+super+Yang-Mills+theory">N=2 D=4 super Yang-Mills theory</a> obtained by <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">compactifying</a> the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>6</mn><mi>d</mi></mrow><annotation encoding="application/x-tex">6d</annotation></semantics></math> M5-brane theory on a <a class="existingWikiWord" href="/nlab/show/Riemann+surface">Riemann surface</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mrow><mi>g</mi><mo>,</mo><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">C_{g,n}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/genus">genus</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> punctures to 2d <a class="existingWikiWord" href="/nlab/show/Liouville+theory">Liouville theory</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mrow><mi>g</mi><mo>,</mo><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">C_{g,n}</annotation></semantics></math>.</p> <p>More generally, this kind of construction allows to describe the 6d (2,0)-theory as a “<a class="existingWikiWord" href="/nlab/show/2d+SCFT">2d SCFT</a> with values in <a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">4d SYM</a>”. See at <em><a class="existingWikiWord" href="/nlab/show/AGT+correspondence">AGT correspondence</a></em> for more on this.</p> <h3 id="twistor_space_description">Twistor space description</h3> <p>Famously the solutions to <a class="existingWikiWord" href="/nlab/show/self-dual+Yang-Mills+theory">self-dual Yang-Mills theory</a> in <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> 4 can be obtained as images of degree-2 cohomology classes under the <a class="existingWikiWord" href="/nlab/show/Penrose-Ward+twistor+transform">Penrose-Ward twistor transform</a>. Since the 6d QFT on the <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> contains a 2-form <a class="existingWikiWord" href="/nlab/show/self-dual+higher+gauge+field">self-dual higher gauge field</a> it seems natural to expect that it can be described by a higher analogy of the twistor transform. For references exploring this idea see at <em><a href="twistor+space#ReferencesApplicationToSelfDual2FormField">twistor space – References – Application to the self-dual 2-form field in 6d</a></em>.</p> <h2 id="related_entries">Related entries</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/little+string+theory">little string theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D%3D6+N%3D%281%2C0%29+SCFT">D=6 N=(1,0) SCFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D%3D6+supergravity">D=6 supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D%3D2+N%3D%282%2C0%29+SCFT">D=2 N=(2,0) SCFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D%3D5+super+Yang-Mills+theory">D=5 super Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/M5-brane+elliptic+genus">M5-brane elliptic genus</a></p> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a> induced via <a class="existingWikiWord" href="/nlab/show/AdS-CFT+correspondence">AdS-CFT correspondence</a></strong></p> <table><thead><tr><th><strong><a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a> perspective via <a class="existingWikiWord" href="/nlab/show/AdS7-CFT6">AdS7-CFT6</a></strong></th><th><a class="existingWikiWord" href="/nlab/show/F-theory">F-theory</a> perspective</th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/11d+supergravity">11d supergravity</a>/<a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo stretchy="false">↓</mo></mrow><annotation encoding="application/x-tex">\;\;\;\;\downarrow</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+compactification">Kaluza-Klein compactification</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">S^4</annotation></semantics></math></td><td style="text-align: left;">compactificationon <a class="existingWikiWord" href="/nlab/show/elliptic+fibration">elliptic fibration</a> followed by <a class="existingWikiWord" href="/nlab/show/T-duality">T-duality</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/7-dimensional+supergravity">7-dimensional supergravity</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo stretchy="false">↓</mo></mrow><annotation encoding="application/x-tex">\;\;\;\;\downarrow</annotation></semantics></math> topological sector</td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/7-dimensional+Chern-Simons+theory">7-dimensional Chern-Simons theory</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo stretchy="false">↓</mo></mrow><annotation encoding="application/x-tex">\;\;\;\;\downarrow </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/AdS-CFT">AdS7-CFT6 holographic duality</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-superconformal+QFT">6d (2,0)-superconformal QFT</a> on the <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> with <a class="existingWikiWord" href="/nlab/show/CFT">conformal invariance</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> worldvolume theory</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo stretchy="false">↓</mo></mrow><annotation encoding="application/x-tex">\;\;\;\; \downarrow</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a> on <a class="existingWikiWord" href="/nlab/show/Riemann+surface">Riemann surface</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/double+dimensional+reduction">double dimensional reduction</a> on <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a>/<a class="existingWikiWord" href="/nlab/show/F-theory">F-theory</a> <a class="existingWikiWord" href="/nlab/show/elliptic+fibration">elliptic fibration</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/N%3D2+D%3D4+super+Yang-Mills+theory">N=2 D=4 super Yang-Mills theory</a> with <a class="existingWikiWord" href="/nlab/show/Montonen-Olive+duality">Montonen-Olive</a> <a class="existingWikiWord" href="/nlab/show/S-duality">S-duality</a> invariance; <a class="existingWikiWord" href="/nlab/show/AGT+correspondence">AGT correspondence</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D3-brane">D3-brane</a> worldvolume theory with type IIB <a class="existingWikiWord" href="/nlab/show/S-duality">S-duality</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo stretchy="false">↓</mo></mrow><annotation encoding="application/x-tex">\;\;\;\;\; \downarrow</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/topological+twist">topological twist</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/topologically+twisted+N%3D2+D%3D4+super+Yang-Mills+theory">topologically twisted N=2 D=4 super Yang-Mills theory</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo stretchy="false">↓</mo></mrow><annotation encoding="application/x-tex">\;\;\;\; \downarrow</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a> on <a class="existingWikiWord" href="/nlab/show/Riemann+surface">Riemann surface</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Bun</mi> <mi>G</mi></msub></mrow><annotation encoding="application/x-tex">Bun_G</annotation></semantics></math>, <a class="existingWikiWord" href="/nlab/show/Donaldson+theory">Donaldson theory</a></td><td style="text-align: left;"></td></tr> </tbody></table> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <table><thead><tr><th><strong><a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a> induced via <a class="existingWikiWord" href="/nlab/show/AdS5-CFT4">AdS5-CFT4</a></strong></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo stretchy="false">↓</mo></mrow><annotation encoding="application/x-tex">\;\;\;\;\downarrow</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+compactification">Kaluza-Klein compactification</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>5</mn></msup></mrow><annotation encoding="application/x-tex">S^5</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo stretchy="false">↓</mo></mrow><annotation encoding="application/x-tex">\;\;\;\; \downarrow</annotation></semantics></math> topological sector</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/5-dimensional+Chern-Simons+theory">5-dimensional Chern-Simons theory</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo stretchy="false">↓</mo></mrow><annotation encoding="application/x-tex">\;\;\;\;\downarrow </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/AdS-CFT">AdS5-CFT4 holographic duality</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/N%3D4+D%3D4+super+Yang-Mills+theory">N=4 D=4 super Yang-Mills theory</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo stretchy="false">↓</mo></mrow><annotation encoding="application/x-tex">\;\;\;\;\; \downarrow</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/topological+twist">topological twist</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/topologically+twisted+N%3D4+D%3D4+super+Yang-Mills+theory">topologically twisted N=4 D=4 super Yang-Mills theory</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo stretchy="false">↓</mo></mrow><annotation encoding="application/x-tex">\;\;\;\; \downarrow</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a> on <a class="existingWikiWord" href="/nlab/show/Riemann+surface">Riemann surface</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Bun</mi> <mi>G</mi></msub></mrow><annotation encoding="application/x-tex">Bun_G</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Loc</mi> <mi>G</mi></msub></mrow><annotation encoding="application/x-tex">Loc_G</annotation></semantics></math>, <a class="existingWikiWord" href="/nlab/show/geometric+Langlands+correspondence">geometric Langlands correspondence</a></td></tr> </tbody></table> </div><div> <table><thead><tr><th><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></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></th><th><a class="existingWikiWord" href="/nlab/show/superconformal+super+Lie+algebra">superconformal super Lie algebra</a></th><th><a class="existingWikiWord" href="/nlab/show/R-symmetry">R-symmetry</a></th><th><a class="existingWikiWord" href="/nlab/show/black+brane">black brane</a> <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> <br /> <a class="existingWikiWord" href="/nlab/show/superconformal+field+theory">superconformal field theory</a> <br /> via <a class="existingWikiWord" href="/nlab/show/AdS-CFT">AdS-CFT</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mn>3</mn><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}3\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}2k+1\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mi>B</mi><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo><mo>≃</mo></mrow><annotation encoding="application/x-tex">\phantom{A}B(k,2) \simeq </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/osp">osp</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">|</mo><mn>4</mn><mo stretchy="false">)</mo><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">(2k+1 \vert 4)\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mi>SO</mi><mo stretchy="false">(</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}SO(2k+1)\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mn>3</mn><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}3\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mn>2</mn><mi>k</mi><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}2k\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mi>D</mi><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo><mo>≃</mo></mrow><annotation encoding="application/x-tex">\phantom{A}D(k,2)\simeq </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/osp">osp</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>k</mi><mo stretchy="false">|</mo><mn>4</mn><mo stretchy="false">)</mo><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">(2k \vert 4)\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mi>SO</mi><mo stretchy="false">(</mo><mn>2</mn><mi>k</mi><mo stretchy="false">)</mo><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}SO(2k)\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a> <br /> <a class="existingWikiWord" href="/nlab/show/D%3D3+SYM">D=3 SYM</a> <br /> <a class="existingWikiWord" href="/nlab/show/BLG+model">BLG model</a> <br /> <a class="existingWikiWord" href="/nlab/show/ABJM+model">ABJM model</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mn>4</mn><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}4\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mi>k</mi><mo>+</mo><mn>1</mn><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}k+1\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mi>A</mi><mo stretchy="false">(</mo><mn>3</mn><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>𝔰𝔩</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">|</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}A(3,k)\simeq \mathfrak{sl}(4 \vert k+1)\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mi>U</mi><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}U(k+1)\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D3-brane">D3-brane</a> <br /> <a class="existingWikiWord" href="/nlab/show/D%3D4+N%3D4+SYM">D=4 N=4 SYM</a> <br /> <a class="existingWikiWord" href="/nlab/show/D%3D4+N%3D2+SYM">D=4 N=2 SYM</a> <br /> <a class="existingWikiWord" href="/nlab/show/D%3D4+N%3D1+SYM">D=4 N=1 SYM</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mn>5</mn><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}5\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mn>1</mn><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}1\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mi>F</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}F(4)\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mi>SO</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}SO(3)\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D4-brane">D4-brane</a> <br /> <a class="existingWikiWord" href="/nlab/show/D%3D5+SYM">D=5 SYM</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mn>6</mn><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}6\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mi>k</mi><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}k\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mi>D</mi><mo stretchy="false">(</mo><mn>4</mn><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo>≃</mo></mrow><annotation encoding="application/x-tex">\phantom{A}D(4,k) \simeq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/osp">osp</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>8</mn><mo stretchy="false">|</mo><mn>2</mn><mi>k</mi><mo stretchy="false">)</mo><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">(8 \vert 2k)\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mi>Sp</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}Sp(k)\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> <br /> <a class="existingWikiWord" href="/nlab/show/D%3D6+N%3D%282%2C0%29+SCFT">D=6 N=(2,0) SCFT</a> <br /> <a class="existingWikiWord" href="/nlab/show/D%3D6+N%3D%281%2C0%29+SCFT">D=6 N=(1,0) SCFT</a></td></tr> </tbody></table> <p>(<a href="supersymmetry#Shnider88">Shnider 88</a>, also <a href="supersymmetry#Nahm78">Nahm 78</a>, see <a href="supersymmetry#Minwalla98">Minwalla 98, section 4.2</a>)</p> </div> <h2 id="references">References</h2> <blockquote> <p>See also the references at <em><a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a></em>.</p> </blockquote> <h3 id="general">General</h3> <p>The first indication of a 6d theory with a self-dual 2-form field appears in</p> <ul> <li id="Witten95"><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, section 1 of <em>Some comments on string dynamics</em> (<a href="http://arxiv.org/abs/hep-th/9507121">hepth/9507121</a>)</li> </ul> <p>Derivation of the abelian 6d theory to low order as the small fluctuations of the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+sigma-model">Green-Schwarz sigma-model</a> of the <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> around a solution embedding as the asymptotic boundary of the <a class="existingWikiWord" href="/nlab/show/AdS-spacetime">AdS-spacetime</a> <a class="existingWikiWord" href="/nlab/show/near-horizon+geometry">near-horizon geometry</a> of a <a class="existingWikiWord" href="/nlab/show/black+brane">black</a> 5-brane is due to</p> <ul> <li id="ClausKalloshProeyen97"><a class="existingWikiWord" href="/nlab/show/Piet+Claus">Piet Claus</a>, <a class="existingWikiWord" href="/nlab/show/Renata+Kallosh">Renata Kallosh</a>, <a class="existingWikiWord" href="/nlab/show/Antoine+Van+Proeyen">Antoine Van Proeyen</a>, <em>M 5-brane and superconformal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,2)</annotation></semantics></math> tensor multiplet in 6 dimensions</em>, Nucl. Phys. B <strong>518</strong> (1998) 117-150 [<a href="https://doi.org/10.1016/S0550-3213(98)00137-0">doi:10.1016/S0550-3213(98)00137-0</a>, <a href="http://arxiv.org/abs/hep-th/9711161">arXiv:hep-th/9711161</a>]</li> </ul> <p>General survey:</p> <ul> <li id="Moore11"> <p><a class="existingWikiWord" href="/nlab/show/Greg+Moore">Greg Moore</a>, <em>On the role of six‐dimensional <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,0)</annotation></semantics></math>-theories in recent developments in Physical Mathematics</em>, talk at <a href="http://www2.physics.uu.se/external/strings2011/">Strings 2011</a> (<a href="http://www.physics.rutgers.edu/~gmoore/Strings2011FinalPDF.pdf">pdf slides</a>)</p> </li> <li id="Moore12"> <p><a class="existingWikiWord" href="/nlab/show/Greg+Moore">Greg Moore</a>, <em>Applications of the six-dimensional (2,0) theories to Physical Mathematics</em>, <a href="http://www.hcm.uni-bonn.de/events/eventpages/felix-klein-lectures/applications-of-the-six-dimensional-20-theories-to-physical-mathematics/">Felix Klein lectures Bonn (2012)</a> (<a href="http://www.physics.rutgers.edu/~gmoore/FelixKleinLectureNotes.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/MooreKleinLectures12.pdf" title="pdf">pdf</a>)</p> </li> <li id="Workshop14"> <p><a class="existingWikiWord" href="/nlab/show/Qiaochu+Yuan">Qiaochu Yuan</a>: <em><a href="https://web.archive.org/web/20150513204032/https://math.berkeley.edu/~qchu/Notes/6d/">lecture notes</a></em> for <em><a href="http://www.math.northwestern.edu/~celliott/workshop/">Mathematical Aspects of Six-Dimensional Quantum Field Theories</a></em>, Berkeley, December 8th-12th, 2014 at the University of California, Berkeley</p> </li> </ul> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/anomaly+cancellation">anomaly cancellation</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Kantaro+Ohmori">Kantaro Ohmori</a>, <a class="existingWikiWord" href="/nlab/show/Hiroyuki+Shimizu">Hiroyuki Shimizu</a>, <a class="existingWikiWord" href="/nlab/show/Yuji+Tachikawa">Yuji Tachikawa</a>, Kazuya Yonekura, <em>Anomaly polynomial of general 6d SCFTs</em>, Progress of Theoretical and Experimental Physics, Volume 2014, Issue 10, October 2014, 103B07 (<a href="https://arxiv.org/abs/1408.5572">arXiv:1408.5572</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hiroyuki+Shimizu">Hiroyuki Shimizu</a>, <em>Aspects of anomalies in 6d superconformal field theories</em>, Tokyo 2018 (<a href="https://inspirehep.net/literature/1802462">spire:1802462</a>, <a class="existingWikiWord" href="/nlab/files/ShimizuAnomaliesIn6dSCFT.pdf" title="pdf">pdf</a>)</p> </li> </ul> <p>Review of construction from <a class="existingWikiWord" href="/nlab/show/F-theory">F-theory</a> <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a>:</p> <ul> <li id="HeckmanRudelius18"><a class="existingWikiWord" href="/nlab/show/Jonathan+Heckman">Jonathan Heckman</a>, <a class="existingWikiWord" href="/nlab/show/Tom+Rudelius">Tom Rudelius</a>, <em>Top Down Approach to 6D SCFTs</em>, J. Phys. A: Math. Theor. <strong>52</strong> (2018) 093001 [<a href="https://arxiv.org/abs/1805.06467">arXiv:1805.06467</a>, <a href="https://doi.org/10.1088/1751-8121/aafc81">doi:10.1088/1751-8121/aafc81</a>]</li> </ul> <p>On the absence of <a class="existingWikiWord" href="/nlab/show/perturbation+theory">perturbation theory</a>:</p> <ul> <li>Chi-Ming Chang, <em>5d and 6d SCFTs Have No Weak Coupling Limit</em>, J. High Energ. Phys. <strong>2019</strong> 16 (2019) [<a href="https://arxiv.org/abs/1810.04169">arXiv:1810.04169</a>, <a href="https://doi.org/10.1007/JHEP09(2019)016">doi:10.1007/JHEP09(2019)016</a>]</li> </ul> <p>New approach to construction of candidate <a class="existingWikiWord" href="/nlab/show/Lagrangian+densities">Lagrangian densities</a> for <a class="existingWikiWord" href="/nlab/show/D%3D6+N%3D%282%2C0%29+SCFTs">D=6 N=(2,0) SCFTs</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Neil+Lambert">Neil Lambert</a>, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,0)</annotation></semantics></math> Lagrangian Structures</em>, Physics Letters B <strong>798</strong> (2019) 134948 [<a href="https://arxiv.org/abs/1908.10752">arXiv:1908.10752</a>, <a href="https://doi.org/10.1016/j.physletb.2019.134948">doi;10.1016/j.physletb.2019.134948</a>]</li> </ul> <h3 id="compactification_to_5d_superyangmills">Compactification to 5d super-Yang-Mills</h3> <p><a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a> on <a class="existingWikiWord" href="/nlab/show/circle">circle</a> <a class="existingWikiWord" href="/nlab/show/fibers">fibers</a> to <a class="existingWikiWord" href="/nlab/show/D%3D5+super+Yang-Mills+theory">D=5 super Yang-Mills theory</a> is discussed in (see also at <a class="existingWikiWord" href="/nlab/show/Perry-Schwarz+Lagrangian">Perry-Schwarz Lagrangian</a>):</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Nathan+Seiberg">Nathan Seiberg</a>, Sec. 7 of: <em>Notes on Theories with 16 Supercharges</em>, Nucl. Phys. Proc. Suppl. 67 (1998) 158-171 [<a href="https://arxiv.org/abs/hep-th/9705117">arXiv:hep-th/9705117</a>]</p> </li> <li id="Douglas11"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Douglas">Michael Douglas</a>, <em>On D=5 super Yang-Mills theory and (2,0) theory</em>, JHEP 1102:011, 2011 (<a href="https://arxiv.org/abs/1012.2880">arXiv:1012.2880</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Neil+Lambert">Neil Lambert</a>, Constantinos Papageorgakis, Maximilian Schmidt-Sommerfeld, <em>M5-Branes, D4-Branes and Quantum 5D super-Yang-Mills</em>, JHEP 1101:083, 2011 (<a href="https://arxiv.org/abs/1012.2882">arXiv:1012.2882</a>)</p> </li> <li id="Witten11"> <p><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, Sections 4 and 5 of <em>Fivebranes and Knots</em> (<a href="http://arxiv.org/abs/1101.3216">arXiv:1101.3216</a>)</p> </li> <li id="Hu13"> <p>Shan Hu, <em>6D <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,0)</annotation></semantics></math> theory and M5 branes: A KK mode approach</em>, 2013 (<a href="http://hdl.handle.net/1969.1/151094">hdl:1969.1/151094</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chris+Hull">Chris Hull</a>, <a class="existingWikiWord" href="/nlab/show/Neil+Lambert">Neil Lambert</a>, <em>Emergent Time and the M5-Brane</em>, JHEP06(2014)016 (<a href="https://arxiv.org/abs/1403.4532">arXiv:1403.4532</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Andreas+Gustavsson">Andreas Gustavsson</a>, <em>Five-dimensional Super-Yang-Mills and its Kaluza-Klein tower</em>. JHEP01(2019)222 (<a href="https://arxiv.org/abs/1812.01897">arXiv:1812.01897</a>)</p> </li> <li id="Lambert19"> <p><a class="existingWikiWord" href="/nlab/show/Neil+Lambert">Neil Lambert</a>, Sec. 3.1 and 3.4.3. of <em>Lessons from M2’s and Hopes for M5’s</em>, <em>Proceedings of the <a href="http://www.maths.dur.ac.uk/lms/">LMS-EPSRC Durham Symposium</a>:</em> <em><a href="http://www.maths.dur.ac.uk/lms/109/index.html">Higher Structures in M-Theory</a>, August 2018</em> Fortschritte der Physik Volume 67, Issue 8-9 2019 (<a href="https://arxiv.org/abs/1903.02825">arXiv:1903.02825</a>, <a href="https://doi.org/10.1002/prop.201910011">doi:10.1002/prop.201910011</a>, <a href="http://www.maths.dur.ac.uk/lms/109/talks/1877lambert.pdf">slides pdf</a>, <a href="http://www.maths.dur.ac.uk/lms/109/movies/1877lamb.mp4">video recording</a>)</p> </li> </ul> <h3 id="CompactificationTo4DSYM">Compactification to 4d super-Yang-Mills</h3> <p><a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><mn>6</mn></mrow><annotation encoding="application/x-tex">D=6</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒩</mi><mo>=</mo><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{N} = (2,0)</annotation></semantics></math>-CFT on <a class="existingWikiWord" href="/nlab/show/torus">torus</a> <a class="existingWikiWord" href="/nlab/show/fibers">fibers</a> to <a class="existingWikiWord" href="/nlab/show/D%3D4+super+Yang-Mills+theory">D=4 super Yang-Mills theory</a> and the related <a class="existingWikiWord" href="/nlab/show/electric-magnetic+duality">electric-magnetic duality</a>/<a class="existingWikiWord" href="/nlab/show/S-duality">S-duality</a> in 4-dimensions:</p> <ul> <li id="Witten07"><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em><a class="existingWikiWord" href="/nlab/show/Conformal+field+theory+in+four+and+six+dimensions">Conformal field theory in four and six dimensions</a></em> in <a class="existingWikiWord" href="/nlab/show/Ulrike+Tillmann">Ulrike Tillmann</a> (ed.) <em>Topology, geometry and quantum field theory</em> LMS Lecture Note Series (2004) (<a href="http://arxiv.org/abs/0712.0157">arXiv:0712.0157</a>)</li> </ul> <p>and the resulting relation to the <a class="existingWikiWord" href="/nlab/show/geometric+Langlands+correspondence">geometric Langlands correspondence</a>:</p> <ul> <li id="Witten09"><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>Geometric Langlands From Six Dimensions</em>, in Peter Kotiuga (ed.) <em>A Celebration of the Mathematical Legacy of Raoul Bott</em>, CRM Proceedings & Lecture Notes <strong>50</strong> AMS (2010) [<a href="http://arxiv.org/abs/0905.2720">arXiv:0905.2720</a>, <a href="https://bookstore.ams.org/crmp-50">ISBN:978-0-8218-4777-0</a>]</li> </ul> <p>For more references on this see at <em><a class="existingWikiWord" href="/nlab/show/N%3D2+D%3D4+super+Yang-Mills+theory">N=2 D=4 super Yang-Mills theory</a></em> the section <em><a href="N%3D2+D%3D4+super+Yang-Mills+theory#ConstructionByCompactificationOf5Branes">References - Constructions from 5-branes</a></em>.</p> <p>Relation to <a class="existingWikiWord" href="/nlab/show/BFSS+matrix+model">BFSS matrix model</a> on <a class="existingWikiWord" href="/nlab/show/tori">tori</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Micha+Berkooz">Micha Berkooz</a>, <a class="existingWikiWord" href="/nlab/show/Moshe+Rozali">Moshe Rozali</a>, <a class="existingWikiWord" href="/nlab/show/Nathan+Seiberg">Nathan Seiberg</a>, <em>Matrix Description of M-theory on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>T</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">T^3</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>T</mi> <mn>5</mn></msup></mrow><annotation encoding="application/x-tex">T^5</annotation></semantics></math></em> (<a href="http://arxiv.org/abs/hep-th/9704089">arXiv:hep-th/9704089</a>)</li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a> specifically of the <a class="existingWikiWord" href="/nlab/show/D%3D6+N%3D%281%2C0%29+SCFT">D=6 N=(1,0) SCFT</a> to <a class="existingWikiWord" href="/nlab/show/D%3D4+N%3D1+super+Yang-Mills">D=4 N=1 super Yang-Mills</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ibrahima+Bah">Ibrahima Bah</a>, <a class="existingWikiWord" href="/nlab/show/Christopher+Beem">Christopher Beem</a>, Nikolay Bobev, Brian Wecht, <em>Four-Dimensional SCFTs from M5-Branes</em> (<a href="http://arxiv.org/abs/1203.0303">arXiv:1203.0303</a>)</p> </li> <li> <p>Shlomo S. Razamat, <a class="existingWikiWord" href="/nlab/show/Cumrun+Vafa">Cumrun Vafa</a>, Gabi Zafrir, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>4</mn><mi>d</mi></mrow><annotation encoding="application/x-tex">4d</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒩</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\mathcal{N} = 1</annotation></semantics></math> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>6</mn><mi>d</mi><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">6d (1,0)</annotation></semantics></math></em>, J. High Energ. Phys. (2017) 2017: 64 (<a href="https://arxiv.org/abs/1610.09178">arXiv:1610.09178</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ibrahima+Bah">Ibrahima Bah</a>, <a class="existingWikiWord" href="/nlab/show/Amihay+Hanany">Amihay Hanany</a>, Kazunobu Maruyoshi, Shlomo S. Razamat, Yuji Tachikawa, Gabi Zafrir, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>4</mn><mi>d</mi></mrow><annotation encoding="application/x-tex">4d</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒩</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\mathcal{N}=1</annotation></semantics></math> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>6</mn><mi>d</mi></mrow><annotation encoding="application/x-tex">6d</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒩</mi><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{N}=(1,0)</annotation></semantics></math> on a torus with fluxes</em> (<a href="https://arxiv.org/abs/1702.04740">arXiv:1702.04740</a>)</p> </li> <li> <p>Hee-Cheol Kim, Shlomo S. Razamat, <a class="existingWikiWord" href="/nlab/show/Cumrun+Vafa">Cumrun Vafa</a>, Gabi Zafrir, <em>E-String Theory on Riemann Surfaces</em>, Fortsch. Phys. (<a href="https://arxiv.org/abs/1709.02496">arXiv:1709.02496</a>)</p> </li> <li> <p>Hee-Cheol Kim, Shlomo S. Razamat, <a class="existingWikiWord" href="/nlab/show/Cumrun+Vafa">Cumrun Vafa</a>, Gabi Zafrir, <em>D-type Conformal Matter and SU/USp Quivers</em>, JHEP06(2018)058 (<a href="https://arxiv.org/abs/1802.00620">arXiv:1802.00620</a>)</p> </li> <li> <p>Hee-Cheol Kim, Shlomo S. Razamat, <a class="existingWikiWord" href="/nlab/show/Cumrun+Vafa">Cumrun Vafa</a>, Gabi Zafrir, <em>Compactifications of ADE conformal matter on a torus</em>, JHEP09(2018)110 (<a href="https://arxiv.org/abs/1806.07620">arXiv:1806.07620</a>)</p> </li> <li> <p>Shlomo S. Razamat, Gabi Zafrir, <em>Compactification of 6d minimal SCFTs on Riemann surfaces</em>, Phys. Rev. D 98, 066006 (2018) (<a href="https://arxiv.org/abs/1806.09196">arXiv:1806.09196</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jin+Chen">Jin Chen</a>, <a class="existingWikiWord" href="/nlab/show/Babak+Haghighat">Babak Haghighat</a>, <a class="existingWikiWord" href="/nlab/show/Shuwei+Liu">Shuwei Liu</a>, <a class="existingWikiWord" href="/nlab/show/Marcus+Sperling">Marcus Sperling</a>, <em>4d <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">\mathcal{N}=1</annotation></semantics></math> from 6d D-type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒩</mi><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{N}=(1,0)</annotation></semantics></math></em>, J. High Energ. Phys. <strong>2020</strong> 152 (2020) [<a href="https://arxiv.org/abs/1907.00536">arXiv:1907.00536</a>, <a href=" https://doi.org/10.1007/JHEP01(2020)152">doi:10.1007/JHEP01(2020)152</a>]</p> </li> </ul> <h3 id="CompactificationTo2dCFT">Compactification to 2d CFT</h3> <p>On <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a> of <a class="existingWikiWord" href="/nlab/show/D%3D6+N%3D%282%2C0%29+SCFT">D=6 N=(2,0) SCFT</a> on <a class="existingWikiWord" href="/nlab/show/4-manifolds">4-manifolds</a> to <a class="existingWikiWord" href="/nlab/show/2d+CFTs">2d CFTs</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Abhijit+Gadde">Abhijit Gadde</a>, <a class="existingWikiWord" href="/nlab/show/Sergei+Gukov">Sergei Gukov</a>, <a class="existingWikiWord" href="/nlab/show/Pavel+Putrov">Pavel Putrov</a>, <em>Fivebranes and 4-manifolds</em>, in: <em>Arbeitstagung Bonn 2013</em>, Progress in Mathematics <strong>319</strong>, Birkhäuser (2016) [<a href="https://arxiv.org/abs/1306.4320">arXiv:1306.4320</a>, <a href="https://doi.org/10.1007/978-3-319-43648-7_7">doi:10.1007/978-3-319-43648-7_7</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Mykola+Dedushenko">Mykola Dedushenko</a>, <a class="existingWikiWord" href="/nlab/show/Sergei+Gukov">Sergei Gukov</a>, <a class="existingWikiWord" href="/nlab/show/Pavel+Putrov">Pavel Putrov</a>, <em>Vertex algebras and 4-manifold invariants</em>, Chapter 11 in: <em>Geometry and Physics: Volume I</em> (2018) 249–318 [<a href="https://arxiv.org/abs/1705.01645">arXiv:1705.01645</a>, <a href="https://doi.org/10.1093/oso/9780198802013.003.0011">doi:10.1093/oso/9780198802013.003.0011</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Boris+Feigin">Boris Feigin</a>, <a class="existingWikiWord" href="/nlab/show/Sergei+Gukov">Sergei Gukov</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>VOA</mi><mo stretchy="false">[</mo><msub><mi>M</mi> <mn>4</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">VOA[M_4]</annotation></semantics></math>, J. Math. Phys. <strong>61</strong> 012302 (2020) [<a href="https://arxiv.org/abs/1806.02470">arXiv:1806.02470</a>, <a href="https://doi.org/10.1063/1.5100059">doi:10.1063/1.5100059</a>]</p> </li> </ul> <p>In relation to <a class="existingWikiWord" href="/nlab/show/M5-brane+elliptic+genus">M5-brane elliptic genus</a>:</p> <ul> <li id="GukovPeiPutrovVafa18"><a class="existingWikiWord" href="/nlab/show/Sergei+Gukov">Sergei Gukov</a>, <a class="existingWikiWord" href="/nlab/show/Du+Pei">Du Pei</a>, <a class="existingWikiWord" href="/nlab/show/Pavel+Putrov">Pavel Putrov</a>, <a class="existingWikiWord" href="/nlab/show/Cumrun+Vafa">Cumrun Vafa</a>, <em>4-manifolds and topological modular forms</em>, J. High Energ. Phys. <strong>2021</strong> 84 (2021) [<a href="https://arxiv.org/abs/1811.07884">arXiv:1811.07884</a>, <a href="https://doi.org/10.1007/JHEP05(2021)084">doi:10.1007/JHEP05(2021)084</a>, <a href="https://inspirehep.net/literature/1704312">spire:1704312</a>]</li> </ul> <p>and in relation to <a class="existingWikiWord" href="/nlab/show/QFT+with+defects">defects</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jin+Chen">Jin Chen</a>, <a class="existingWikiWord" href="/nlab/show/Wei+Cui">Wei Cui</a>, <a class="existingWikiWord" href="/nlab/show/Babak+Haghighat">Babak Haghighat</a>, <a class="existingWikiWord" href="/nlab/show/Yi-Nan+Wang">Yi-Nan Wang</a>, <em>SymTFTs and Duality Defects from 6d SCFTs on 4-manifolds</em>, JHEP <strong>2023</strong> 208 (2023) [<a href="https://arxiv.org/abs/2305.09734">arXiv:2305.09734</a>, <a href="https://doi.org/10.1007/JHEP11(2023)208">doi:10.1007/JHEP11(2023)208</a>]</li> </ul> <h3 id="ade_classification">ADE classification</h3> <p>Discussion of the <a class="existingWikiWord" href="/nlab/show/ADE+classification">ADE classification</a> of the 6d theories includes, after (<a href="#Witten95">Witten95</a>)</p> <ul> <li> <p>Julie D. Blum, <a class="existingWikiWord" href="/nlab/show/Kenneth+Intriligator">Kenneth Intriligator</a>, <em>New Phases of String Theory and 6d RG Fixed Points via Branes at Orbifold Singularities</em>, Nucl.Phys.B506:199-222,1997 (<a href="http://arxiv.org/abs/hep-th/9705044">arXiv:hep-th/9705044</a>)</p> </li> <li id="HeckmannMorrisonVafa13"> <p><a class="existingWikiWord" href="/nlab/show/Jonathan+Heckman">Jonathan Heckman</a>, <a class="existingWikiWord" href="/nlab/show/David+Morrison">David Morrison</a>, <a class="existingWikiWord" href="/nlab/show/Cumrun+Vafa">Cumrun Vafa</a>, <em>On the Classification of 6D SCFTs and Generalized ADE Orbifolds</em> (<a href="http://arxiv.org/abs/1312.5746">arXiv:1312.5746</a>)</p> </li> </ul> <h3 id="ModelsAndSpecialProperties">Models and special properties</h3> <p>Realization of the 6d theory in <a class="existingWikiWord" href="/nlab/show/F-theory">F-theory</a> is discussed in (<a href="#HeckmannMorrisonVafa13">Heckmann-Morrison-Vafa 13</a>).</p> <p>A proposal for related higher nonabelian differential form data is made in</p> <ul> <li id="SSW11"><a class="existingWikiWord" href="/nlab/show/Henning+Samtleben">Henning Samtleben</a>, <a class="existingWikiWord" href="/nlab/show/Ergin+Sezgin">Ergin Sezgin</a>, Robert Wimmer, <em>(1,0) superconformal models in six dimensions</em>, J. High Energ. Phys. <strong>2011</strong> 62 (2011) [<a href="http://arxiv.org/abs/1108.4060">arXiv:1108.4060</a>, <a href="https://doi.org/10.1007/JHEP12(2011)062">doi:10.1007/JHEP12(2011)062</a>]</li> </ul> <p>Since by <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a> every nonabelian <a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>/<a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a> gives rise to some kind of nonabelian 1-bundle on <a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a> it is clear that some parts (but not all) of the nonabelian gerbe theory on the 5-brane has an equivalent reformulation in terms of ordinary gauge theory on the <a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a> of the 5-brane and possibly for gauge group the <a class="existingWikiWord" href="/nlab/show/loop+group">loop group</a> of the original gauge group.</p> <p>Comments along these lines have been made in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Andreas+Gustavsson">Andreas Gustavsson</a>, <em>Selfdual strings and loop space Nahm equations</em> (<a href="http://arxiv.org/abs/0802.3456">arXiv:0802.3456</a>).</li> </ul> <p>In fact, via the <a class="existingWikiWord" href="/nlab/show/strict+2-group">strict 2-group</a> version of the <a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a> there is a local gauge in which the loop group variables appear already before transgression of the 5-brane gerbe to loop space. This is discussed from a <a class="existingWikiWord" href="/nlab/show/holographic+principle">holographic</a> point of view in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/7d+Chern-Simons+theory+and+the+5-brane">7d Chern-Simons theory and the 5-brane</a></em></li> </ul> <h3 id="ReferencesOnTheHolographicDual">On the holographic dual</h3> <p>The basics of the relation of the 6d theory to a 7d theory under <a class="existingWikiWord" href="/nlab/show/AdS-CFT">AdS-CFT</a> is reviewed <a class="existingWikiWord" href="/nlab/show/holographic+principle">holographic duality</a></p> <ul> <li id="Maldacena"><a class="existingWikiWord" href="/nlab/show/Juan+Maldacena">Juan Maldacena</a>, <em>The Large N limit of superconformal field theories and supergravity</em>, Adv. Theor. Math. Phys. 2:231, 1998, <a href="http://arxiv.org/abs/hep-th/9711200">hep-th/9711200</a>; <em>Wilson loops in Large N field theories</em>, Phys. Rev. Lett. <strong>80</strong> (1998) 4859, <a href="http://arxiv.org/abs/hep-th/9803002">hep-th/9803002</a></li> </ul> <p>The argument that the abelian <a class="existingWikiWord" href="/nlab/show/7d+Chern-Simons+theory">7d Chern-Simons theory</a> of a <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">3-connection</a> yields this way the <a class="existingWikiWord" href="/nlab/show/conformal+blocks">conformal blocks</a> of the abelian <a class="existingWikiWord" href="/nlab/show/self-dual+higher+gauge+theory">self-dual higher gauge theory</a> of the 6d theory on a <em>single</em> brane is due to</p> <ul> <li id="WittenI"> <p><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>Five-Brane Effective Action In M-Theory</em> J. Geom. Phys.22:103-133,1997 (<a href="http://arxiv.org/abs/hep-th/9610234">arXiv:hep-th/9610234</a>)</p> </li> <li id="Witten98"> <p><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>AdS/CFT Correspondence And Topological Field Theory</em> JHEP 9812:012,1998 (<a href="http://arxiv.org/abs/hep-th/9812012">arXiv:hep-th/9812012</a>)</p> </li> </ul> <p>The nonabelian generalization of this 7d action functional that follows from taking the quantum corrections (<a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a> and flux quantization) of the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a> into account is discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/7d+Chern-Simons+theory+and+the+5-brane">7d Chern-Simons theory and the 5-brane</a></em></li> </ul> <p>See also</p> <ul> <li id="HEGK"> <p><a class="existingWikiWord" href="/nlab/show/Eric+D%27Hoker">Eric D'Hoker</a>, John Estes, Michael Gutperle, Darya Krym,</p> <p><em>Exact Half-BPS Flux Solutions in M-theory I Local Solutions</em> (<a href="http://arxiv.org/abs/0806.0605">arXiv:0806.0605</a>)</p> <p><em>Exact Half-BPS Flux Solutions in M-theory II: Global solutions asymptotic to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>AdS</mi> <mn>7</mn></msub><mo>×</mo><msup><mi>S</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">AdS_7 \times S^4</annotation></semantics></math> (<a href="http://arxiv.org/abs/0810.4647">arXiv:0810.4647</a>)</em></p> </li> </ul> <h3 id="solitonic_1brane_excitations">Solitonic 1-brane excitations</h3> <ul> <li id="GGT"> <p><a class="existingWikiWord" href="/nlab/show/Jerome+Gauntlett">Jerome Gauntlett</a>, <a class="existingWikiWord" href="/nlab/show/Joaquim+Gomis">Joaquim Gomis</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Townsend">Paul Townsend</a>, <em>BPS Bounds for Worldvolume Branes</em> (<a href="http://arxiv.org/abs/hep-th/9711205">arXiv:hep-th/9711205</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Paul+Howe">Paul Howe</a>, <a class="existingWikiWord" href="/nlab/show/Neil+Lambert">Neil Lambert</a>, <a class="existingWikiWord" href="/nlab/show/Peter+West">Peter West</a>, <em>The Threebrane Soliton of the M-Fivebrane</em> (<a href="http://arxiv.org/abs/hep-th/9710033">arXiv:hep-th/9710033</a>)</p> </li> </ul> <div> <h3 id="TheoryXAsAnFQFTReferences">The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><mn>6</mn></mrow><annotation encoding="application/x-tex">D=6</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒩</mi><mo>=</mo><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{N}=(2,0)</annotation></semantics></math> SCFT as an extended functorial field theory</h3> <p>On the (conjectural) suggestion to view at least some aspects of the <a class="existingWikiWord" href="/nlab/show/D%3D6+N%3D%282%2C0%29+SCFT">D=6 N=(2,0) SCFT</a> (such as its <a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a> or its image as a <a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a> under the <a class="existingWikiWord" href="/nlab/show/AGT+correspondence">AGT correspondence</a>) as a <a class="existingWikiWord" href="/nlab/show/functorial+field+theory">functorial field theory</a> given by a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> on a suitable <a class="existingWikiWord" href="/nlab/show/cobordism+category">cobordism category</a>, or rather as an <a class="existingWikiWord" href="/nlab/show/extended+TQFT">extended</a> such <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a>, given by an <a class="existingWikiWord" href="/nlab/show/n-functor">n-functor</a> (at least a <a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a> on a <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-category+of+cobordisms">2-category of cobordisms</a>):</p> <ul> <li id="Witten09"> <p><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, Section 1 of: <em>Geometric Langlands From Six Dimensions</em>, in Peter Kotiuga (ed.) <em>A Celebration of the Mathematical Legacy of Raoul Bott</em>, CRM Proceedings & Lecture Notes Volume: 50, AMS 2010 (<a href="http://arxiv.org/abs/0905.2720">arXiv:0905.2720</a>, <a href="https://bookstore.ams.org/crmp-50">ISBN:978-0-8218-4777-0</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <em><a class="existingWikiWord" href="/nlab/show/4-3-2+8-7-6">4-3-2 8-7-6</a></em>, talk at <em><a href="https://people.maths.ox.ac.uk/tillmann/ASPECTS.html">ASPECTS of Topology</a></em> Dec 2012 (<a href="https://people.maths.ox.ac.uk/tillmann/ASPECTSfreed.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Freed432876.pdf" title="pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, p. 32 of: <em>The cobordism hypothesis</em>, Bulletin of the American Mathematical Society 50 (2013), pp. 57-92, (<a href="http://arxiv.org/abs/1210.5100">arXiv:1210.5100</a>, <a href="https://doi.org/10.1090/S0273-0979-2012-01393-9">doi:10.1090/S0273-0979-2012-01393-9</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <a class="existingWikiWord" href="/nlab/show/Constantin+Teleman">Constantin Teleman</a>: <em>Relative quantum field theory</em>, Commun. Math. Phys. <strong>326</strong> (2014) 459–476 [<a href="https://doi.org/10.1007/s00220-013-1880-1">doi:10.1007/s00220-013-1880-1</a>, <a href="https://arxiv.org/abs/1212.1692">arXiv:1212.1692</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/David+Ben-Zvi">David Ben-Zvi</a>: <em>Theory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒳</mi></mrow><annotation encoding="application/x-tex">\mathcal{X}</annotation></semantics></math> and Geometric Representation Theory</em>, talks at <em><a href="http://www.ihes.fr/~celliott/workshop/">Mathematical Aspects of Six-Dimensional Quantum Field Theories</a></em> IHES 2014, notes by <a class="existingWikiWord" href="/nlab/show/Qiaochu+Yuan">Qiaochu Yuan</a> (<a class="existingWikiWord" href="/nlab/files/BenZvi-TheoryX-I.pdf" title="pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/BenZvi-TheoryX-II.pdf" title="pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/BenZvi-TheoryX-III.pdf" title="pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/David+Ben-Zvi">David Ben-Zvi</a>, <em>Algebraic geometry of topological field theories</em>, talk at <em><a href="https://www.msri.org/workshops/689">Reimagining the Foundations of Algebraic Topology April 07, 2014 - April 11, 2014</a></em> (<a href="https://www.msri.org/workshops/689/schedules/18216">web video</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lukas+M%C3%BCller">Lukas Müller</a>, <em>Extended Functorial Field Theories and Anomalies in Quantum Field Theories</em> (<a href="https://arxiv.org/abs/2003.08217">arXiv:2003.08217</a>)</p> </li> </ul> </div></body></html> </div> <div class="revisedby"> <p> Last revised on August 28, 2024 at 07:16:01. 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