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17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> nLab heterotic string theory </h1> <div class="navigation"> <a href='#navEnd'>Skip the Navigation Links</a> | <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/10433/#Item_10" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="string_theory">String theory</h4> <div class="hide"><div> <ul> <li> <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/books+about+string+theory">books about string theory</a> </li> </ul> <h3 id="ingredients">Ingredients</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a>, <ul> <li><a class="existingWikiWord" href="/nlab/show/CFT">CFT</a>, <a class="existingWikiWord" href="/nlab/show/perturbation+theory">perturbation theory</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/effective+QFT">effective background QFT</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>, <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a>, <a class="existingWikiWord" href="/nlab/show/quantum+gravity">quantum gravity</a></li> </ul> </li> </ul> <h3 id="critical_string_models">Critical string models</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a>, <a class="existingWikiWord" href="/nlab/show/differential+string+structure">differential string structure</a>, </li> <li> <a class="existingWikiWord" href="/nlab/show/dual+heterotic+string+theory">dual heterotic string theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+fivebrane+structure">differential fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/type+IIA+string+theory">type IIA string theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/type+IIB+string+theory">type IIB string theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/F-theory">F-theory</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/string+field+theory">string field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/duality+in+string+theory">duality in string theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/T-duality">T-duality</a>, <a class="existingWikiWord" href="/nlab/show/mirror+symmetry">mirror symmetry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/S-duality">S-duality</a>, <a class="existingWikiWord" href="/nlab/show/electric-magnetic+duality">electric-magnetic duality</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/U-duality">U-duality</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/open%2Fclosed+string+duality">open/closed string duality</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/AdS%2FCFT+correspondence">AdS/CFT correspondence</a>, <a class="existingWikiWord" href="/nlab/show/holographic+principle">holographic principle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a>, <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%C5%99ava-Witten+theory">Hořava-Witten theory</a></li> </ul> </li> </ul> <h3 id="extended_objects">Extended objects</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/brane">brane</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/D0-brane">D0-brane</a>, <a class="existingWikiWord" href="/nlab/show/D2-brane">D2-brane</a>, <a class="existingWikiWord" href="/nlab/show/D4-brane">D4-brane</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/D1-brane">D1-brane</a>, <a class="existingWikiWord" href="/nlab/show/D3-brane">D3-brane</a>, <a class="existingWikiWord" href="/nlab/show/D5-brane">D5-brane</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/RR-field">RR-field</a>, <a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/NS-brane">NS-brane</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>, <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/B2-field">B2-field</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/NS5-brane">NS5-brane</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/B6-field">B6-field</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/M-brane">M-brane</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/C3-field">C3-field</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/ABJM+theory">ABJM theory</a>, <a class="existingWikiWord" href="/nlab/show/BLG+model">BLG model</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/C6-field">C6-field</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a> </li> </ul> </li> </ul> </li> </ul> <h3 id="topological_strings">Topological strings</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a>, <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+M-theory">topological M-theory</a> </li> </ul> <h2 id="backgrounds">Backgrounds</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/target+space">target space</a>, <a class="existingWikiWord" href="/nlab/show/background+gauge+field">background gauge field</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/twisted+smooth+cohomology+in+string+theory">twisted smooth cohomology in string theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/landscape+of+string+theory+vacua">landscape of string theory vacua</a> </li> </ul> <h2 id="phenomenology">Phenomenology</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/string+phenomenology">string phenomenology</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/moduli+stabilization">moduli stabilization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/G%E2%82%82-MSSM">G₂-MSSM</a> </li> </ul> </li> </ul> <div> <a href="/nlab/edit/string+theory+-+contents">Edit this sidebar</a> </div></div></div> <h4 id="supergeometry">Super-Geometry</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a> and (<a class="existingWikiWord" href="/nlab/show/synthetic+differential+supergeometry">synthetic</a> ) <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a> <h2 id="background">Background</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/graded+object">graded object</a> </li> </ul> <h2 id="introductions">Introductions</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+superalgebra">geometry of physics – superalgebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+supergeometry">geometry of physics – supergeometry</a> </li> </ul> <h2 id="superalgebra">Superalgebra</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/super+commutative+monoid">super commutative monoid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+abelian+group">super abelian group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+ring">super ring</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/supercommutative+ring">supercommutative ring</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/exterior+ring">exterior ring</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Clifford+ring">Clifford ring</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+module">super module</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+vector+space">super vector space</a>, <a class="existingWikiWord" href="/nlab/show/SVect">SVect</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+algebra">super algebra</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/supercommutative+algebra">supercommutative algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/exterior+algebra">exterior algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Clifford+algebra">Clifford algebra</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/superdeterminant">superdeterminant</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">super Poincare Lie algebra</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/chain+complex+of+super+vector+spaces">chain complex of super vector spaces</a> (<a class="existingWikiWord" href="/nlab/show/model+structure+on+chain+complexes+of+super+vector+spaces">model structure</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/differential+graded-commutative+superalgebra">differential graded-commutative superalgebra</a> (<a class="existingWikiWord" href="/nlab/show/model+structure+on+differential+graded-commutative+superalgebras">model structure</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/super+L-infinity+algebra">super L-infinity algebra</a> </li> </ul> <h2 id="supergeometry">Supergeometry</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/superpoint">superpoint</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+Cartesian+space">super Cartesian space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/supermanifold">supermanifold</a>, <a class="existingWikiWord" href="/nlab/show/SDiff">SDiff</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/NQ-supermanifold">NQ-supermanifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+vector+bundle">super vector bundle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/complex+supermanifold">complex supermanifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Euclidean+supermanifold">Euclidean supermanifold</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+spacetime">super spacetime</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Minkowski+spacetime">super Minkowski spacetime</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/integration+over+supermanifolds">integration over supermanifolds</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Berezin+integral">Berezin integral</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+Lie+group">super Lie group</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/super+Lie+group">super translation group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+Euclidean+group">super Euclidean group</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+%E2%88%9E-groupoid">super ∞-groupoid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+formal+smooth+%E2%88%9E-groupoid">super formal smooth ∞-groupoid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+line+2-bundle">super line 2-bundle</a> </li> </ul> <h2 id="supersymmetry">Supersymmetry</h2> <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/division+algebra+and+supersymmetry">division algebra and supersymmetry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">super Poincare Lie algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/supermultiplet">supermultiplet</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/BPS+state">BPS state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/M-theory+super+Lie+algebra">M-theory super Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/type+II+super+Lie+algebra">type II super Lie algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a>, <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a> </li> </ul> <h2 id="supersymmetric_field_theory">Supersymmetric field theory</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/superfield">superfield</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/supersymmetric+quantum+mechanics">supersymmetric quantum mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/adinkra">adinkra</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">super Yang-Mills theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/gauged+supergravity">gauged supergravity</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/superstring+theory">superstring theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a> </li> </ul> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometric+model+for+elliptic+cohomology">geometric model for elliptic cohomology</a></li> </ul> <div> <a href="/nlab/edit/supergeometry+-+contents">Edit this sidebar</a> </div></div></div> <h4 id="physics">Physics</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/physics">physics</a>, <a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a>, <a class="existingWikiWord" href="/nlab/show/philosophy+of+physics">philosophy of physics</a> <h2 id="surveys_textbooks_and_lecture_notes">Surveys, textbooks and lecture notes</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">(higher) category theory and physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/books+and+reviews+in+mathematical+physics">books and reviews</a>, <a class="existingWikiWord" href="/nlab/show/physics+resources">physics resources</a> </li> </ul> <hr /> <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory (physics)</a>, <a class="existingWikiWord" href="/nlab/show/model+%28physics%29">model (physics)</a> <a class="existingWikiWord" href="/nlab/show/experiment">experiment</a>, <a class="existingWikiWord" href="/nlab/show/measurement">measurement</a>, <a class="existingWikiWord" href="/nlab/show/computable+physics">computable physics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/mechanics">mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/mass">mass</a>, <a class="existingWikiWord" href="/nlab/show/charge">charge</a>, <a class="existingWikiWord" href="/nlab/show/momentum">momentum</a>, <a class="existingWikiWord" href="/nlab/show/angular+momentum">angular momentum</a>, <a class="existingWikiWord" href="/nlab/show/moment+of+inertia">moment of inertia</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/dynamics+on+Lie+groups">dynamics on Lie groups</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/rigid+body+dynamics">rigid body dynamics</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Lagrangian+mechanics">Lagrangian mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a>, <a class="existingWikiWord" href="/nlab/show/state">state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange equations</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-symplectic+manifold">n-symplectic manifold</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/smooth+Lorentzian+manifold">smooth Lorentzian manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/special+relativity">special relativity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/dilaton+gravity">dilaton gravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a> </li> </ul> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/classical+field+theory">Classical field theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/classical+physics">classical physics</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+mechanics">classical mechanics</a></li> <li><a class="existingWikiWord" href="/nlab/show/waves">waves</a> and <a class="existingWikiWord" href="/nlab/show/optics">optics</a></li> <li><a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">Quantum Mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+mechanics+in+terms+of+dagger-compact+categories">in terms of ∞-compact categories</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hamiltonian+operator">Hamiltonian operator</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantization">Quantization</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/path+integral">path integral quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/semiclassical+approximation">semiclassical approximation</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">Quantum Field Theory</a> <ul> <li> Axiomatizations <ul> <li> <a class="existingWikiWord" href="/nlab/show/AQFT">algebraic QFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+net">local net</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/boson">boson</a>, <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/FQFT">functorial QFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem </li> <li> <a class="existingWikiWord" href="/nlab/show/extended+topological+quantum+field+theory">extended topological quantum field theory</a> </li> </ul> </li> </ul> </li> <li> Tools <ul> <li> <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-function+theory">geometric ∞-function theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+%28in+particle+phyiscs%29">models</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/fields+and+quanta+-+table">fields and quanta</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/GUT">Grand Unified Theories</a>, <a class="existingWikiWord" href="/nlab/show/MSSM">MSSM</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/on-shell+recursion">on-shell recursion</a>, <a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></li> </ul> </li> </ul> </li> <li> Structural phenomena <ul> <li> <a class="existingWikiWord" href="/nlab/show/universality+class">universality class</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/instanton">instanton</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spontaneously+broken+symmetry">spontaneously broken symmetry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein mechanism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/integrable+systems">integrable systems</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/holonomic+quantum+fields">holonomic quantum fields</a> </li> </ul> </li> <li> Types of quantum field thories <ul> <li> <a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory and elliptic cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/CFT">CFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a>, <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>, <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixing</a> </li> <li> examples <ul> <li><a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a>, <a class="existingWikiWord" href="/nlab/show/QED">QED</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/electric+charge">electric charge</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>, <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spinors+in+Yang-Mills+theory">spinors in Yang-Mills theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a></li> <li><a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a></li> <li><a class="existingWikiWord" href="/nlab/show/RR+field">RR field</a></li> <li><a class="existingWikiWord" href="/nlab/show/first-order+formulation+of+gravity">first-order formulation of gravity</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/general+covariance">general covariance</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gravity+as+a+BF-theory">gravity as a BF-theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/particle">particle</a>, <a class="existingWikiWord" href="/nlab/show/relativistic+particle">relativistic particle</a>, <a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental particle</a>, <a class="existingWikiWord" href="/nlab/show/spinning+particle">spinning particle</a>, <a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>, <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/membrane">membrane</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/AKSZ+theory">AKSZ theory</a> </li> </ul> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/string+theory">String Theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+theory+results+applied+elsewhere">string theory results applied elsewhere</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/number+theory+and+physics">number theory and physics</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Riemann+hypothesis+and+physics">Riemann hypothesis and physics</a></li> </ul> </li> </ul> <div> <a href="/nlab/edit/physicscontents">Edit this sidebar</a> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#compactifications'>Compactifications</a></li> <li><a href='#enhanced_supersymmetry'>Enhanced supersymmetry</a></li> <li><a href='#GaugeGroups'>Gauge groups</a></li> <li><a href='#Dualities'>Dualities</a></li> <ul> <li><a href='#duality_with_type_i_string_theory'>Duality with type I string theory</a></li> <li><a href='#duality_with_type_ii_string_theory'>Duality with type II string theory</a></li> <li><a href='#duality_with_mtheory'>Duality with M-theory</a></li> <li><a href='#duality_with_ftheory'>Duality with F-theory</a></li> </ul> <li><a href='#partition_function_and_witten_genus'>Partition function and Witten genus</a></li> <li><a href='#GeneralGaugeBackgroundsAndParameterizedWZWModels'>General gauge backgrounds and parameterized WZW models</a></li> <li><a href='#SuperspaceFormulation'>Superspace formulation</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#orbifold_and_orientifold_compactifications'>Orbifold and orientifold compactifications</a></li> <li><a href='#HeteroticStringPhenomenologyReferences'>Heterotic string phenomenology</a></li> <ul> <li><a href='#HeteroticStringPhenomenologyReferencesForE8'>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub><mo>×</mo><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">E_8 \times E_8</annotation></semantics></math>-heterotic string</a></li> <li><a href='#HeteroticStringPhenomenologyReferencesForSemiSpin32'>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SemiSpin</mi><mo stretchy="false">(</mo><mn>32</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SemiSpin(32)</annotation></semantics></math>-heterotic string</a></li> <li><a href='#HeteroticStringPhenomenologyReferencesForSO16'>The “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo><mo>×</mo><mi>SO</mi><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(16)\times SO(16)</annotation></semantics></math>”-heterotic string</a></li> </ul> <li><a href='#HigherGaugeTheoryOfTheGreenSchwarzMechanismReferences'>Higher gauge theory of the Green-Schwarz mechanism</a></li> <li><a href='#superspace_formulation_of_heterotic_supergravity'>Superspace formulation of Heterotic supergravity</a></li> <li><a href='#in_elliptic_cohomology'>In elliptic cohomology</a></li> <li><a href='#general_flux_backgrounds_and_parameterized_wzw_models'>General flux backgrounds and parameterized WZW models</a></li> <li><a href='#on_elliptic_fibrations'>On elliptic fibrations</a></li> <li><a href='#dualities_2'>Dualities</a></li> <ul> <li><a href='#with_type_i_superstring_theory'>With type I superstring theory</a></li> <li><a href='#DualityWithFTheory'>With <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-theory</a></li> </ul> <li><a href='#FlavorOnSingleHeteroticM5BranesReferences'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(2)</annotation></semantics></math>-flavor symmetry on heterotic M5-branes</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> In <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> a <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <a class="existingWikiWord" href="/nlab/show/landscape+of+string+theory+vacua">vacuum</a> is encoded by a <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> 2-dimensional <a class="existingWikiWord" href="/nlab/show/SCFT">SCFT</a>. In heterotic string theory that SCFT is assumed to be the sum of a supersymmetric chiral piece and a non-supersymmetric piece (therefore “heterotic”). <h2 id="properties">Properties</h2> <h3 id="compactifications">Compactifications</h3> An effective <a class="existingWikiWord" href="/nlab/show/target+space">target space</a> <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> induced from a given heterotic 2d <a class="existingWikiWord" href="/nlab/show/CFT">CFT</a> <a class="existingWikiWord" href="/nlab/show/sigma+model">sigma model</a> that has a <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>M</mi> <mn>4</mn></msup><mo>×</mo><msup><mi>Y</mi> <mn>6</mn></msup></mrow><annotation encoding="application/x-tex">M^4 \times Y^6</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>M</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">M^4</annotation></semantics></math> the 4-dimensional <a class="existingWikiWord" href="/nlab/show/Minkowski+space">Minkowski space</a> that is experimentally observed locally (say on the scale of a particle accelerator) has <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">N= 1</annotation></semantics></math> global <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a> precisely if the remaining 6-dimensional <a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Y</mi> <mn>6</mn></msup></mrow><annotation encoding="application/x-tex">Y^6</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifold">Calabi-Yau manifold</a>. See the <a href="#ReferencesNEqOne">references below</a>. Since global <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">N=1</annotation></semantics></math> supersymmetry for a long time has been considered a promising phenomenological model in high energy physics, this fact has induced a lot of interest in <a class="existingWikiWord" href="/nlab/show/heterotic+string+theory+on+CY3-manifolds">heterotic string theory on CY3-manifolds</a>. <h3 id="enhanced_supersymmetry">Enhanced supersymmetry</h3> A priori the <a class="existingWikiWord" href="/nlab/show/worldsheet">worldsheet</a> <a class="existingWikiWord" href="/nlab/show/2d+SCFT">2d SCFT</a> describing the quantum heterotic string has <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N=(1,0)</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a>. Precisely if the corresponding <a class="existingWikiWord" href="/nlab/show/target+space">target space</a> <a class="existingWikiWord" href="/nlab/show/effective+field+theory">effective field theory</a> has <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">N=1</annotation></semantics></math> supersymmetry does the worldsheet theory enhance to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N=(2,0)</annotation></semantics></math> supersymmetry. See at <a class="existingWikiWord" href="/nlab/show/2d+%282%2C0%29-superconformal+QFT">2d (2,0)-superconformal QFT</a> and at <a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifolds+and+supersymmetry">Calabi-Yau manifolds and supersymmetry</a> for more on this. <h3 id="GaugeGroups">Gauge groups</h3> Precisely two (<a class="existingWikiWord" href="/nlab/show/isomorphism+classes">isomorphism classes</a> of) <a class="existingWikiWord" href="/nlab/show/gauge+groups">gauge groups</a> are consistent (give <a class="existingWikiWord" href="/nlab/show/quantum+anomaly+cancellation">quantum anomaly cancellation</a>) while preserving <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a>: one is the <a class="existingWikiWord" href="/nlab/show/direct+product+group">direct product group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub><mo>×</mo><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">E_8 \times E_8</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/exceptional+Lie+group">exceptional Lie group</a> <a class="existingWikiWord" href="/nlab/show/E8">E8</a> with itself, the other is in fact the <a class="existingWikiWord" href="/nlab/show/semi-spin+group">semi-spin group</a> <a class="existingWikiWord" href="/nlab/show/SemiSpin%2832%29"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>SemiSpin</mi> <mo stretchy="false">(</mo> <mn>32</mn> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">SemiSpin(32)</annotation> </semantics> </math></a>. If the supersymmetry requirement is dropped, then there is a third option, which <a class="existingWikiWord" href="/nlab/show/Lie+algebra">locally</a> looks like the product of the <a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo><mo>×</mo><mi>SO</mi><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(16)\times SO(16)</annotation></semantics></math>, as first described by <a href="#AGMV86het">Alvarez-Gaumé, Ginsparg, Moore, & Vafa 1986</a> and by <a href="#DH86het">Dixon & Harvey 1986</a>. It was suggested bu <a href="#SW87anom">Schellekens & Warner 1987</a> that this theory could be regarded as some sort of “difference” (a <a class="existingWikiWord" href="/nlab/show/correspondence">correspondence</a>, or <a class="existingWikiWord" href="/nlab/show/duality+in+string+theory">duality</a>) between the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub><mo>×</mo><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">E_8\times E_8</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/SemiSpin%2832%29"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mtext>SemiSpin</mtext> <mo stretchy="false">(</mo> <mn>32</mn> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">\text{SemiSpin}(32)</annotation> </semantics> </math></a> theories, by looking at how the relevant representations of these two theories assemble into those appearing in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo><mo>×</mo><mi>SO</mi><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(16)\times SO(16)</annotation></semantics></math> theory. The global character of the gauge group is not <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo><mo>×</mo><mi>SO</mi><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(16)\times SO(16)</annotation></semantics></math>, however. <a href="#McIn99spin">McInnes 1999</a> proposed that the actual gauge group is a quotient of <a class="existingWikiWord" href="/nlab/show/spin+group">spin groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mtext>Spin</mtext><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo><mo>×</mo><mtext>Spin</mtext><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{Spin}(16) \times \text{Spin}(16)</annotation></semantics></math> by a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/action">action</a> (not corresponding to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mtext>SemiSpin</mtext><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo><mo>×</mo><mtext>SemiSpin</mtext><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{SemiSpin}(16)\times \text{SemiSpin}(16)</annotation></semantics></math>). This proposal allows to identify <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mtext>Spin</mtext><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo><mo>×</mo><mtext>Spin</mtext><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{Spin} (16) \times \text{Spin}(16)/ \mathbb{Z}_2</annotation></semantics></math> as a subgroup of both <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub><mo>×</mo><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">E_8\times E_8</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mtext>SemiSpin</mtext><mo stretchy="false">(</mo><mn>32</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{SemiSpin}(32)</annotation></semantics></math>, where the author’s purpose for this is identifying the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mtext>Spin</mtext><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo><mo>×</mo><mtext>Spin</mtext><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{Spin} (16) \times \text{Spin} (16)/\mathbb{Z}_2</annotation></semantics></math> theory as the realization of the <a class="existingWikiWord" href="/nlab/show/T-duality">T-duality</a> between the supersymmetric heterotic strings. <h3 id="Dualities">Dualities</h3> Some <a class="existingWikiWord" href="/nlab/show/dualities+in+string+theory">dualities in string theory</a> involving the heterotic string: <h4 id="duality_with_type_i_string_theory">Duality with type I string theory</h4> <ul> <li><a class="existingWikiWord" href="/nlab/show/duality+between+heterotic+and+type+I+string+theory">duality between heterotic and type I string theory</a></li> </ul> <h4 id="duality_with_type_ii_string_theory">Duality with type II string theory</h4> See <a class="existingWikiWord" href="/nlab/show/duality+between+heterotic+and+type+II+string+theory">duality between heterotic and type II string theory</a>. <h4 id="duality_with_mtheory">Duality with M-theory</h4> See <a class="existingWikiWord" href="/nlab/show/duality+between+heterotic+string+theory+and+M-theory">duality between heterotic string theory and M-theory</a> <h4 id="duality_with_ftheory">Duality with F-theory</h4> See <a class="existingWikiWord" href="/nlab/show/duality+between+heterotic+string+theory+and+F-theory">duality between heterotic string theory and F-theory</a> and see <a href="#DualityWithFTheory">references below</a>. <h3 id="partition_function_and_witten_genus">Partition function and Witten genus</h3> <div> <a class="existingWikiWord" href="/nlab/show/partition+functions">partition functions</a> in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> as <a class="existingWikiWord" href="/nlab/show/index">indices</a>/<a class="existingWikiWord" href="/nlab/show/genera">genera</a>/<a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">orientations</a> in <a class="existingWikiWord" href="/nlab/show/generalized+cohomology+theory">generalized cohomology theory</a>: <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><a class="existingWikiWord" href="/nlab/show/partition+function">partition function</a> in <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>-dimensional QFT</th><th><a class="existingWikiWord" href="/nlab/show/supercharge">supercharge</a></th><th><a class="existingWikiWord" href="/nlab/show/index">index</a> in <a class="existingWikiWord" href="/nlab/show/cohomology+theory">cohomology theory</a></th><th><a class="existingWikiWord" href="/nlab/show/genus">genus</a></th><th>logarithmic coefficients of <a class="existingWikiWord" href="/nlab/show/Hirzebruch+series">Hirzebruch series</a></th><th></th></tr></thead><tbody><tr><td style="text-align: left;">0</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;">push-forward in <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>: <a class="existingWikiWord" href="/nlab/show/integration+of+differential+forms">integration of differential forms</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a></td></tr> <tr><td style="text-align: left;">1</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/spinning+particle">spinning particle</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/KO">KO</a><a class="existingWikiWord" href="/nlab/show/K-theory">-theory</a> <a class="existingWikiWord" href="/nlab/show/index">index</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/A-hat+genus">A-hat genus</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Bernoulli+numbers">Bernoulli numbers</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Atiyah-Bott-Shapiro+orientation">Atiyah-Bott-Shapiro orientation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mi>Spin</mi><mo>→</mo><mi>KO</mi></mrow><annotation encoding="application/x-tex">M Spin \to KO</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;">endpoint of <a class="existingWikiWord" href="/nlab/show/2d+Poisson-Chern-Simons+theory">2d Poisson-Chern-Simons theory</a> string</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%5Ec+Dirac+operator">Spin^c Dirac operator</a> twisted by <a class="existingWikiWord" href="/nlab/show/prequantum+line+bundle">prequantum line bundle</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a> of boundary <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a>/<a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Todd+genus">Todd genus</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Bernoulli+numbers">Bernoulli numbers</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Atiyah-Bott-Shapiro+orientation">Atiyah-Bott-Shapiro orientation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><msup><mi>Spin</mi> <mi>c</mi></msup><mo>→</mo><mi>KU</mi></mrow><annotation encoding="application/x-tex">M Spin^c \to KU</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;">endpoint of <a class="existingWikiWord" href="/nlab/show/type+II+superstring">type II superstring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%5Ec+Dirac+operator">Spin^c Dirac operator</a> twisted by <a class="existingWikiWord" href="/nlab/show/Chan-Paton+gauge+field">Chan-Paton gauge field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D-brane+charge">D-brane charge</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Todd+genus">Todd genus</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Bernoulli+numbers">Bernoulli numbers</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Atiyah-Bott-Shapiro+orientation">Atiyah-Bott-Shapiro orientation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><msup><mi>Spin</mi> <mi>c</mi></msup><mo>→</mo><mi>KU</mi></mrow><annotation encoding="application/x-tex">M Spin^c \to KU</annotation></semantics></math></td></tr> <tr><td style="text-align: left;">2</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+II+superstring">type II superstring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Dirac-Ramond+operator">Dirac-Ramond operator</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/superstring">superstring</a> <a class="existingWikiWord" href="/nlab/show/partition+function">partition function</a> in NS-R sector</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Ochanine+elliptic+genus">Ochanine elliptic genus</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO+orientation+of+elliptic+cohomology">SO orientation of elliptic cohomology</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/heterotic+superstring">heterotic superstring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Dirac-Ramond+operator">Dirac-Ramond operator</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/superstring">superstring</a> <a class="existingWikiWord" href="/nlab/show/partition+function">partition function</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Witten+genus">Witten genus</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Eisenstein+series">Eisenstein series</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/string+orientation+of+tmf">string orientation of tmf</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/self-dual+string">self-dual string</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M5-brane+charge">M5-brane charge</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">3</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/w4-orientation+of+EO%282%29-theory">w4-orientation of EO(2)-theory</a></td></tr> </tbody></table> </div> <h3 id="GeneralGaugeBackgroundsAndParameterizedWZWModels">General gauge backgrounds and parameterized WZW models</h3> The traditional construction of the <a class="existingWikiWord" href="/nlab/show/worldsheet">worldsheet</a> theory of the heterotic string produces via the <a class="existingWikiWord" href="/nlab/show/current+algebra">current algebra</a> of the left-moving worldsheet <a class="existingWikiWord" href="/nlab/show/fermions">fermions</a> only those <a class="existingWikiWord" href="/nlab/show/E8">E8</a>-<a class="existingWikiWord" href="/nlab/show/background+gauge+fields">background gauge fields</a> which are reducible to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">Spin(16)/\mathbb{Z}_2</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+connections">principal connections</a> (<a href="#DistlerSharpe10">Distler-Sharpe 10, sections 2-4</a>). But it is known that, for instance, the <a class="existingWikiWord" href="/nlab/show/duality+between+F-theory+and+heterotic+string+theory">duality between F-theory and heterotic string theory</a> produces more general gauge backgrounds (<a href="#DistlerSharpe10">Distler-Sharpe 10, section 5</a>). In (<a href="#DistlerSharpe10">Distler-Sharpe 10, section 7</a>), following (<a href="#GatesSiegel88">Gates-Siegel 88</a>), it is argued that the way to fix this is to consider <a class="existingWikiWord" href="/nlab/show/parameterized+WZW+models">parameterized WZW models</a>, parameterized over the <a class="existingWikiWord" href="/nlab/show/E8">E8</a>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> over <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>. This does allow the incorporation of all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">E_8</annotation></semantics></math>-background gauge fields, and the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+anomaly">Green-Schwarz anomaly</a> (and its cancellation) of the heterotic string now comes out as being equivalently the <a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a> (and its lifting) for such a parameterized WZW term to exist. Moreover, where the traditional construction only produces level-1 <a class="existingWikiWord" href="/nlab/show/current+algebras">current algebras</a>, this construction accommodates all levels, and it is argued (<a href="#DistlerSharpe10">Distler-Sharpe 10, section 8.5</a>) that the <a class="existingWikiWord" href="/nlab/show/elliptic+genus">elliptic genus</a> of the resulting <a class="existingWikiWord" href="/nlab/show/parameterized+WZW+models">parameterized WZW models</a> are the <a class="existingWikiWord" href="/nlab/show/equivariant+elliptic+genera">equivariant elliptic genera</a> found by Liu and Ando (<a href="#Ando07">Ando 07</a>). However, presently questions remain concerning formulating a <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> for strings propagating on the total space of the bundle, as it is only the chiral part of the geometric WZW model that appears in the heterotic string. (…) <h3 id="SuperspaceFormulation">Superspace formulation</h3> The <a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a> <a class="existingWikiWord" href="/nlab/show/field+strength">strength</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mrow><mi>α</mi><mi>β</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">F_{\alpha \beta} = 0</annotation></semantics></math> (<a href="#Witten86">Witten 86</a>, <a href="#BonoraBregolaLechnerPastiTonin87">Bonora-Bregola-Lechner-Pasti-Tonin 87, above (2.7)</a>, <a href="#BonoraBregolaLechnerPastiTonin87">Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.13)</a>). <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mrow><mi>a</mi><mi>α</mi></mrow></msub><mo>=</mo><msub><mi>Γ</mi> <mrow><mi>a</mi><mi>α</mi><mi>β</mi></mrow></msub><msup><mi>χ</mi> <mi>β</mi></msup></mrow><annotation encoding="application/x-tex">F_{a \alpha} = \Gamma_{a \alpha \beta} \chi^\beta</annotation></semantics></math> (<a href="#Witten86">Witten 86 (8)</a>, <a href="#AtickDharRatra86">Atick-Dhar-Ratra 86, (4.14)</a>, <a href="#BonoraPastiTonin87">Bonora-Pasti-Tonin 87, below (11)</a>, <a href="#BonoraBregolaLechnerPastiTonin87">Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.27)</a>). Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>χ</mi> <mi>α</mi></msup></mrow><annotation encoding="application/x-tex">\chi^\alpha</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/gaugino">gaugino</a>. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msub><mo>=</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo stretchy="false">(</mo><msub><mi>Γ</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msub><msub><mo stretchy="false">)</mo> <mi>α</mi></msub><msup><mrow></mrow> <mi>β</mi></msup><msub><mi>D</mi> <mi>β</mi></msub><msup><mi>χ</mi> <mi>α</mi></msup></mrow><annotation encoding="application/x-tex">F_{a b} = \tfrac{1}{4} (\Gamma_{a b})_\alpha{}^\beta D_\beta \chi^\alpha</annotation></semantics></math> (<a href="#BonoraPastiTonin87">Bonora-Pasti-Tonin 87, below (11)</a>, <a href="#BonoraBregolaLechnerPastiTonin87">Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.28)</a>) <a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>D</mi> <mi>a</mi></msup><msub><mi>Γ</mi> <mi>a</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mi>α</mi><mi>β</mi></mrow></msub><msup><mi>χ</mi> <mi>β</mi></msup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> (D^a \Gamma_a)_{\alpha\beta} \chi^\beta =0 </annotation></semantics></math> (<a href="#BonoraBregolaLechnerPastiTonin87">Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.30)</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>b</mi></msup><msub><mi>F</mi> <mrow><mi>b</mi><mi>a</mi></mrow></msub><mo>+</mo><msub><mi>T</mi> <mi>a</mi></msub><msup><mrow></mrow> <mrow><mi>b</mi><mi>c</mi></mrow></msup><msub><mi>F</mi> <mrow><mi>b</mi><mi>c</mi></mrow></msub><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">(</mo><msub><mi>Γ</mi> <mi>a</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mi>α</mi><mi>β</mi></mrow></msub><msup><mi>χ</mi> <mi>α</mi></msup><msup><mi>χ</mi> <mi>β</mi></msup><mo>−</mo><msup><mi>χ</mi> <mi>α</mi></msup><msub><mi>L</mi> <mrow><mi>α</mi><mi>a</mi></mrow></msub></mrow><annotation encoding="application/x-tex">D^b F_{b a} + T_a{}^{b c} F_{b c} = - (\Gamma_a)_{\alpha \beta} \chi^\alpha \chi^\beta - \chi^\alpha L_{\alpha a}</annotation></semantics></math> (<a href="#BonoraBregolaLechnerPastiTonin87">Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.31)</a>) (where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mrow><mi>α</mi><mi>a</mi></mrow></msub></mrow><annotation encoding="application/x-tex">L_{\alpha a}</annotation></semantics></math> is defined by (2.20) there…) <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> The <a class="existingWikiWord" href="/nlab/show/B-field">B-field</a> <a class="existingWikiWord" href="/nlab/show/field+strength">strength</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mrow><mi>α</mi><mi>β</mi><mi>γ</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">H_{\alpha \beta \gamma} = 0</annotation></semantics></math> (<a href="#AtickDharRatra86">Atick-Dhar-Ratra 86, (4.2)</a>, <a href="#BonoraPastiTonin87">Bonora-Pasti-Tonin 87, (15)</a>, <a href="#BonoraBregolaLechnerPastiTonin87">Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.14)</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mrow><mi>a</mi><mi>α</mi><mi>β</mi></mrow></msub><mo>=</mo><mi>ϕ</mi><msub><mi>Γ</mi> <mrow><mi>a</mi><mi>α</mi><mi>β</mi></mrow></msub></mrow><annotation encoding="application/x-tex">H_{a \alpha \beta} = \phi \Gamma_{a \alpha \beta}</annotation></semantics></math> (<a href="#AtickDharRatra86">Atick-Dhar-Ratra 86, (4.19)</a>, <a href="#BonoraPastiTonin87">Bonora-Pasti-Tonin 87, (15)</a>, <a href="#BonoraBregolaLechnerPastiTonin87">Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.15)</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>≔</mo><msub><mi>D</mi> <mi>α</mi></msub><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\rho \coloneqq D_\alpha \phi</annotation></semantics></math> (<a href="#AtickDharRatra86">Atick-Dhar-Ratra 86, (4.20)</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mrow><mi>a</mi><mi>b</mi><mi>α</mi></mrow></msub><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>Γ</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msub><msub><mo></mo><mi>α</mi></msub><msup><mrow></mrow> <mi>β</mi></msup><msub><mi>ρ</mi> <mi>β</mi></msub></mrow><annotation encoding="application/x-tex">H_{a b \alpha} = -\tfrac{1}{2} \Gamma_{a b }_\alpha{}^\beta \rho_\beta</annotation></semantics></math> (<a href="#AtickDharRatra86">Atick-Dhar-Ratra 86, (4.21)</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></msub><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mstyle displaystyle="false"><mfrac><mn>3</mn><mn>2</mn></mfrac></mstyle><mi>ϕ</mi><msub><mi>T</mi> <mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></msub><mo>+</mo><mstyle displaystyle="false"><mfrac><mrow><msub><mi>c</mi> <mn>1</mn></msub></mrow><mn>4</mn></mfrac></mstyle><mo stretchy="false">(</mo><msub><mi>Γ</mi> <mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></msub><msub><mo stretchy="false">)</mo> <mrow><mi>α</mi><mi>β</mi></mrow></msub><mi>tr</mi><mo stretchy="false">(</mo><msup><mi>χ</mi> <mi>α</mi></msup><msup><mi>χ</mi> <mi>β</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H_{a b c} = - \tfrac{3}{2} \phi T_{a b c} + \tfrac{c_1}{4} (\Gamma_{a b c})_{\alpha \beta} tr(\chi^\alpha \chi^\beta)</annotation></semantics></math> (<a href="#AtickDharRatra86">Atick-Dhar-Ratra 86, (4.22)</a>) According to (<a href="#BonoraBregolaLechnerPastiTonin90">Bonora-Bregola-Lechner-Pasti-Tonin 90</a>) in fact all these constraints follow from just <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>T</mi> <mrow><mi>α</mi><mi>β</mi></mrow> <mi>a</mi></msubsup><mo>∝</mo><msubsup><mi>Γ</mi> <mrow><mi>α</mi><mi>β</mi></mrow> <mi>a</mi></msubsup></mrow><annotation encoding="application/x-tex">T^a_{\alpha \beta} \propto \Gamma^a_{\alpha \beta}</annotation></semantics></math>, up to field redefinition. See also at <a class="existingWikiWord" href="/nlab/show/torsion+constraints+in+supergravity">torsion constraints in supergravity</a>. <h2 id="related_concepts">Related concepts</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/SemiSpin%2816%29">SemiSpin(16)</a>, <a class="existingWikiWord" href="/nlab/show/SemiSpin%2832%29">SemiSpin(32)</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/small+instanton">small instanton</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/E-string">E-string</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> <ul> <li> heterotic string theory <ul> <li> <a class="existingWikiWord" href="/nlab/show/2d+SCFT">2d SCFT</a>, <ul> <li><a class="existingWikiWord" href="/nlab/show/2-spectral+triple">2-spectral triple</a>, <a class="existingWikiWord" href="/nlab/show/Dirac-Ramond+operator">Dirac-Ramond operator</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/2d+%282%2C0%29-superconformal+QFT">2d (2,0)-superconformal QFT</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/dual+heterotic+string+theory">dual heterotic string theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/heterotic+string+theory+on+CY3-manifolds">heterotic string theory on CY3-manifolds</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Witten+genus">Witten genus</a>, <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theories+and+tmf">(2,1)-dimensional Euclidean field theories and tmf</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/heterotic+line+bundle">heterotic line bundle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/type+0+string+theory">type 0 string theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/landscape+of+string+theory+vacua">landscape of string theory vacua</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/supersymmetry+and+Calabi-Yau+manifolds">supersymmetry and Calabi-Yau manifolds</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/string+field+theory">string field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Ho%C5%99ava-Witten+theory">Hořava-Witten theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/AdS-CFT+correspondence">AdS-CFT correspondence</a> </li> <li> <a href="string+theory+FAQ#DoesSTPredictSupersymmetry">string theory FAQ – Does string theory predict supersymmetry?</a> </li> </ul> <h2 id="references">References</h2> <h3 id="general">General</h3> Heterotic strings were introduced in <ul> <li> <a class="existingWikiWord" href="/nlab/show/David+Gross">David Gross</a>, <a class="existingWikiWord" href="/nlab/show/Jeffrey+Harvey">Jeffrey Harvey</a>, <a class="existingWikiWord" href="/nlab/show/Emil+Martinec">Emil Martinec</a>, <a class="existingWikiWord" href="/nlab/show/Ryan+Rohm">Ryan Rohm</a>: Heterotic string theory (I). The free heterotic string Nucl. Phys. B 256 (1985), 253 (<a href="https://doi.org/10.1016/0550-3213(85)90394-3">doi:10.1016/0550-3213(85)90394-3</a>) Heterotic string theory (II). The interacting heterotic string , Nucl. Phys. B 267 (1986), 75 (<a href="https://doi.org/10.1016/0550-3213(86)90146-X">doi:10.1016/0550-3213(86)90146-X</a>) </li> <li id="CHSW85"> <a class="existingWikiWord" href="/nlab/show/Philip+Candelas">Philip Candelas</a>, <a class="existingWikiWord" href="/nlab/show/Gary+Horowitz">Gary Horowitz</a>, <a class="existingWikiWord" href="/nlab/show/Andrew+Strominger">Andrew Strominger</a>, <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, Vacuum configurations for superstrings, Nuclear Physics B Volume 258, 1985, Pages 46-74 Nucl. Phys. B 258, 46 (1985) (<a href="https://doi.org/10.1016/0550-3213(85)90602-9">doi:10.1016/0550-3213(85)90602-9</a>) </li> <li id="Schellekens91"> <a class="existingWikiWord" href="/nlab/show/Bert+Schellekens">Bert Schellekens</a>, Classification of Ten-Dimensional Heterotic Strings, Phys.Lett. B277 (1992) 277-284 (<a href="http://arxiv.org/abs/hep-th/9112006">arXiv:hep-th/9112006</a>) </li> </ul> Relation to <a class="existingWikiWord" href="/nlab/show/Niemeier+lattices">Niemeier lattices</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Wolfgang+Lerche">Wolfgang Lerche</a>, <a class="existingWikiWord" href="/nlab/show/Dieter+L%C3%BCst">Dieter Lüst</a>, <a class="existingWikiWord" href="/nlab/show/Adrian+Norbert+Schellekens">Adrian Norbert Schellekens</a>, Ten-dimensional heterotic strings from Niemeier lattices, Physics Letters B 181 1–2 (1986) 71-75 [<a href="https://doi.org/10.1016/0370-2693(86)91257-8">doi:10.1016/0370-2693(86)91257-8</a>, <a href="https://inspirehep.net/literature/233203">inspire:233203</a>]</li> </ul> Textbook accounts: <ul> <li id="CastellaniDAuriaFre91"> <a class="existingWikiWord" href="/nlab/show/Leonardo+Castellani">Leonardo Castellani</a>, <a class="existingWikiWord" href="/nlab/show/Riccardo+D%27Auria">Riccardo D'Auria</a>, <a class="existingWikiWord" href="/nlab/show/Pietro+Fr%C3%A9">Pietro Fré</a>, vol 3 (which is part 6) of <a class="existingWikiWord" href="/nlab/show/Supergravity+and+Superstrings+-+A+Geometric+Perspective">Supergravity and Superstrings - A Geometric Perspective</a>, World Scientific (1991) </li> <li> <a class="existingWikiWord" href="/nlab/show/Joseph+Polchinski">Joseph Polchinski</a>, volume II, section 11 of <a class="existingWikiWord" href="/nlab/show/String+theory">String theory</a>, </li> <li> <a class="existingWikiWord" href="/nlab/show/Eric+D%27Hoker">Eric D'Hoker</a>, String theory – lecture 8: Heterotic strings in part 3 (p. 941 of volume II) of <a class="existingWikiWord" href="/nlab/show/Pierre+Deligne">Pierre Deligne</a>, P. Etingof, <a class="existingWikiWord" href="/nlab/show/Dan+Freed">Dan Freed</a>, L. Jeffrey, <a class="existingWikiWord" href="/nlab/show/David+Kazhdan">David Kazhdan</a>, <a class="existingWikiWord" href="/nlab/show/John+Morgan">John Morgan</a>, D.R. Morrison and <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, eds. . <a class="existingWikiWord" href="/nlab/show/Quantum+Fields+and+Strings">Quantum Fields and Strings</a>, A course for mathematicians, 2 vols. Amer. Math. Soc. Providence 1999. (<a href="http://www.math.ias.edu/qft">web version</a>) </li> </ul> See also: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Eric+Sharpe">Eric Sharpe</a>, Recent developments in heterotic compactifications, AMS/IP Stud. Adv. Math. 44 (2008) 209-230 (<a href="https://arxiv.org/abs/0801.4080">arXiv:0801.4080</a>) </li> <li> Andrea Fontanella, Tomas Ortin, On the supersymmetric solutions of the Heterotic Superstring effective action (<a href="https://arxiv.org/abs/1910.08496">arxiv:1910.08496</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Ioannis+Florakis">Ioannis Florakis</a>, <a class="existingWikiWord" href="/nlab/show/John+Rizos">John Rizos</a>: Free Fermionic Constructions of Heterotic Strings [<a href="https://arxiv.org/abs/2407.07034">arXiv:2407.07034</a>] </li> </ul> Relation to <a class="existingWikiWord" href="/nlab/show/Donaldson-Thomas+theory">Donaldson-Thomas theory</a> and <a class="existingWikiWord" href="/nlab/show/quiver+gauge+theory">quiver gauge theory</a>: <ul> <li id="HeLee12"><a class="existingWikiWord" href="/nlab/show/Yang-Hui+He">Yang-Hui He</a>, Seung-Joo Lee, Quiver Structure of Heterotic Moduli, J. High Energ. Phys. (2012) 2012: 119 (<a href="https://arxiv.org/abs/1208.3004">arXiv:1208.3004</a>)</li> </ul> Discussion of <a class="existingWikiWord" href="/nlab/show/higher+curvature+corrections">higher curvature corrections</a>: <ul> <li> Eric Lescano, Carmen Núñez, Jesús A. Rodríguez, Supersymmetry, T-duality and Heterotic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\alpha'</annotation></semantics></math>-corrections (<a href="https://arxiv.org/abs/2104.09545">arXiv:2104.09545</a>) </li> <li> Hao-Yuan Chang, <a class="existingWikiWord" href="/nlab/show/Ergin+Sezgin">Ergin Sezgin</a>, Yoshiaki Tanii, Dimensional reduction of higher derivative heterotic supergravity (<a href="https://arxiv.org/abs/2110.13163">arXiv:2110.13163</a>) </li> </ul> On non-supersymmetric <a class="existingWikiWord" href="/nlab/show/branes">branes</a> in heterotic string theory: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Justin+Kaidi">Justin Kaidi</a>, <a class="existingWikiWord" href="/nlab/show/Kantaro+Ohmori">Kantaro Ohmori</a>, <a class="existingWikiWord" href="/nlab/show/Yuji+Tachikawa">Yuji Tachikawa</a>, <a class="existingWikiWord" href="/nlab/show/Kazuya+Yonekura">Kazuya Yonekura</a>, Non-supersymmetric heterotic branes [<a href="https://arxiv.org/abs/2303.17623">arXiv:2303.17623</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Justin+Kaidi">Justin Kaidi</a>, Non-Supersymmetric Heterotic Branes, talk at TH String Theory Seminar (Nov 2023) [<a href="https://cds.cern.ch/record/2881994">cds:2881994</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Justin+Kaidi">Justin Kaidi</a>, <a class="existingWikiWord" href="/nlab/show/Yuji+Tachikawa">Yuji Tachikawa</a>, <a class="existingWikiWord" href="/nlab/show/Kazuya+Yonekura">Kazuya Yonekura</a>: On non-supersymmetric heterotic branes [<a href="https://arxiv.org/abs/2411.04344">arXiv:2411.04344</a>] </li> </ul> <h3 id="orbifold_and_orientifold_compactifications">Orbifold and orientifold compactifications</h3> Heterotic strings on <a class="existingWikiWord" href="/nlab/show/orbifolds">orbifolds</a>: <ul> <li id="DixonHarveyVafaWitten85"> <a class="existingWikiWord" href="/nlab/show/Lance+Dixon">Lance Dixon</a>, <a class="existingWikiWord" href="/nlab/show/Jeff+Harvey">Jeff Harvey</a>, <a class="existingWikiWord" href="/nlab/show/Cumrun+Vafa">Cumrun Vafa</a>, <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, Strings on orbifolds, Nuclear Physics B Volume 261, 1985, Pages 678-686 (<a href="https://doi.org/10.1016/0550-3213(85)90593-0">doi:10.1016/0550-3213(85)90593-0</a>) </li> <li id="DixonHarveyVafaWitten86"> <a class="existingWikiWord" href="/nlab/show/Lance+Dixon">Lance Dixon</a>, <a class="existingWikiWord" href="/nlab/show/Jeff+Harvey">Jeff Harvey</a>, <a class="existingWikiWord" href="/nlab/show/Cumrun+Vafa">Cumrun Vafa</a>, <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, Strings on orbifolds (II), Nuclear Physics B Volume 274, Issue 2, 15 September 1986, Pages 285-314 (<a href="https://doi.org/10.1016/0550-3213(86)90287-7">doi:10.1016/0550-3213(86)90287-7</a>) </li> <li> Joel Giedt, Heterotic Orbifolds (<a href="https://arxiv.org/abs/hep-ph/0204315">arXiv:hep-ph/0204315</a>) </li> <li> Kang-Sin Choi, Spectra of Heterotic Strings on Orbifolds, Nucl. Phys. B708: 194-214, 2005 (<a href="https://arxiv.org/abs/hep-th/0405195">arXiv:hep-th/0405195</a>) </li> </ul> Specifically on <a class="existingWikiWord" href="/nlab/show/ADE-singularities">ADE-singularities</a>: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Paul+Aspinwall">Paul Aspinwall</a>, <a class="existingWikiWord" href="/nlab/show/David+Morrison">David Morrison</a>, Point-like Instantons on K3 Orbifolds, Nucl. Phys. B503 (1997) 533-564 (<a href="https://arxiv.org/abs/hep-th/9705104">arXiv:hep-th/9705104</a>) </li> <li id="Witten99"> <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, Heterotic String Conformal Field Theory And A-D-E Singularities, JHEP 0002:025, 2000 (<a href="https://arxiv.org/abs/hep-th/9909229">arXiv:hep-th/9909229</a>) </li> </ul> A kind of unusual boundary condition for heterotic strings, (analogous to open <a class="existingWikiWord" href="/nlab/show/M5-branes">M5-branes</a> ending in <a class="existingWikiWord" href="/nlab/show/Yang+monopoles">Yang monopoles</a> on <a class="existingWikiWord" href="/nlab/show/M9-branes">M9-branes</a>): <ul> <li><a class="existingWikiWord" href="/nlab/show/Joseph+Polchinski">Joseph Polchinski</a>, Open Heterotic Strings, JHEP 0609 (2006) 082 (<a href="http://arxiv.org/abs/hep-th/0510033">arXiv:hep-th/0510033</a>)</li> </ul> <div> <h3 id="HeteroticStringPhenomenologyReferences">Heterotic string phenomenology</h3> The historical origin of all <a class="existingWikiWord" href="/nlab/show/string+phenomenology">string phenomenology</a> is the <a class="existingWikiWord" href="/nlab/show/top-down+model+building">top-down</a> <a class="existingWikiWord" href="/nlab/show/GUT">GUT</a>-model building in <a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a> due to <ul> <li id="CHSW85"><a class="existingWikiWord" href="/nlab/show/Philip+Candelas">Philip Candelas</a>, <a class="existingWikiWord" href="/nlab/show/Gary+Horowitz">Gary Horowitz</a>, <a class="existingWikiWord" href="/nlab/show/Andrew+Strominger">Andrew Strominger</a>, <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, Vacuum configurations for superstrings, Nuclear Physics B Volume 258, 1985, Pages 46-74 Nucl. Phys. B 258, 46 (1985) [<a href="https://doi.org/10.1016/0550-3213(85)90602-9">doi:10.1016/0550-3213(85)90602-9</a>]</li> </ul> Review and exposition: <ul> <li id="Witten02"> <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, Quest For Unification, Heinrich Hertz lecture at <a href="http://www.desy.de/susy02/">SUSY 2002</a> at DESY, Hamburg [<a href="http://arxiv.org/abs/hep-ph/0207124">arXiv:hep-ph/0207124</a>] </li> <li id="Nilles11"> <a class="existingWikiWord" href="/nlab/show/Hans-Peter+Nilles">Hans-Peter Nilles</a>, Strings, Exceptional Groups and Grand Unification, talk at <a href="https://indico.cern.ch/event/112851/">Planck 2011</a> [<a href="http://www.th.physik.uni-bonn.de/nilles/db/HPtalks/114planck.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Nilles11GUT.pdf" title="pdf">pdf</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Saul+Ramos-Sanchez">Saul Ramos-Sanchez</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Ratz">Michael Ratz</a>, Heterotic Orbifold Models, in <a class="existingWikiWord" href="/nlab/show/Handbook+of+Quantum+Gravity">Handbook of Quantum Gravity</a>, Springer (2024) [<a href="https://doi.org/10.1007/978-981-19-3079-9">doi:10.1007/978-981-19-3079-9</a>, <a href="https://arxiv.org/abs/2401.03125">arXiv:2401.03125</a>] </li> </ul> <h4 id="HeteroticStringPhenomenologyReferencesForE8">The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub><mo>×</mo><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">E_8 \times E_8</annotation></semantics></math>-heterotic string</h4> The following articles claim the existence of exact realization of the <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a> and <a class="existingWikiWord" href="/nlab/show/matter">matter</a>-content of the <a class="existingWikiWord" href="/nlab/show/MSSM">MSSM</a> in <a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a> on <a class="existingWikiWord" href="/nlab/show/orbifolds">orbifolds</a> (not yet checking <a class="existingWikiWord" href="/nlab/show/Yukawa+couplings">Yukawa couplings</a>): <ul> <li> <a class="existingWikiWord" href="/nlab/show/Volker+Braun">Volker Braun</a>, <a class="existingWikiWord" href="/nlab/show/Yang-Hui+He">Yang-Hui He</a>, <a class="existingWikiWord" href="/nlab/show/Burt+Ovrut">Burt Ovrut</a>, <a class="existingWikiWord" href="/nlab/show/Tony+Pantev">Tony Pantev</a>, A Heterotic Standard Model, Phys. Lett. B618 : 252-258 2005 (<a href="http://arxiv.org/abs/hep-th/0501070">arXiv:hep-th/0501070</a>) </li> <li> Wilfried Buchmuller, Koichi Hamaguchi, Oleg Lebedev, <a class="existingWikiWord" href="/nlab/show/Michael+Ratz">Michael Ratz</a>, Supersymmetric Standard Model from the Heterotic String, Phys. Rev. Lett. 96 121602 (2006) ([<a href="https://doi.org/10.1103/PhysRevLett.96.121602">doi:v</a><a href="https://arxiv.org/abs/hep-ph/0511035">arXiv:hep-ph/0511035</a>] </li> <li id="BraunHeOvrutPantev06"> <a class="existingWikiWord" href="/nlab/show/Volker+Braun">Volker Braun</a>, <a class="existingWikiWord" href="/nlab/show/Yang-Hui+He">Yang-Hui He</a>, <a class="existingWikiWord" href="/nlab/show/Burt+Ovrut">Burt Ovrut</a>, <a class="existingWikiWord" href="/nlab/show/Tony+Pantev">Tony Pantev</a>, The Exact MSSM Spectrum from String Theory, JHEP 0605:043, 2006 (<a href="http://arxiv.org/abs/hep-th/0512177">arXiv:hep-th/0512177</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Vincent+Bouchard">Vincent Bouchard</a>, <a class="existingWikiWord" href="/nlab/show/Ron+Donagi">Ron Donagi</a>, An SU(5) Heterotic Standard Model, Phys. Lett. B633:783-791,2006 (<a href="http://arxiv.org/abs/hep-th/0512149">arXiv:hep-th/0512149</a>) </li> </ul> A computer search through the “<a class="existingWikiWord" href="/nlab/show/landscape+of+string+theory+vacua">landscape</a>” of <a class="existingWikiWord" href="/nlab/show/Calabi-Yau+varieties">Calabi-Yau varieties</a> showed severeal hundreds more such exact heterotic standard models (about one billionth of all CYs searched, and most of them arising as <a class="existingWikiWord" href="/nlab/show/SU%285%29">SU(5)</a>-<a class="existingWikiWord" href="/nlab/show/GUTs">GUTs</a>): general computational theory: <ul> <li><a class="existingWikiWord" href="/nlab/show/Lara+Anderson">Lara Anderson</a>, <a class="existingWikiWord" href="/nlab/show/Yang-Hui+He">Yang-Hui He</a>, <a class="existingWikiWord" href="/nlab/show/Andre+Lukas">Andre Lukas</a>, Heterotic Compactification, An Algorithmic Approach, JHEP 0707:049, 2007 (<a href="https://arxiv.org/abs/hep-th/0702210">arXiv:hep-th/0702210</a>)</li> </ul> using <a class="existingWikiWord" href="/nlab/show/heterotic+line+bundle+models">heterotic line bundle models</a>: <ul> <li id="AndersonGrayLukasPalti11"> <a class="existingWikiWord" href="/nlab/show/Lara+Anderson">Lara Anderson</a>, <a class="existingWikiWord" href="/nlab/show/James+Gray">James Gray</a>, <a class="existingWikiWord" href="/nlab/show/Andre+Lukas">Andre Lukas</a>, <a class="existingWikiWord" href="/nlab/show/Eran+Palti">Eran Palti</a>, Two Hundred Heterotic Standard Models on Smooth Calabi-Yau Threefolds, Phys. Rev. D 84, 106005 (2011) (<a href="http://arxiv.org/abs/1106.4804">arXiv:1106.4804</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Lara+Anderson">Lara Anderson</a>, <a class="existingWikiWord" href="/nlab/show/James+Gray">James Gray</a>, <a class="existingWikiWord" href="/nlab/show/Andre+Lukas">Andre Lukas</a>, <a class="existingWikiWord" href="/nlab/show/Eran+Palti">Eran Palti</a>, Heterotic Line Bundle Standard Models JHEP06(2012)113 (<a href="https://arxiv.org/abs/1202.1757">arXiv:1202.1757</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Lara+Anderson">Lara Anderson</a>, <a class="existingWikiWord" href="/nlab/show/Andrei+Constantin">Andrei Constantin</a>, <a class="existingWikiWord" href="/nlab/show/James+Gray">James Gray</a>, <a class="existingWikiWord" href="/nlab/show/Andre+Lukas">Andre Lukas</a>, <a class="existingWikiWord" href="/nlab/show/Eran+Palti">Eran Palti</a>, A Comprehensive Scan for Heterotic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(5)</annotation></semantics></math> GUT models, JHEP01(2014)047 (<a href="https://arxiv.org/abs/1307.4787">arXiv:1307.4787</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Yang-Hui+He">Yang-Hui He</a>, Seung-Joo Lee, <a class="existingWikiWord" href="/nlab/show/Andre+Lukas">Andre Lukas</a>, Chuang Sun, Heterotic Model Building: 16 Special Manifolds, J. High Energ. Phys. 2014, 77 (2014) (<a href="http://arxiv.org/abs/1309.0223">arXiv:1309.0223</a>) </li> <li> Stefan Groot Nibbelink, Orestis Loukas, Fabian Ruehle, Patrick K.S. Vaudrevange, Infinite number of MSSMs from heterotic line bundles?, Phys. Rev. D 92, 046002 (2015) (<a href="https://arxiv.org/abs/1506.00879">arXiv:1506.00879</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Andreas+Braun">Andreas Braun</a>, Callum R. Brodie, <a class="existingWikiWord" href="/nlab/show/Andre+Lukas">Andre Lukas</a>, Heterotic Line Bundle Models on Elliptically Fibered Calabi-Yau Three-folds, JHEP04 (2018) 087 (<a href="https://arxiv.org/abs/1706.07688">arXiv:1706.07688</a>) </li> <li id="CHE18"> <a class="existingWikiWord" href="/nlab/show/Andrei+Constantin">Andrei Constantin</a>, <a class="existingWikiWord" href="/nlab/show/Yang-Hui+He">Yang-Hui He</a>, <a class="existingWikiWord" href="/nlab/show/Andre+Lukas">Andre Lukas</a>, Counting String Theory Standard Models, Physics Letters B Volume 792, 10 May 2019, Pages 258-262 (<a href="https://arxiv.org/abs/1810.00444">arXiv:1810.00444</a>) </li> <li> Alon E. Faraggi, Glyn Harries, Benjamin Percival, John Rizos, Towards machine learning in the classification of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>×</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2 \times \mathbb{Z}_2</annotation></semantics></math> orbifold compactifications (<a href="https://arxiv.org/abs/1901.04448">arXiv:1901.04448</a>) </li> <li> Magdalena Larfors, Robin Schneider, Explore and Exploit with Heterotic Line Bundle Models, Fortschritte der Physik Vol 86 Nr. 5 (<a href="https://arxiv.org/abs/2003.04817">arXiv:2003.04817</a>) </li> </ul> The resulting database of <a class="existingWikiWord" href="/nlab/show/heterotic+line+bundle+models">heterotic line bundle models</a> is here: <ul> <li><a class="existingWikiWord" href="/nlab/show/Lara+Anderson">Lara Anderson</a>, <a class="existingWikiWord" href="/nlab/show/James+Gray">James Gray</a>, <a class="existingWikiWord" href="/nlab/show/Andre+Lukas">Andre Lukas</a>, <a class="existingWikiWord" href="/nlab/show/Eran+Palti">Eran Palti</a>, Heterotic standard model database (<a href="http://www-thphys.physics.ox.ac.uk/projects/CalabiYau/linebundlemodels/index.html.">web</a>)</li> </ul> Review includes <ul> <li> <a class="existingWikiWord" href="/nlab/show/Lara+Anderson">Lara Anderson</a>, New aspects of heterotic geometry and phenomenology, talk at <a href="http://wwwth.mpp.mpg.de/conf/strings2012/">Strings2012</a>, Munich 2012 (<a href="http://wwwth.mpp.mpg.de/members/strings/strings2012/strings_files/program/Talks/Wednesday/Anderson.pdf">pdf</a>) </li> <li id="He18"> <a class="existingWikiWord" href="/nlab/show/Yang-Hui+He">Yang-Hui He</a>, Deep-learning the landscape, talk at <a href="https://ims.nus.edu.sg/events/2018/wstring/wk.php">String and M-Theory: The new geometry of the 21st century</a> (<a href="https://ims.nus.edu.sg/events/2018/wstring/files/yang.pdf">pdf slides</a>, <a href="https://www.youtube.com/watch?v=x3ThgBgkPlE">video recording</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Yang-Hui+He">Yang-Hui He</a>, Calabi-Yau Spaces in the String Landscape (<a href="https://arxiv.org/abs/2006.16623">arXiv:2006.16623</a>) </li> </ul> Computation of <a class="existingWikiWord" href="/nlab/show/metrics">metrics</a> on these Calabi-Yau compactifications (eventually needed for computing their induced <a class="existingWikiWord" href="/nlab/show/Yukawa+couplings">Yukawa couplings</a>) is started in <ul> <li><a class="existingWikiWord" href="/nlab/show/Volker+Braun">Volker Braun</a>, Tamaz Brelidze, <a class="existingWikiWord" href="/nlab/show/Michael+Douglas">Michael Douglas</a>, <a class="existingWikiWord" href="/nlab/show/Burt+Ovrut">Burt Ovrut</a>, Calabi-Yau Metrics for Quotients and Complete Intersections, JHEP 0805:080, 2008 (<a href="https://arxiv.org/abs/0712.3563">arXiv:0712.3563</a>)</li> </ul> and via <a class="existingWikiWord" href="/nlab/show/machine+learning">machine learning</a>: <ul> <li> Andrei Constantin, Cristofero S. Fraser-Taliente, Thomas R. Harvey, <a class="existingWikiWord" href="/nlab/show/Andre+Lukas">Andre Lukas</a>, <a class="existingWikiWord" href="/nlab/show/Burt+Ovrut">Burt Ovrut</a>, Computation of Quark Masses from String Theory [<a href="https://arxiv.org/abs/2402.01615">arXiv:2402.01615</a>] </li> <li> Per Berglund, Giorgi Butbaia, <a class="existingWikiWord" href="/nlab/show/Tristan+H%C3%BCbsch">Tristan Hübsch</a>, Vishnu Jejjala, Damián Mayorga Peña, Challenger Mishra, Justin Tan: Precision String Phenomenology [<a href="https://arxiv.org/abs/2407.13836">arXiv:2407.13836</a>] </li> </ul> This “heterotic standard model” has a “hidden sector” copy of the actual standard model, more details of which are discussed here: <ul> <li><a class="existingWikiWord" href="/nlab/show/Volker+Braun">Volker Braun</a>, <a class="existingWikiWord" href="/nlab/show/Yang-Hui+He">Yang-Hui He</a>, <a class="existingWikiWord" href="/nlab/show/Burt+Ovrut">Burt Ovrut</a>, Supersymmetric Hidden Sectors for Heterotic Standard Models (<a href="http://arxiv.org/abs/1301.6767">arXiv:1301.6767</a>)</li> </ul> The issue of <a class="existingWikiWord" href="/nlab/show/moduli+stabilization">moduli stabilization</a> in these kinds of models is discussed in <ul> <li> Michele Cicoli, Senarath de Alwis, Alexander Westphal, Heterotic Moduli Stabilization (<a href="http://arxiv.org/abs/1304.1809">arXiv:1304.1809</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Lara+Anderson">Lara Anderson</a>, <a class="existingWikiWord" href="/nlab/show/James+Gray">James Gray</a>, <a class="existingWikiWord" href="/nlab/show/Andre+Lukas">Andre Lukas</a>, <a class="existingWikiWord" href="/nlab/show/Burt+Ovrut">Burt Ovrut</a>, Vacuum Varieties, Holomorphic Bundles and Complex Structure Stabilization in Heterotic Theories (<a href="http://arxiv.org/abs/1304.2704">arXiv:1304.2704</a>) </li> </ul> Principles singling out heterotic models with three generations of fundamental particles are discussed in: <ul> <li><a class="existingWikiWord" href="/nlab/show/Philip+Candelas">Philip Candelas</a>, <a class="existingWikiWord" href="/nlab/show/Xenia+de+la+Ossa">Xenia de la Ossa</a>, <a class="existingWikiWord" href="/nlab/show/Yang-Hui+He">Yang-Hui He</a>, Balazs Szendroi, Triadophilia: A Special Corner in the Landscape, Adv.Theor.Math.Phys.12:2,2008 (<a href="http://arxiv.org/abs/0706.3134">arXiv:0706.3134</a>)</li> </ul> Discussion of non-<a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetric</a>: <a class="existingWikiWord" href="/nlab/show/GUT">GUT</a> models: <ul> <li>Alon E. Faraggi, Viktor G. Matyas, Benjamin Percival, Classification of Non-Supersymmetric Pati-Salam Heterotic String Models (<a href="https://arxiv.org/abs/2011.04113">arXiv:2011.04113</a>)</li> </ul> See also: <ul> <li id="AngelantonjFlorakis18"><a class="existingWikiWord" href="/nlab/show/Carlo+Angelantonj">Carlo Angelantonj</a>, <a class="existingWikiWord" href="/nlab/show/Ioannis+Florakis">Ioannis Florakis</a>, GUT Scale Unification in Heterotic Strings, Physics Letters B 789 (2019) 496-501 [<a href="https://doi.org/10.1016/j.physletb.2018.12.054">doi:10.1016/j.physletb.2018.12.054</a><a href="https://arxiv.org/abs/1812.06915">arXiv:1812.06915</a>]</li> </ul> <h4 id="HeteroticStringPhenomenologyReferencesForSemiSpin32">The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SemiSpin</mi><mo stretchy="false">(</mo><mn>32</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SemiSpin(32)</annotation></semantics></math>-heterotic string</h4> Discussion of <a class="existingWikiWord" href="/nlab/show/string+phenomenology">string phenomenology</a> for the <a class="existingWikiWord" href="/nlab/show/SemiSpin%2832%29">SemiSpin(32)</a>-<a class="existingWikiWord" href="/nlab/show/heterotic+string">heterotic string</a> (see also at <a href="type+I+string+theory#Phenomenology">type I phenomenology</a>): <ul> <li> Kang-Sin Choi, Stefan Groot Nibbelink, Michele Trapletti, Heterotic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>32</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(32)</annotation></semantics></math> model building in four dimensions, JHEP 0412:063, 2004 (<a href="https://arxiv.org/abs/hep-th/0410232">arXiv:hep-th/0410232</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Hans-Peter+Nilles">Hans-Peter Nilles</a>, <a class="existingWikiWord" href="/nlab/show/Saul+Ramos-Sanchez">Saul Ramos-Sanchez</a>, Patrick K.S. Vaudrevange, Akin Wingerter, Exploring the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>32</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(32)</annotation></semantics></math> Heterotic String, JHEP 0604:050 (2006) [<a href="https://doi.org/10.1088/1126-6708/2006/04/050">doi:10.1088/1126-6708/2006/04/050</a><a href="https://arxiv.org/abs/hep-th/0603086">arXiv:hep-th/0603086</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Saul+Ramos-Sanchez">Saul Ramos-Sanchez</a>, Towards Low Energy Physics from the Heterotic String, Fortsch. Phys. 10 (2009) 907-1036 [<a href="https://doi.org/10.1002/prop.200900073">doi:10.1002/prop.200900073</a><a href="https://arxiv.org/abs/0812.3560">arXiv:0812.3560</a>] (<a class="existingWikiWord" href="/nlab/show/heterotic+string">heterotic</a> <a class="existingWikiWord" href="/nlab/show/SemiSpin%2832%29">SemiSpin(32)</a> <a class="existingWikiWord" href="/nlab/show/R-parity">R-parity</a> <a class="existingWikiWord" href="/nlab/show/MSSM">MSSM</a> <a class="existingWikiWord" href="/nlab/show/string+theory+vacua">vacua</a>) </li> <li id="Yamatsu17"> Naoki Yamatsu, String-Inspired Special Grand Unification, Progress of Theoretical and Experimental Physics, Volume 2017, Issue 10, 1 (<a href="https://arxiv.org/abs/1708.02078">arXiv:1708.02078</a>, <a href="https://doi.org/10.1093/ptep/ptx135">doi:10.1093/ptep/ptx135</a>) </li> <li> Jihn E. Kim, Grand unfication models from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>32</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(32)</annotation></semantics></math> heterotic string (<a href="https://arxiv.org/abs/2008.00367">arXov:2008.00367</a>) </li> </ul> On <a class="existingWikiWord" href="/nlab/show/heterotic+line+bundle+models">heterotic line bundle models</a>: <ul> <li>Hajime Otsuka, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>32</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(32)</annotation></semantics></math> heterotic line bundle models, (<a href="https://arxiv.org/abs/1801.03684">arXiv:1801.03684</a>)</li> </ul> <h4 id="HeteroticStringPhenomenologyReferencesForSO16">The “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo><mo>×</mo><mi>SO</mi><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(16)\times SO(16)</annotation></semantics></math>”-heterotic string</h4> This non-<a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetric</a> string theory was first described in: <ul> <li id="AGMV86het"> <a class="existingWikiWord" href="/nlab/show/Luis+Alvarez-Gaum%C3%A9">Luis Alvarez-Gaumé</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Ginsparg">Paul Ginsparg</a>, <a class="existingWikiWord" href="/nlab/show/Gregory+Moore">Gregory Moore</a> and <a class="existingWikiWord" href="/nlab/show/Cumrun+Vafa">Cumrun Vafa</a>, An <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo><mo>×</mo><mi>O</mi><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(16)\times O(16)</annotation></semantics></math> heterotic string, Phys. Lett. B 171 (1986) 155 [<a href="https://doi.org/10.1016/0370-2693(86)91524-8">doi:10.1016/0370-2693(86)91524-8</a>] </li> <li id="DH86het"> <a class="existingWikiWord" href="/nlab/show/Lance+Dixon">Lance Dixon</a> and <a class="existingWikiWord" href="/nlab/show/Jeffrey+Harvey">Jeffrey Harvey</a>, String Theories in Ten-Dimensions Without Space-Time Supersymmetry, Nucl. Phys. B 274 (1986) 93 [<a href="https://doi.org/10.1016/0550-3213(86)90619-X">doi:10.1016/0550-3213(86)90619-X</a>] </li> </ul> A proposal on what the correct global character of the <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a> is appears in: <ul> <li id="McIn99spin"><a class="existingWikiWord" href="/nlab/show/Brett+McInnes">Brett McInnes</a>, The Semispin Groups in String Theory, J. Math. Phys. 40 (1999) 4699-4712 [<a href="https://arxiv.org/abs/hep-th/9906059">arXiv:hep-th/9906059</a>, <a href="https://doi.org/10.1063/1.532999">doi:10.1063/1.532999</a>]</li> </ul> A suggestion that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo><mo>×</mo><mi>SO</mi><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(16)\times SO(16)</annotation></semantics></math> heterotic string is a the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub><mo>×</mo><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">E_8\times E_8</annotation></semantics></math> string “minus” the <a class="existingWikiWord" href="/nlab/show/semispin+group">semispin group</a> <a class="existingWikiWord" href="/nlab/show/SemiSpin%2832%29"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mtext>Spin</mtext> <mo stretchy="false">(</mo> <mn>32</mn> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <msub><mi>ℤ</mi> <mn>2</mn></msub> </mrow> <annotation encoding="application/x-tex">\text{Spin}(32)/\mathbb{Z}_2</annotation> </semantics> </math></a>: <ul> <li id="SW87anom"><a class="existingWikiWord" href="/nlab/show/Adrian+Norbert+Schellekens">Adrian Norbert Schellekens</a>, and <a class="existingWikiWord" href="/nlab/show/Nicholas+Warner">Nicholas Warner</a> Anomalies, characters and strings, Nuclear Physics B 287 (1987) 317-361 [<a href="https://doi.org/10.1016/0550-3213(87)90108-8">doi:10.1016/0550-3213(87)90108-8</a>]</li> </ul> </div><div> <h3 id="HigherGaugeTheoryOfTheGreenSchwarzMechanismReferences">Higher gauge theory of the Green-Schwarz mechanism</h3> Discussion of <a class="existingWikiWord" href="/nlab/show/higher+gauge+theory">higher gauge theory</a> modeling the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanisms">Green-Schwarz mechanisms</a> for <a class="existingWikiWord" href="/nlab/show/anomaly+cancellation">anomaly cancellation</a> in <a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a>, on <a class="existingWikiWord" href="/nlab/show/M5-branes">M5-branes</a>, and in related systems in terms of some kind of <a class="existingWikiWord" href="/nlab/show/nonabelian+differential+cohomology">nonabelian differential cohomology</a> (ordered by arXiv time-stamp): <ul> <li> <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, pp. 13 in: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra connections and applications to String- and Chern-Simons <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>-transport, in Quantum Field Theory, Birkhäuser (2009) 303-424 [<a href="https://arxiv.org/abs/0801.3480">arXiv:0801.3480</a>, <a href="https://doi.org/10.1007/978-3-7643-8736-5_17">doi:10.1007/978-3-7643-8736-5_17</a>] </li> <li id="SatiSchreiberStasheff12"> <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, <a class="existingWikiWord" href="/schreiber/show/Twisted+Differential+String+and+Fivebrane+Structures">Twisted Differential String and Fivebrane Structures</a>, Comm. Math. Phys. 315 (2012) 169-213 (<a href="https://arxiv.org/abs/0910.4001">arXiv:0910.4001</a>, <a href="https://link.springer.com/article/10.1007/s00220-012-1510-3">doi:10.1007/s00220-012-1510-3</a>) <blockquote> (via <a class="existingWikiWord" href="/nlab/show/adjusted+Weil+algebras">adjusted Weil algebras</a>, see there for more) </blockquote> </li> <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>, §3.7, §3.8 in: Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory, Adv. Theor. Math. Phys. 18 (2014) 229-321 [<a href="https://arxiv.org/abs/1201.5277">arXiv:1201.5277</a>, <a href="https://projecteuclid.org/journals/advances-in-theoretical-and-mathematical-physics/volume-18/issue-2/Multiple-M5-branes-String-2-connections-and-7d-nonabelian-Chern/atmp/1414414836.full">euclid:atmp/1414414836</a>] </li> <li id="FSS15"> <a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/schreiber/show/The+moduli+3-stack+of+the+C-field">The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>E</mi> <mn>8</mn></msub> </mrow> <annotation encoding="application/x-tex">E_8</annotation> </semantics> </math> moduli 3-stack of the C-field in M-theory</a>, Comm. Math. Phys. 333 1 (2015) 117-151 [<a href="https://arxiv.org/abs/1202.2455">arXiv:1202.2455</a>, <a href="https://doi.org/10.1007/s00220-014-2228-1">doi:10.1007/s00220-014-2228-1</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Clay+Cordova">Clay Cordova</a>, <a class="existingWikiWord" href="/nlab/show/Thomas+Dumitrescu">Thomas Dumitrescu</a>, <a class="existingWikiWord" href="/nlab/show/Kenneth+Intriligator">Kenneth Intriligator</a>: Exploring 2-Group Global Symmetries, J. High Energ. Phys. 2019 184 (2019) (<a href="https://arxiv.org/abs/1802.04790">arXiv:1802.04790</a>, <a href="https://doi.org/10.1007/JHEP02(2019)184">doi:10.1007/JHEP02(2019)184</a>) </li> <li id="BeniniCordovaHsin19"> <a class="existingWikiWord" href="/nlab/show/Francesco+Benini">Francesco Benini</a>, <a class="existingWikiWord" href="/nlab/show/Clay+Cordova">Clay Cordova</a>, <a class="existingWikiWord" href="/nlab/show/Po-Shen+Hsin">Po-Shen Hsin</a>, On 2-Group Global Symmetries and Their Anomalies, J. High Energ. Phys. 2019 118 (2019) (<a href="https://arxiv.org/abs/1803.09336">arXiv:1803.09336</a>, <a href="https://doi.org/10.1007/JHEP03(2019)118">doi:10.1007/JHEP03(2019)118</a>) </li> <li id="FSS22"> <a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/schreiber/show/Twistorial+Cohomotopy+implies+Green-Schwarz+anomaly+cancellation">Twistorial Cohomotopy implies Green-Schwarz anomaly cancellation</a>, Reviews in Mathematical Physics 34 05 (2022) 2250013 [<a href="https://doi.org/10.1142/S0129055X22500131">doi:10.1142/S0129055X22500131</a>, <a href="https://arxiv.org/abs/2008.08544">arXiv:2008.08544</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Clay+Cordova">Clay Cordova</a>, <a class="existingWikiWord" href="/nlab/show/Thomas+T.+Dumitrescu">Thomas T. Dumitrescu</a>, <a class="existingWikiWord" href="/nlab/show/Kenneth+Intriligator">Kenneth Intriligator</a>, 2-Group Global Symmetries and Anomalies in Six-Dimensional Quantum Field Theories, J. High Energ. Phys. 2021 252 (2021) [<a href="https://doi.org/10.1007/JHEP04(2021)252">doi:10.1007/JHEP04(2021)252</a>, <a href="https://arxiv.org/abs/2009.00138">arXiv:2009.00138</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Michele+Del+Zotto">Michele Del Zotto</a>, <a class="existingWikiWord" href="/nlab/show/Kantaro+Ohmori">Kantaro Ohmori</a>, 2-Group Symmetries of 6D Little String Theories and T-Duality, Annales Henri Poincaré 22 (2021) 2451–2474 [<a href="https://arxiv.org/abs/2009.03489">arXiv:2009.03489</a>, <a href="https://doi.org/10.1007/s00023-021-01018-3">doi:10.1007/s00023-021-01018-3</a>] </li> <li id="SS20"> <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, The character map in equivariant twistorial Cohomotopy implies the Green-Schwarz mechanism with heterotic M5-branes [<a href="https://arxiv.org/abs/2011.06533">arXiv:2011.06533</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, §2.9 in: <a class="existingWikiWord" href="/schreiber/show/M%2FF-Theory+as+Mf-Theory">M/F-Theory as Mf-Theory</a>, Reviews in Mathematical Physics 35 10 (2023) [<a href="https://doi.org/10.1142/S0129055X23500289">doi:10.1142/S0129055X23500289</a>, <a href="https://arxiv.org/abs/2103.01877">arXiv:2103.01877</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Yasunori+Lee">Yasunori Lee</a>, <a class="existingWikiWord" href="/nlab/show/Kantaro+Ohmori">Kantaro Ohmori</a>, <a class="existingWikiWord" href="/nlab/show/Yuji+Tachikawa">Yuji Tachikawa</a>, Matching higher symmetries across Intriligator-Seiberg duality, J. High Energ. Phys. 2021 114 (2021) [<a href="https://arxiv.org/abs/2108.05369">arXiv:2108.05369</a>, <a href="https://doi.org/10.1007/JHEP10(2021)114">doi:10.1007/JHEP10(2021)114</a>] </li> <li> Monica Jinwoo Kang, Sungkyung Kang, Central extensions of higher groups: Green-Schwarz mechanism and 2-connections [<a href="https://arxiv.org/abs/2311.14666">arXiv:2311.14666</a>] </li> </ul> </div> <h3 id="superspace_formulation_of_heterotic_supergravity">Superspace formulation of Heterotic supergravity</h3> Discussion of heterotic supergravity in terms of <a class="existingWikiWord" href="/nlab/show/superspace">superspace</a> includes the following. One solution of the heterotic superspace <a class="existingWikiWord" href="/nlab/show/Bianchi+identities">Bianchi identities</a> is due to <ul> <li id="AtickDharRatra86"> <a class="existingWikiWord" href="/nlab/show/Joseph+Atick">Joseph Atick</a>, Avinash Dhar, and Bharat Ratra, Superspace formulation of ten-dimensional N=1 supergravity coupled to N=1 super Yang-Mills theory, Phys. Rev. D 33, 2824, 1986 (<a href="https://doi.org/10.1103/PhysRevD.33.2824">doi.org/10.1103/PhysRevD.33.2824</a>) </li> <li id="Witten86"> <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, Twistor-like transform in ten dimensions, Nuclear Physics B Volume 266, Issue 2, 17 March 1986 </li> </ul> A second solution is due to <a class="existingWikiWord" href="/nlab/show/Bengt+Nilsson">Bengt Nilsson</a>, <a class="existingWikiWord" href="/nlab/show/Renata+Kallosh">Renata Kallosh</a> and others <ul> <li><a class="existingWikiWord" href="/nlab/show/Bengt+Nilsson">Bengt Nilsson</a>, Simple 10-dimensional supergravity in superspace, Nucl. Phys. B188 (1981) 176 (<a href="https://doi.org/10.1016/0550-3213(81)90111-5">doi:10.1016/0550-3213(81)90111-5</a>)</li> </ul> These two solutions are supposed to be equivalent under field redefinition. See also at <a class="existingWikiWord" href="/nlab/show/torsion+constraints+in+supergravity">torsion constraints in supergravity</a>. Further references include these: <ul> <li id="BonoraPastiTonin87"> <a class="existingWikiWord" href="/nlab/show/Loriano+Bonora">Loriano Bonora</a>, <a class="existingWikiWord" href="/nlab/show/Paolo+Pasti">Paolo Pasti</a>, <a class="existingWikiWord" href="/nlab/show/Mario+Tonin">Mario Tonin</a>, Superspace formulation of 10D SUGRA+SYM theory a la Green-Schwarz, Physics Letters B Volume 188, Issue 3, 16 April 1987, Pages 335–339 (<a href="http://dx.doi.org/10.1016/0370-2693(87)91392-X">doi:10.1016/0370-2693(87)91392-X</a>) </li> <li id="BonoraBregolaLechnerPastiTonin87"> <a class="existingWikiWord" href="/nlab/show/Loriano+Bonora">Loriano Bonora</a>, M. Bregola, <a class="existingWikiWord" href="/nlab/show/Kurt+Lechner">Kurt Lechner</a>, <a class="existingWikiWord" href="/nlab/show/Paolo+Pasti">Paolo Pasti</a>, <a class="existingWikiWord" href="/nlab/show/Mario+Tonin">Mario Tonin</a>, Anomaly-free supergravity and super-Yang-Mills theories in ten dimensions, Nuclear Physics B 296 4 (1988) [<a href="http://dx.doi.org/10.1016/0550-3213(88)90402-6">doi:10.1016/0550-3213(88)90402-6</a>, <a href="https://inspirehep.net/literature/247764">inspire:247764</a>] </li> <li id="BonoraBregolaLechnerPastiTonin90"> <a class="existingWikiWord" href="/nlab/show/Loriano+Bonora">Loriano Bonora</a>, M. Bregola; <a class="existingWikiWord" href="/nlab/show/Kurt+Lechner">Kurt Lechner</a>, <a class="existingWikiWord" href="/nlab/show/Paolo+Pasti">Paolo Pasti</a>, <a class="existingWikiWord" href="/nlab/show/Mario+Tonin">Mario Tonin</a>, A discussion of the constraints in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">N=1</annotation></semantics></math> SUGRA-SYM in 10-D, International Journal of Modern Physics A, February 1990, Vol. 05, No. 03 : pp. 461-477 (<a href="https://doi.org/10.1142/S0217751X90000222">doi:10.1142/S0217751X90000222</a>) </li> <li> <a href="#CastellaniDAuriaFre91">Castellani-D’Auria-Fre 91, vol 3, part 6</a> </li> <li id="BBDFLPPRRTZ"> L. Bonora, M. Bregola, R. D’Auria, P. Fré <a class="existingWikiWord" href="/nlab/show/Kurt+Lechner">Kurt Lechner</a>, <a class="existingWikiWord" href="/nlab/show/Paolo+Pasti">Paolo Pasti</a>, I. Pesando, M. Raciti, F. Riva, <a class="existingWikiWord" href="/nlab/show/Mario+Tonin">Mario Tonin</a> and D. Zanon, Some remarks on the supersymmetrization of the Lorentz Chern-Simons form in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">D = 10</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">N= 1</annotation></semantics></math> supergravity theories, Physics Letters B 277 (1992) (<a class="existingWikiWord" href="/nlab/files/BonoraSuperGS.pdf" title="pdf">pdf</a>) </li> <li id="LechnerTonin08"> <a class="existingWikiWord" href="/nlab/show/Kurt+Lechner">Kurt Lechner</a>, <a class="existingWikiWord" href="/nlab/show/Mario+Tonin">Mario Tonin</a>, Superspace formulations of ten-dimensional supergravity, JHEP 0806:021,2008 (<a href="https://arxiv.org/abs/0802.3869">arXiv:0802.3869</a>) </li> </ul> <h3 id="in_elliptic_cohomology">In elliptic cohomology</h3> For more mathematically precise discussion in the context of <a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a> and the <a class="existingWikiWord" href="/nlab/show/Witten+genus">Witten genus</a> see also the references at <a href="Witten+genus#TwistedWittenGenus">Witten genus – Heterotic (twisted) Witten genus, loop group representations and parameterized WZW models</a>. <h3 id="general_flux_backgrounds_and_parameterized_wzw_models">General flux backgrounds and parameterized WZW models</h3> Discussion of heterotic strings whoe <a class="existingWikiWord" href="/nlab/show/current+algebra">current algebra</a>-sector is parameterized by a <a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> originates with <ul> <li id="GatesSiegel88"> <a class="existingWikiWord" href="/nlab/show/Jim+Gates">Jim Gates</a>, <a class="existingWikiWord" href="/nlab/show/Warren+Siegel">Warren Siegel</a>, Leftons, Rightons, Nonlinear <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-Models, and Superstrings, Phys.Lett. B206 (1988) 631 (<a href="https://inspirehep.net/record/251286/">spire</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Jim+Gates">Jim Gates</a>, Strings, superstrings, and two-dimensional lagrangian field theory, pp. 140-184 in Z. Haba, J. Sobczyk (eds.) Functional integration, geometry, and strings, proceedings of the XXV Winter School of Theoretical Physics, Karpacz, Poland (Feb. 1989), , Birkhäuser, 1989. </li> <li id="GKKS91"> <a class="existingWikiWord" href="/nlab/show/Jim+Gates">Jim Gates</a>, S. Ketov, S. Kozenko, O. Solovev, Lagrangian chiral coset construction of heterotic string theories in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1,0)</annotation></semantics></math> superspace, Nucl.Phys. B362 (1991) 199-231 (<a href="http://inspirehep.net/record/314337/?ln=en">spire</a>) </li> </ul> and is further expanded on in <ul> <li id="DistlerSharpe10"><a class="existingWikiWord" href="/nlab/show/Jacques+Distler">Jacques Distler</a>, <a class="existingWikiWord" href="/nlab/show/Eric+Sharpe">Eric Sharpe</a>, Heterotic compactifications with principal bundles for general groups and general levels, Adv. Theor. Math. Phys. 14:335-398, 2010 (<a href="http://arxiv.org/abs/hep-th/0701244">arXiv:hep-th/0701244</a>)</li> </ul> reviewed in <ul> <li id="Sharpe08"><a class="existingWikiWord" href="/nlab/show/Eric+Sharpe">Eric Sharpe</a>, Recent developments in heterotic compactifications, in <a class="existingWikiWord" href="/nlab/show/Eric+Sharpe">Eric Sharpe</a>, <a class="existingWikiWord" href="/nlab/show/Arthur+Greenspoon">Arthur Greenspoon</a>, <a href="http://www.ams.org/bookstore-getitem/item=AMSIP-44">Advances in String Theory: The First Sowers Workshop in Theoretical Physics</a>, AMS 2008 (<a href="http://www.phys.vt.edu/sowers/talks/sharpe-sowers.pdf">pdf slides (39-49)</a>)</li> </ul> The relation of this to <a class="existingWikiWord" href="/nlab/show/equivariant+elliptic+cohomology">equivariant elliptic cohomology</a> is amplified in <ul> <li id="Ando07"><a class="existingWikiWord" href="/nlab/show/Matthew+Ando">Matthew Ando</a>, Equivariant elliptic cohomology and the Fibered WZW models of Distler and Sharpe, <a href="http://www.math.ucsb.edu/~drm/GTPseminar/2007-fall.php">talk 2007</a> (<a href="http://www.math.ucsb.edu/~drm/GTPseminar/notes/20071026-ando/20071026-malmendier.pdf">lecture notes pdf</a>)</li> </ul> <h3 id="on_elliptic_fibrations">On elliptic fibrations</h3> Compactified on an <a class="existingWikiWord" href="/nlab/show/elliptic+curve">elliptic curve</a> or, more generally, <a class="existingWikiWord" href="/nlab/show/elliptic+fibration">elliptic fibration</a>, heterotic string compactifictions are controled by a choice holomorphic <a class="existingWikiWord" href="/nlab/show/stable+bundle">stable bundle</a> on the compact space. Dually this is an <a class="existingWikiWord" href="/nlab/show/F-theory">F-theory</a> compactification on a <a class="existingWikiWord" href="/nlab/show/K3">K3</a>-bundles. The basis of this story is discussed in <ul> <li>Robert Friedman, <a class="existingWikiWord" href="/nlab/show/John+Morgan">John Morgan</a>, <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, Vector Bundles And F-Theory (<a href="http://arxiv.org/abs/hep-th/9701162">arXiv:hep-th/9701162</a>)</li> </ul> A more formal discussion is in <ul> <li>B. Andreas and D. Hernandez Ruiperez, Adv. Theor. Math. Phys. Volume 7, Number 5 (2003), 751-786 Comments on N = 1 Heterotic String Vacua (<a href="http://projecteuclid.org/euclid.atmp/1111510429">project Euclid</a>)</li> </ul> <h3 id="dualities_2">Dualities</h3> <h4 id="with_type_i_superstring_theory">With type I superstring theory</h4> The original conjecture is due to <ul> <li id="Witten05"><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, section 5 of <a class="existingWikiWord" href="/nlab/show/String+Theory+Dynamics+In+Various+Dimensions">String Theory Dynamics In Various Dimensions</a>, Nucl.Phys.B443:85-126 (1995) (<a href="http://arxiv.org/abs/hep-th/9503124">arXiv:hep-th/9503124</a>)</li> </ul> More details are then in <ul> <li id="PolchinskiWitten96"><a class="existingWikiWord" href="/nlab/show/Joseph+Polchinski">Joseph Polchinski</a>, <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, Evidence for Heterotic - Type I String Duality, Nucl.Phys.B460:525-540,1996 (<a href="http://arxiv.org/abs/hep-th/9510169">arXiv:hep-th/9510169</a>)</li> </ul> <h4 id="DualityWithFTheory">With <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-theory</h4> The <a class="existingWikiWord" href="/nlab/show/duality+between+F-theory+and+heterotic+string+theory">duality between F-theory and heterotic string theory</a> originates in <ul> <li id="Sen96"> <a class="existingWikiWord" href="/nlab/show/Ashoke+Sen">Ashoke Sen</a>, F-theory and Orientifolds (<a href="http://arxiv.org/abs/hep-th/9605150">arXiv:hep-th/9605150</a>) </li> <li id="FriedmanMorganWitten97"> Robert Friedman, <a class="existingWikiWord" href="/nlab/show/John+Morgan">John Morgan</a>, <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, Vector Bundles And F Theory (<a href="http://arxiv.org/abs/hep-th/9701162">arXiv:hep-th/9701162</a>) </li> </ul> Reviews include <ul> <li id="Donagi98"> <a class="existingWikiWord" href="/nlab/show/Ron+Donagi">Ron Donagi</a>, ICMP lecture on heterotic/F-theory duality (<a href="http://arxiv.org/abs/hep-th/9802093">arXiv:hep-th/9802093</a>) </li> <li> Björn Andreas, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">N=1</annotation></semantics></math> Heterotic/F-theory duality PhD thesis (<a href="http://edoc.hu-berlin.de/dissertationen/physik/andreas-bjoern/PDF/Andreas.pdf">pdf</a>) </li> </ul> <div> <h3 id="FlavorOnSingleHeteroticM5BranesReferences"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(2)</annotation></semantics></math>-flavor symmetry on heterotic M5-branes</h3> Emergence of <a class="existingWikiWord" href="/nlab/show/SU%282%29">SU(2)</a> <a class="existingWikiWord" href="/nlab/show/flavor+%28particle+physics%29">flavor</a>-<a class="existingWikiWord" href="/nlab/show/chiral+symmetry">symmetry</a> on single <a class="existingWikiWord" href="/nlab/show/M5-branes">M5-branes</a> in <a class="existingWikiWord" href="/nlab/show/heterotic+M-theory+on+ADE-orbifolds">heterotic M-theory on ADE-orbifolds</a> (in the <a class="existingWikiWord" href="/nlab/show/D%3D6+N%3D%281%2C0%29+SCFT">D=6 N=(1,0) SCFT</a> on <a class="existingWikiWord" href="/nlab/show/small+instantons">small instantons</a> in <a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a>): <ul> <li> Abhijit Gadde, Babak Haghighat, <a class="existingWikiWord" href="/nlab/show/Joonho+Kim">Joonho Kim</a>, <a class="existingWikiWord" href="/nlab/show/Seok+Kim">Seok Kim</a>, Guglielmo Lockhart, <a class="existingWikiWord" href="/nlab/show/Cumrun+Vafa">Cumrun Vafa</a>, Section 4.2 of: 6d String Chains, J. High Energ. Phys. 2018, 143 (2018) (<a href="https://arxiv.org/abs/1504.04614">arXiv:1504.04614</a>, <a href="https://doi.org/10.1007/JHEP02(2018)143">doi:10.1007/JHEP02(2018)143</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Kantaro+Ohmori">Kantaro Ohmori</a>, Section 2.3.1 of: Six-Dimensional Superconformal Field Theories and Their Torus Compactifications, Springer Theses 2018 (<a href="https://www.springer.com/gp/book/9789811330919">springer:book/9789811330919</a>) </li> </ul> Argument for this by translation under <a class="existingWikiWord" href="/nlab/show/duality+between+M-theory+and+type+IIA+string+theory">duality between M-theory and type IIA string theory</a> to <a class="existingWikiWord" href="/nlab/show/half+NS5-brane">half NS5-brane</a>/<a class="existingWikiWord" href="/nlab/show/D6%2FD8-brane+bound+state">D6/D8-brane bound state</a> systems in <a class="existingWikiWord" href="/nlab/show/type+I%27+string+theory">type I' string theory</a>: Reviewed in: <ul> <li>Santiago Cabrera, <a class="existingWikiWord" href="/nlab/show/Amihay+Hanany">Amihay Hanany</a>, Marcus Sperling, Section 2.3 of: Magnetic Quivers, Higgs Branches, and 6d <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> Theories, JHEP06(2019)071, JHEP07(2019)137 (<a href="https://arxiv.org/abs/1904.12293">arXiv:1904.12293</a>)</li> </ul> The emergence of flavor in these <a class="existingWikiWord" href="/nlab/show/half+NS5-brane">half NS5-brane</a>/<a class="existingWikiWord" href="/nlab/show/D6%2FD8-brane+bound+state">D6/D8-brane bound state</a> systems, due to the semi-infinite extension of the <a class="existingWikiWord" href="/nlab/show/D6-branes">D6-branes</a> making them act as <a class="existingWikiWord" href="/nlab/show/flavor+branes">flavor branes</a>: <ul> <li id="HananyZaffaroni98"> <a class="existingWikiWord" href="/nlab/show/Amihay+Hanany">Amihay Hanany</a>, <a class="existingWikiWord" href="/nlab/show/Alberto+Zaffaroni">Alberto Zaffaroni</a>, Branes and Six Dimensional Supersymmetric Theories, Nucl.Phys. B529 (1998) 180-206 (<a href="https://arxiv.org/abs/hep-th/9712145">arXiv:hep-th/9712145</a>) </li> <li id="BrunnerKarch97"> <a class="existingWikiWord" href="/nlab/show/Ilka+Brunner">Ilka Brunner</a>, <a class="existingWikiWord" href="/nlab/show/Andreas+Karch">Andreas Karch</a>, Branes at Orbifolds versus Hanany Witten in Six Dimensions, JHEP 9803:003, 1998 (<a href="https://arxiv.org/abs/hep-th/9712143">arXiv:hep-th/9712143</a>) </li> </ul> Reviewed in: <ul> <li id="ApruzziFazzi17">Fabio Apruzzi, <a class="existingWikiWord" href="/nlab/show/Marco+Fazzi">Marco Fazzi</a>, Section 2.1 of: <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 stretchy="false">/</mo><msub><mi>CFT</mi> <mn>6</mn></msub></mrow><annotation encoding="application/x-tex">AdS_7/CFT_6</annotation></semantics></math> with orientifolds, J. High Energ. Phys. (2018) 2018: 124 (<a href="https://arxiv.org/abs/1712.03235">arXiv:1712.03235</a>)</li> </ul> See also: <ul> <li><a class="existingWikiWord" href="/nlab/show/Amihay+Hanany">Amihay Hanany</a>, Noppadol Mekareeya, The Small <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">E_8</annotation></semantics></math> Instanton and the Kraft Procesi Transition, JHEP07 (2018) 098 (<a href="https://arxiv.org/abs/1801.01129">arXiv:1801.01129</a>)</li> </ul> </div></body></html> </div> <div class="revisedby"> Last revised on November 8, 2024 at 05:24:43. 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