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1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> nLab heterotic string theory </h1> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='string_theory'>String theory</h4> <div class='hide'> <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/conformal+field+theory'>CFT</a>, <a class='existingWikiWord' href='/nlab/show/perturbation+theory'>perturbation theory</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/effective+quantum+field+theory'>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+II+string+theory'>type IIA string theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/type+II+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-CFT+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/D%3D11+N%3D1+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/Kalb-Ramond+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/supergravity+C-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/D%3D6+N%3D%282%2C0%29+SCFT'>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+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> <h4 id='supergeometry'>Super-Geometry</h4> <div class='hide'> <a class='existingWikiWord' href='/nlab/show/super+algebra'>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+set'>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/super+vector+space'>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+Poincar%C3%A9+Lie+algebra'>super Poincare Lie algebra</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/chain+complex+in+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+algebra'>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_2'>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/supergroup'>super Lie group</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/supergroup'>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+infinity-groupoid'>super ∞-groupoid</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/super+formal+smooth+infinity-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+Poincar%C3%A9+Lie+algebra'>super Poincare Lie algebra</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/super+multiplet'>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+supersymmetry+algebra'>M-theory super Lie algebra</a>, <a class='existingWikiWord' href='/nlab/show/type+II+supersymmetry+algebra'>type II super Lie algebra</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/D%3D11+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/string+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> <h4 id='physics'>Physics</h4> <div class='hide'> <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+%28in+theoretical+physics%29'>model (physics)</a> <a class='existingWikiWord' href='/nlab/show/experimental+observation'>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/Hamiltonian+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+density'>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+density'>Lagrangian</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/phase+space'>covariant phase space</a>, <a class='existingWikiWord' href='/nlab/show/Euler-Lagrange+equation'>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/symplectic+Lie+n-algebroid'>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+space'>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'>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/wave'>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+information+theory+via+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'>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+inequality'>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/causally+local+net+of+observables'>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/functorial+field+theory'>functorial QFT</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/cobordism'>cobordism</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/%28infinity%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+functorial+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+infinity-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+%28physics%29'>phenomenology</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/model+%28in+theoretical+physics%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+system'>integrable systems</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/holonomic+quantum+field'>holonomic quantum fields</a> </li> </ul> </li> <li> Types of quantum field thories <ul> <li> <a class='existingWikiWord' href='/nlab/show/topological+quantum+field+theory'>TQFT</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/D%3D2+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/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%7C1%29-dimensional+Euclidean+field+theory'>(2,1)-dimensional Euclidean field theory and elliptic cohomology</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/conformal+field+theory'>CFT</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/Wess-Zumino-Witten+model'>WZW model</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/D%3D6+N%3D%282%2C0%29+SCFT'>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/quantum+electrodynamics'>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/The+Dirac+Electron'>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-Fr%C3%A9-Regge+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+sigma-model'>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> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#properties'>Properties</a><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><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><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><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a><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><ul><li><a href='#HeteroticStringPhenomenologyReferencesForE8'>The <math class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_1' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_2' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_3' xmlns='http://www.w3.org/1998/Math/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><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><ul><li><a href='#with_type_i_superstring_theory'>With type I superstring theory</a></li><li><a href='#DualityWithFTheory'>With <math class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math>-theory</a></li></ul></li><li><a href='#FlavorOnSingleHeteroticM5BranesReferences'><math class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_5' xmlns='http://www.w3.org/1998/Math/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></li></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/conformal+field+theory'>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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_6' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_7' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_8' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_9' xmlns='http://www.w3.org/1998/Math/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+variety'>Calabi-Yau manifold</a>. See the <a href='#ReferencesNEqOne'>references below</a>. Since global <math class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_10' xmlns='http://www.w3.org/1998/Math/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/worldvolume'>worldsheet</a> <a class='existingWikiWord' href='/nlab/show/D%3D2+SCFT'>2d SCFT</a> describing the quantum heterotic string has <math class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_11' xmlns='http://www.w3.org/1998/Math/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+quantum+field+theory'>effective field theory</a> has <math class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_12' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_13' xmlns='http://www.w3.org/1998/Math/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/D%3D2+N%3D%282%2C0%29+SCFT'>2d (2,0)-superconformal QFT</a> and at <a class='existingWikiWord' href='/nlab/show/supersymmetry+and+Calabi-Yau+manifolds'>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+class'>isomorphism classes</a> of) <a class='existingWikiWord' href='/nlab/show/gauge+group'>gauge groups</a> are consistent (give <a class='existingWikiWord' href='/nlab/show/quantum+anomaly'>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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_14' xmlns='http://www.w3.org/1998/Math/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/E%E2%82%88'>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'>$SemiSpin(32)$</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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_15' xmlns='http://www.w3.org/1998/Math/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/span'>correspondence</a>, or <a class='existingWikiWord' href='/nlab/show/duality+in+string+theory'>duality</a>) between the <math class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_16' xmlns='http://www.w3.org/1998/Math/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'>$\text{SemiSpin}(32)$</a> theories, by looking at how the relevant representations of these two theories assemble into those appearing in the <math class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_17' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_18' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_19' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_20' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_21' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_22' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_23' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_24' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_25' xmlns='http://www.w3.org/1998/Math/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/duality+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+type+I+and+heterotic+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/Ho%C5%99ava-Witten+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+M%2FF-theory+and+heterotic+string+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> <a class='existingWikiWord' href='/nlab/show/partition+function'>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/genus'>genera</a>/<a class='existingWikiWord' href='/nlab/show/orientation+in+generalized+cohomology'>orientations</a> in <a class='existingWikiWord' href='/nlab/show/generalized+cohomology'>generalized cohomology theory</a>: <table><thead><tr><th><math class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_26' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_27' xmlns='http://www.w3.org/1998/Math/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/generalized+cohomology'>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/genus'>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-theory'>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+number'>Bernoulli numbers</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/K-orientation'>Atiyah-Bott-Shapiro orientation</a> <math class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_28' xmlns='http://www.w3.org/1998/Math/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/extended+geometric+quantization+of+2d+Chern-Simons+theory'>2d Poisson-Chern-Simons theory</a> string</td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/spin%E1%B6%9C+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+states+%28in+geometric+quantization%29'>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+class'>Todd genus</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/Bernoulli+number'>Bernoulli numbers</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/K-orientation'>Atiyah-Bott-Shapiro orientation</a> <math class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_29' xmlns='http://www.w3.org/1998/Math/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+string+theory'>type II superstring</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/spin%E1%B6%9C+Dirac+operator'>Spin^c Dirac operator</a> twisted by <a class='existingWikiWord' href='/nlab/show/Chan-Paton+bundle'>Chan-Paton gauge field</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/D-brane'>D-brane charge</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/Todd+class'>Todd genus</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/Bernoulli+number'>Bernoulli numbers</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/K-orientation'>Atiyah-Bott-Shapiro orientation</a> <math class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_30' xmlns='http://www.w3.org/1998/Math/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+string+theory'>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/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+string+theory'>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> <h3 id='GeneralGaugeBackgroundsAndParameterizedWZWModels'>General gauge backgrounds and parameterized WZW models</h3> The traditional construction of the <a class='existingWikiWord' href='/nlab/show/worldvolume'>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/fermion'>fermions</a> only those <a class='existingWikiWord' href='/nlab/show/E%E2%82%88'>E8</a>-<a class='existingWikiWord' href='/nlab/show/background+field'>background gauge fields</a> which are reducible to <math class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_31' xmlns='http://www.w3.org/1998/Math/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+connection'>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+M%2FF-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+model'>parameterized WZW models</a>, parameterized over the <a class='existingWikiWord' href='/nlab/show/E%E2%82%88'>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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_32' xmlns='http://www.w3.org/1998/Math/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+mechanism'>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+algebra'>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+model'>parameterized WZW models</a> are the <a class='existingWikiWord' href='/nlab/show/equivariant+elliptic+genus'>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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_33' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_34' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_35' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_36' xmlns='http://www.w3.org/1998/Math/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/Euler-Lagrange+equation'>equations of motion</a>: <math class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_37' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_38' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_39' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace'></mspace></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math> The <a class='existingWikiWord' href='/nlab/show/Kalb-Ramond+field'>B-field</a> <a class='existingWikiWord' href='/nlab/show/field+strength'>strength</a>: <math class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_41' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_42' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_43' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_44' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_45' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_46' xmlns='http://www.w3.org/1998/Math/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/D%3D2+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/D%3D2+N%3D%282%2C0%29+SCFT'>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/D%3D11+N%3D1+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/Adrian+Norbert+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+lattice'>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 [, inspire:233203]</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/Daniel+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 [[arXiv:2407.07034](https://arxiv.org/abs/2407.07034)] </li> </ul> Relation to <a class='existingWikiWord' href='/nlab/show/Donaldson-Thomas+invariant'>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+correction'>higher curvature corrections</a>: <ul> <li> Eric Lescano, Carmen Núñez, Jesús A. Rodríguez, Supersymmetry, T-duality and Heterotic <math class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_47' xmlns='http://www.w3.org/1998/Math/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/brane'>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 [[arXiv:2303.17623](https://arxiv.org/abs/2303.17623)] </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) [[cds:2881994](https://cds.cern.ch/record/2881994)] </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 [[arXiv:2411.04344](https://arxiv.org/abs/2411.04344)] </li> <li> Masaki Fukuda, Shun K. Kobayashi, Kento Watanabe, Kazuya Yonekura: Black <math class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math>-Branes in Heterotic String Theory [[arXiv:2412.02277](https://arxiv.org/abs/2412.02277)] </li> </ul> Concise formulation of the <a class='existingWikiWord' href='/nlab/show/Lagrangian+density'>Lagrangian density</a> of heterotic/type I <a class='existingWikiWord' href='/nlab/show/D%3D10+supergravity'>D=10 supergravity</a>, including all <a class='existingWikiWord' href='/nlab/show/fermion'>fermionic</a> terms, via <a class='existingWikiWord' href='/nlab/show/generalized+geometry'>generalized geometry</a>: <ul> <li> Julian Kupka, <a class='existingWikiWord' href='/nlab/show/Charles+Strickland-Constable'>Charles Strickland-Constable</a>, <a class='existingWikiWord' href='/nlab/show/Fridrich+Valach'>Fridrich Valach</a>: Direct derivation of gauged <math class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒩</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>\mathcal{N}=1</annotation></semantics></math> supergravity in ten dimensions to all orders in fermions [[arXiv:2410.16046](https://arxiv.org/abs/2410.16046)] </li> <li> Julian Kupka, <a class='existingWikiWord' href='/nlab/show/Charles+Strickland-Constable'>Charles Strickland-Constable</a>, <a class='existingWikiWord' href='/nlab/show/Fridrich+Valach'>Fridrich Valach</a>: Batalin-Vilkovisky formulation of the <math class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒩</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>\mathcal{N}=1</annotation></semantics></math> supergravity in ten dimensions [[arXiv:2501.18008](https://arxiv.org/abs/2501.18008)] <blockquote> (<a class='existingWikiWord' href='/nlab/show/BV-BRST+formalism'>BV-formulation</a>) </blockquote> </li> </ul> <h3 id='orbifold_and_orientifold_compactifications'>Orbifold and orientifold compactifications</h3> Heterotic strings on <a class='existingWikiWord' href='/nlab/show/orbifold'>orbifolds</a>: <ul> <li id='DixonHarveyVafaWitten85'> <a class='existingWikiWord' href='/nlab/show/Lance+Dixon'>Lance Dixon</a>, <a class='existingWikiWord' href='/nlab/show/Jeffrey+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/Jeffrey+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+singularity'>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-brane'>M5-branes</a> ending in <a class='existingWikiWord' href='/nlab/show/Yang+monopole'>Yang monopoles</a> on <a class='existingWikiWord' href='/nlab/show/M9-brane'>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> <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/bottom-up+and+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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_51' xmlns='http://www.w3.org/1998/Math/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/orbifold'>orbifolds</a> (not yet checking <a class='existingWikiWord' href='/nlab/show/Yukawa+coupling'>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+variety'>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/GUT'>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'>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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_52' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_53' xmlns='http://www.w3.org/1998/Math/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'>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/metric+space'>metrics</a> on these Calabi-Yau compactifications (eventually needed for computing their induced <a class='existingWikiWord' href='/nlab/show/Yukawa+coupling'>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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_54' xmlns='http://www.w3.org/1998/Math/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+theory'>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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_55' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_56' xmlns='http://www.w3.org/1998/Math/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+theory'>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/perturbative+string+theory+vacuum'>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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_57' xmlns='http://www.w3.org/1998/Math/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'>heterotic line bundle models</a>: <ul> <li>Hajime Otsuka, <math class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_58' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_59' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_60' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_61' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_62' xmlns='http://www.w3.org/1998/Math/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/semi-spin+group'>semispin group</a> <a class='existingWikiWord' href='/nlab/show/SemiSpin%2832%29'>$\text{Spin}(32)/\mathbb{Z}_2$</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> <h3 id='HigherGaugeTheoryOfTheGreenSchwarzMechanismReferences'>Higher gauge theory of the Green-Schwarz mechanism</h3> Discussion of <a class='existingWikiWord' href='/nlab/show/higher+gauge+field'>higher gauge theory</a> modeling the <a class='existingWikiWord' href='/nlab/show/Green-Schwarz+mechanism'>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-brane'>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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_63' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_64' xmlns='http://www.w3.org/1998/Math/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' title='schreiber'>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+algebra'>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' title='schreiber'>The $E_8$ 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' title='schreiber'>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+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' title='schreiber'>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> <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+identity'>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) [, inspire:247764] </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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_65' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi><mo>=</mo><mn>10</mn></mrow><annotation encoding='application/x-tex'>D = 10</annotation></semantics></math> <math class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_67' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_68' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_69' xmlns='http://www.w3.org/1998/Math/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+surface'>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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_70' xmlns='http://www.w3.org/1998/Math/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+M%2FF-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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_71' xmlns='http://www.w3.org/1998/Math/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> <h3 id='FlavorOnSingleHeteroticM5BranesReferences'><math class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_72' xmlns='http://www.w3.org/1998/Math/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/flavour+%28particle+physics%29'>flavor</a>-<a class='existingWikiWord' href='/nlab/show/chiral+perturbation+theory'>symmetry</a> on single <a class='existingWikiWord' href='/nlab/show/M5-brane'>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+instanton'>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/NS5-brane'>half NS5-brane</a>/<a class='existingWikiWord' href='/nlab/show/D6-D8-brane+bound+state'>D6/D8-brane bound state</a> systems in <a class='existingWikiWord' href='/nlab/show/type+I+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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_73' xmlns='http://www.w3.org/1998/Math/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/NS5-brane'>half NS5-brane</a>/<a class='existingWikiWord' href='/nlab/show/D6-D8-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-brane'>D6-branes</a> making them act as <a class='existingWikiWord' href='/nlab/show/color+branes+and+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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_74' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_0c53c7a5f5cd8eef117fd963c8f8236e54ced595_75' xmlns='http://www.w3.org/1998/Math/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>  </div>  </body> </html>