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1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> type II string theory </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/2917/#Item_1" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" 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field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a>,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/CFT">CFT</a>, <a class="existingWikiWord" href="/nlab/show/perturbation+theory">perturbation theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+QFT">effective background QFT</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>, <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a>, <a class="existingWikiWord" href="/nlab/show/quantum+gravity">quantum gravity</a></li> </ul> </li> </ul> <h3 id="critical_string_models">Critical string models</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a>, <a class="existingWikiWord" href="/nlab/show/differential+string+structure">differential string structure</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dual+heterotic+string+theory">dual heterotic string theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+fivebrane+structure">differential fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/type+IIA+string+theory">type IIA string theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/type+IIB+string+theory">type IIB string theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/F-theory">F-theory</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+field+theory">string field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/duality+in+string+theory">duality in string theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/T-duality">T-duality</a>, <a class="existingWikiWord" href="/nlab/show/mirror+symmetry">mirror symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/S-duality">S-duality</a>, <a class="existingWikiWord" href="/nlab/show/electric-magnetic+duality">electric-magnetic duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/U-duality">U-duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open%2Fclosed+string+duality">open/closed string duality</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/AdS%2FCFT+correspondence">AdS/CFT correspondence</a>, <a class="existingWikiWord" href="/nlab/show/holographic+principle">holographic principle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a>, <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%C5%99ava-Witten+theory">Hořava-Witten theory</a></li> </ul> </li> </ul> <h3 id="extended_objects">Extended objects</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/brane">brane</a></p> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/D0-brane">D0-brane</a>, <a class="existingWikiWord" href="/nlab/show/D2-brane">D2-brane</a>, <a class="existingWikiWord" href="/nlab/show/D4-brane">D4-brane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D1-brane">D1-brane</a>, <a class="existingWikiWord" href="/nlab/show/D3-brane">D3-brane</a>, <a class="existingWikiWord" href="/nlab/show/D5-brane">D5-brane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/RR-field">RR-field</a>, <a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/NS-brane">NS-brane</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>, <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/B2-field">B2-field</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/NS5-brane">NS5-brane</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/B6-field">B6-field</a></li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/M-brane">M-brane</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/C3-field">C3-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ABJM+theory">ABJM theory</a>, <a class="existingWikiWord" href="/nlab/show/BLG+model">BLG model</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/C6-field">C6-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a></p> </li> </ul> </li> </ul> </li> </ul> <h3 id="topological_strings">Topological strings</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a>, <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+M-theory">topological M-theory</a></p> </li> </ul> <h2 id="backgrounds">Backgrounds</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/target+space">target space</a>, <a class="existingWikiWord" href="/nlab/show/background+gauge+field">background gauge field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+smooth+cohomology+in+string+theory">twisted smooth cohomology in string theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/landscape+of+string+theory+vacua">landscape of string theory vacua</a></p> </li> </ul> <h2 id="phenomenology">Phenomenology</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/string+phenomenology">string phenomenology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/moduli+stabilization">moduli stabilization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/G%E2%82%82-MSSM">G₂-MSSM</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/string+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="supergeometry">Super-Geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a></strong> and (<a class="existingWikiWord" href="/nlab/show/synthetic+differential+supergeometry">synthetic</a> ) <strong><a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a></strong></p> <h2 id="background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/graded+object">graded object</a></p> </li> </ul> <h2 id="introductions">Introductions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+superalgebra">geometry of physics – superalgebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+supergeometry">geometry of physics – supergeometry</a></p> </li> </ul> <h2 id="superalgebra">Superalgebra</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/super+commutative+monoid">super commutative monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+abelian+group">super abelian group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+ring">super ring</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supercommutative+ring">supercommutative ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exterior+ring">exterior ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Clifford+ring">Clifford ring</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+module">super module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+vector+space">super vector space</a>, <a class="existingWikiWord" href="/nlab/show/SVect">SVect</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+algebra">super algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supercommutative+algebra">supercommutative algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exterior+algebra">exterior algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Clifford+algebra">Clifford algebra</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/superdeterminant">superdeterminant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">super Poincare Lie algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+complex+of+super+vector+spaces">chain complex of super vector spaces</a> (<a class="existingWikiWord" href="/nlab/show/model+structure+on+chain+complexes+of+super+vector+spaces">model structure</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+graded-commutative+superalgebra">differential graded-commutative superalgebra</a> (<a class="existingWikiWord" href="/nlab/show/model+structure+on+differential+graded-commutative+superalgebras">model structure</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+L-infinity+algebra">super L-infinity algebra</a></p> </li> </ul> <h2 id="supergeometry">Supergeometry</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/superpoint">superpoint</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Cartesian+space">super Cartesian space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supermanifold">supermanifold</a>, <a class="existingWikiWord" href="/nlab/show/SDiff">SDiff</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/NQ-supermanifold">NQ-supermanifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+vector+bundle">super vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+supermanifold">complex supermanifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+supermanifold">Euclidean supermanifold</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+spacetime">super spacetime</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Minkowski+spacetime">super Minkowski spacetime</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integration+over+supermanifolds">integration over supermanifolds</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Berezin+integral">Berezin integral</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Lie+group">super Lie group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Lie+group">super translation group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Euclidean+group">super Euclidean group</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+%E2%88%9E-groupoid">super ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+formal+smooth+%E2%88%9E-groupoid">super formal smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+line+2-bundle">super line 2-bundle</a></p> </li> </ul> <h2 id="supersymmetry">Supersymmetry</h2> <p><a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/division+algebra+and+supersymmetry">division algebra and supersymmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">super Poincare Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supermultiplet">supermultiplet</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BPS+state">BPS state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/M-theory+super+Lie+algebra">M-theory super Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/type+II+super+Lie+algebra">type II super Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a>, <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a></p> </li> </ul> <h2 id="supersymmetric_field_theory">Supersymmetric field theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/superfield">superfield</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supersymmetric+quantum+mechanics">supersymmetric quantum mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adinkra">adinkra</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">super Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/gauged+supergravity">gauged supergravity</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/superstring+theory">superstring theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a></p> </li> </ul> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometric+model+for+elliptic+cohomology">geometric model for elliptic cohomology</a></li> </ul> <div> <p> <a href="/nlab/edit/supergeometry+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#effective_qft'>Effective QFT</a></li> <li><a href='#anomaly_cancellation'>Anomaly cancellation</a></li> <li><a href='#background_fields_and_orientifolding'>Background fields and orientifolding</a></li> <li><a href='#dualities'>Dualities</a></li> <ul> <li><a href='#duality_with_heterotic_string_theory'>Duality with heterotic string theory</a></li> <li><a href='#duality_with_mtheory'>Duality with M-theory</a></li> <li><a href='#duality_with_ftheory'>Duality with F-theory</a></li> </ul> <li><a href='#holographic_dual'>Holographic dual</a></li> <li><a href='#partition_function_and_elliptic_genus'>Partition function and elliptic genus</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#strings_and_sduality'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p,q)</annotation></semantics></math>-Strings and S-duality</a></li> <li><a href='#quantum_anomalies'>Quantum anomalies</a></li> <li><a href='#classical_solutions__vacua'>Classical solutions / vacua</a></li> <li><a href='#holography'>Holography</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>Perturbative <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> is defined in terms of certain classes of 2d <a class="existingWikiWord" href="/nlab/show/CFT">CFT</a>s. Depending on which class that is, one speaks of different <em>types</em> of string theory.</p> <ul> <li> <p>In <em>type II string theory</em> the CFTs in question are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1,1)</annotation></semantics></math>-supersymmetric and defined on <a class="existingWikiWord" href="/nlab/show/orientation">oriented</a> <a class="existingWikiWord" href="/nlab/show/worldsheet">worldsheet</a>s;</p> </li> <li> <p>In <em><a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a></em> the CFTs in question are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,1)</annotation></semantics></math>-supersymmetric and defined on <a class="existingWikiWord" href="/nlab/show/orientation">oriented</a> <a class="existingWikiWord" href="/nlab/show/worldsheet">worldsheet</a>s;</p> </li> <li> <p>In …</p> </li> </ul> <p>The Type II string theory referred to here is defined on spaces in <a class="existingWikiWord" href="/nlab/show/Lorentzian+geometry">Lorentzian geometry</a>. It is thought that Type II (and the other string theories) can also be defined in spaces with other metric signatures, such as <a class="existingWikiWord" href="/nlab/show/type+II%2A+string+theory">type II* string theory</a>. See <a class="existingWikiWord" href="/nlab/show/non-Lorentzian+type+II+string+theory">non-Lorentzian type II string theory</a>.</p> <h2 id="properties">Properties</h2> <h3 id="effective_qft">Effective QFT</h3> <p>The <a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a> of type II string theory containts –besides <a class="existingWikiWord" href="/nlab/show/type+II+supergravity">type II supergravity</a> – the <a class="existingWikiWord" href="/nlab/show/self-dual+higher+gauge+theory">self-dual higher gauge theory</a> of <a class="existingWikiWord" href="/nlab/show/RR-fields">RR-fields</a> and <a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a>s.</p> <h3 id="anomaly_cancellation">Anomaly cancellation</h3> <p>Apart from the <a class="existingWikiWord" href="/nlab/show/Weyl+anomaly">Weyl anomaly</a>, which cancels for 10-dimensional <a class="existingWikiWord" href="/nlab/show/target+space">target space</a>s, the <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> of the <a class="existingWikiWord" href="/nlab/show/string">string</a>-<a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> also in general has an <em><a href="http://ncatlab.org/nlab/show/quantum+anomaly#AnomalousActionFunctional">anomalous action functional</a></em> , for two reasons:</p> <ol> <li> <p>The <a class="existingWikiWord" href="/nlab/show/holonomy">higher holonomy</a> of the higher <a class="existingWikiWord" href="/nlab/show/background+gauge+field">background gauge field</a>s is in general not a function, but a <a class="existingWikiWord" href="/nlab/show/section">section</a> of a <a class="existingWikiWord" href="/nlab/show/line+bundle">line bundle</a>;</p> </li> <li> <p>The fermionic <a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a> over the <a class="existingWikiWord" href="/nlab/show/worldsheet">worldsheet</a>-<a class="existingWikiWord" href="/nlab/show/spinor">spinor</a>s of the <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a> produces as section of a <a class="existingWikiWord" href="/nlab/show/Pfaffian+line+bundle">Pfaffian line bundle</a>.</p> </li> </ol> <p>In order for the action functional to be well-defined, the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> of these different anomaly <a class="existingWikiWord" href="/nlab/show/line+bundle">line bundle</a>s over the bosonic <a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a> must have trivial class (as <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">bundles with connection</a>, even). This gives rise to various further anomaly cancellation conditions:</p> <p>For the open <a class="existingWikiWord" href="/nlab/show/type+II+string">type II string</a> the condition is known as the <a class="existingWikiWord" href="/nlab/show/Freed-Witten+anomaly+cancellation">Freed-Witten anomaly cancellation</a> condition: it says that the restriction of the <a class="existingWikiWord" href="/nlab/show/B-field">B-field</a> to any <a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a> must consistute the twist of a <a class="existingWikiWord" href="/nlab/show/twisted+spin%5Ec+structure">twisted spin^c structure</a> on the brane.</p> <p>A more detailed analysis of these type II anomalies is in (<a href="#DFMI">DFMI</a>) and (<a href="#DFMII">DFMII</a>).</p> <h3 id="background_fields_and_orientifolding">Background fields and orientifolding</h3> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/cohomology+theories">cohomology theories</a> of <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> on <a class="existingWikiWord" href="/nlab/show/orientifolds">orientifolds</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a></th><th><a class="existingWikiWord" href="/nlab/show/B-field">B-field</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-field <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli</a></th><th><a class="existingWikiWord" href="/nlab/show/RR-field">RR-field</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/bosonic+string">bosonic string</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/line+2-bundle">line 2-bundle</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><msup><mi>ℤ</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">H\mathbb{Z}^3</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+II+superstring">type II superstring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+line+2-bundle">super line 2-bundle</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Pic</mi><mo stretchy="false">(</mo><mi>KU</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">Pic(KU)//\mathbb{Z}_2</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/KR-theory">KR-theory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>KR</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">KR^\bullet</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+IIA+superstring">type IIA superstring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+line+2-bundle">super line 2-bundle</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>KU</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B GL_1(KU)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/complex+K-theory">KU-theory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>KU</mi> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex">KU^0</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+IIB+superstring">type IIB superstring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+line+2-bundle">super line 2-bundle</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>KU</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B GL_1(KU)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/complex+K-theory">KU-theory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>KU</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">KU^1</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+I+superstring">type I superstring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+line+2-bundle">super line 2-bundle</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Pic</mi><mo stretchy="false">(</mo><mi>KU</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">Pic(KU)//\mathbb{Z}_2</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/KO-theory">KO-theory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>KO</mi></mrow><annotation encoding="application/x-tex">KO</annotation></semantics></math></td></tr> <tr><td style="text-align: left;">type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>I</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde I</annotation></semantics></math> superstring</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+line+2-bundle">super line 2-bundle</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Pic</mi><mo stretchy="false">(</mo><mi>KU</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">Pic(KU)//\mathbb{Z}_2</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/KSC-theory">KSC-theory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>KSC</mi></mrow><annotation encoding="application/x-tex">KSC</annotation></semantics></math></td></tr> </tbody></table> </div> <ul> <li><a class="existingWikiWord" href="/nlab/show/intersecting+D-brane+models">intersecting D-brane models</a></li> </ul> <h3 id="dualities">Dualities</h3> <p>Some <a class="existingWikiWord" href="/nlab/show/dualities+in+string+theory">dualities in string theory</a> involving <a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a>:</p> <h4 id="duality_with_heterotic_string_theory">Duality with heterotic string theory</h4> <p>See <em><a class="existingWikiWord" href="/nlab/show/duality+between+type+II+and+heterotic+string+theory">duality between type II and heterotic string theory</a></em>.</p> <h4 id="duality_with_mtheory">Duality with M-theory</h4> <p>See <em><a class="existingWikiWord" href="/nlab/show/duality+between+type+IIA+string+theory+and+M-theory">duality between type IIA string theory and M-theory</a></em>.</p> <h4 id="duality_with_ftheory">Duality with F-theory</h4> <p>See <em><a class="existingWikiWord" href="/nlab/show/F-theory">F-theory</a></em>.</p> <h3 id="holographic_dual">Holographic dual</h3> <p>By a <a class="existingWikiWord" href="/nlab/show/holographic+principle">holographic principle</a> realized in this case as <a class="existingWikiWord" href="/nlab/show/AdS%2FCFT+correspondence">AdS/CFT correspondence</a> (see the references there), type II string theory is supposed to be dual to 4-dimensional <a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">super Yang-Mills theory</a>.</p> <h3 id="partition_function_and_elliptic_genus">Partition function and elliptic genus</h3> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/partition+functions">partition functions</a> in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> as <a class="existingWikiWord" href="/nlab/show/index">indices</a>/<a class="existingWikiWord" href="/nlab/show/genera">genera</a>/<a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">orientations</a> in <a class="existingWikiWord" href="/nlab/show/generalized+cohomology+theory">generalized cohomology theory</a></strong>:</p> <table><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math></th><th><a class="existingWikiWord" href="/nlab/show/partition+function">partition function</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>-dimensional QFT</th><th><a class="existingWikiWord" href="/nlab/show/supercharge">supercharge</a></th><th><a class="existingWikiWord" href="/nlab/show/index">index</a> in <a class="existingWikiWord" href="/nlab/show/cohomology+theory">cohomology theory</a></th><th><a class="existingWikiWord" href="/nlab/show/genus">genus</a></th><th>logarithmic coefficients of <a class="existingWikiWord" href="/nlab/show/Hirzebruch+series">Hirzebruch series</a></th><th></th></tr></thead><tbody><tr><td style="text-align: left;">0</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;">push-forward in <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>: <a class="existingWikiWord" href="/nlab/show/integration+of+differential+forms">integration of differential forms</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a></td></tr> <tr><td style="text-align: left;">1</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/spinning+particle">spinning particle</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/KO">KO</a><a class="existingWikiWord" href="/nlab/show/K-theory">-theory</a> <a class="existingWikiWord" href="/nlab/show/index">index</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/A-hat+genus">A-hat genus</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Bernoulli+numbers">Bernoulli numbers</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Atiyah-Bott-Shapiro+orientation">Atiyah-Bott-Shapiro orientation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mi>Spin</mi><mo>→</mo><mi>KO</mi></mrow><annotation encoding="application/x-tex">M Spin \to KO</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;">endpoint of <a class="existingWikiWord" href="/nlab/show/2d+Poisson-Chern-Simons+theory">2d Poisson-Chern-Simons theory</a> string</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%5Ec+Dirac+operator">Spin^c Dirac operator</a> twisted by <a class="existingWikiWord" href="/nlab/show/prequantum+line+bundle">prequantum line bundle</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a> of boundary <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a>/<a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Todd+genus">Todd genus</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Bernoulli+numbers">Bernoulli numbers</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Atiyah-Bott-Shapiro+orientation">Atiyah-Bott-Shapiro orientation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><msup><mi>Spin</mi> <mi>c</mi></msup><mo>→</mo><mi>KU</mi></mrow><annotation encoding="application/x-tex">M Spin^c \to KU</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;">endpoint of <a class="existingWikiWord" href="/nlab/show/type+II+superstring">type II superstring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%5Ec+Dirac+operator">Spin^c Dirac operator</a> twisted by <a class="existingWikiWord" href="/nlab/show/Chan-Paton+gauge+field">Chan-Paton gauge field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D-brane+charge">D-brane charge</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Todd+genus">Todd genus</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Bernoulli+numbers">Bernoulli numbers</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Atiyah-Bott-Shapiro+orientation">Atiyah-Bott-Shapiro orientation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><msup><mi>Spin</mi> <mi>c</mi></msup><mo>→</mo><mi>KU</mi></mrow><annotation encoding="application/x-tex">M Spin^c \to KU</annotation></semantics></math></td></tr> <tr><td style="text-align: left;">2</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+II+superstring">type II superstring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Dirac-Ramond+operator">Dirac-Ramond operator</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/superstring">superstring</a> <a class="existingWikiWord" href="/nlab/show/partition+function">partition function</a> in NS-R sector</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Ochanine+elliptic+genus">Ochanine elliptic genus</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO+orientation+of+elliptic+cohomology">SO orientation of elliptic cohomology</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/heterotic+superstring">heterotic superstring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Dirac-Ramond+operator">Dirac-Ramond operator</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/superstring">superstring</a> <a class="existingWikiWord" href="/nlab/show/partition+function">partition function</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Witten+genus">Witten genus</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Eisenstein+series">Eisenstein series</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/string+orientation+of+tmf">string orientation of tmf</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/self-dual+string">self-dual string</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M5-brane+charge">M5-brane charge</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">3</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/w4-orientation+of+EO%282%29-theory">w4-orientation of EO(2)-theory</a></td></tr> </tbody></table> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dual+heterotic+string+theory">dual heterotic string theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Witten+genus">Witten genus</a>, <a class="existingWikiWord" href="/nlab/show/string+orientation+of+tmf">string orientation of tmf</a></p> </li> </ul> </li> <li> <p><strong>type II string theory</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/massive+type+IIA+string+theory">massive type IIA string theory</a></li> </ul> <p><a class="existingWikiWord" href="/nlab/show/Diaconescu-Moore-Witten+anomaly">Diaconescu-Moore-Witten anomaly</a></p> <p><a class="existingWikiWord" href="/nlab/show/type+II+geometry">type II geometry</a></p> <p><a class="existingWikiWord" href="/nlab/show/elliptic+genus">elliptic genus</a></p> <p>4d <a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">super Yang-Mills theory</a></p> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/type+I+string+theory">type I string theory</a></strong></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/type+0+string+theory">type 0 string theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/landscape+of+string+theory+vacua">landscape of string theory vacua</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="general">General</h3> <p>Physics textbook accounts include</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Joseph+Polchinski">Joseph Polchinski</a>, <em><a class="existingWikiWord" href="/nlab/show/String+theory">String theory</a></em>, volume II, section 10</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eric+D%27Hoker">Eric D'Hoker</a>, <em>String theory – lecture 7: Free superstrings</em> , in part 3 of</p> <p><a class="existingWikiWord" href="/nlab/show/Pierre+Deligne">Pierre Deligne</a>, <a class="existingWikiWord" href="/nlab/show/Pavel+Etingof">Pavel Etingof</a>, <a class="existingWikiWord" href="/nlab/show/Dan+Freed">Dan Freed</a>, L. Jeffrey,</p> </li> </ul> <p><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. <em><a class="existingWikiWord" href="/nlab/show/Quantum+Fields+and+Strings">Quantum Fields and Strings</a>, A course for mathematicians</em>, 2 vols. Amer. Math. Soc. Providence 1999. (<a href="http://www.math.ias.edu/qft">web version</a>)</p> <p>A comprehensive discussion of the (<a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential</a>) <a class="existingWikiWord" href="/nlab/show/cohomology">cohomological</a> nature of general type II/type I <a class="existingWikiWord" href="/nlab/show/orientifold">orientifold</a> backgrounds is in</p> <ul> <li id="DFM09"><a class="existingWikiWord" href="/nlab/show/Jacques+Distler">Jacques Distler</a>, <a class="existingWikiWord" href="/nlab/show/Dan+Freed">Dan Freed</a>, <a class="existingWikiWord" href="/nlab/show/Greg+Moore">Greg Moore</a>, <em>Orientifold Précis</em> in: <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a> (eds.) <em><a class="existingWikiWord" href="/schreiber/show/Mathematical+Foundations+of+Quantum+Field+and+Perturbative+String+Theory">Mathematical Foundations of Quantum Field and Perturbative String Theory</a></em> Proceedings of Symposia in Pure Mathematics, AMS (2011) (<a href="http://arxiv.org/abs/0906.0795">arXiv:0906.0795</a>, <a href="http://www.ma.utexas.edu/users/dafr/bilbao.pdf">slides</a>)</li> </ul> <p>with details in</p> <ul> <li id="Freed"> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <em>Lectures on twisted K-theory and orientifolds</em> (<a class="existingWikiWord" href="/nlab/files/FreedESI2012.pdf" title="pdf">pdf</a>)</p> </li> <li id="DistlerFreedMooreII"> <p><a class="existingWikiWord" href="/nlab/show/Jacques+Distler">Jacques Distler</a>, <a class="existingWikiWord" href="/nlab/show/Dan+Freed">Dan Freed</a>, <a class="existingWikiWord" href="/nlab/show/Greg+Moore">Greg Moore</a>, <em>Spin structures and superstrings</em>, Surveys in Differential Geometry, Volume 15 (2010) (<a href="http://arxiv.org/abs/1007.4581">arXiv:1007.4581</a>, <a href="http://dx.doi.org/10.4310/SDG.2010.v15.n1.a4">doi:10.4310/SDG.2010.v15.n1.a4</a>)</p> </li> </ul> <p>Related lecture notes / slides include</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Jacques+Distler">Jacques Distler</a>, <em>Orientifolds and Twisted KR-Theory</em> (2008) (<a href="http://www.perimeterinstitute.ca/pdf/files/731c5f3a-928f-453a-b569-db5c574d2a6c.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <em>Dirac charge quantiation, K-theory, and orientifolds</em>, talk at a workshop <em>Mathematical methods in general relativity and quantum field theories</em>, November, 2009 (<a href="http://www.ma.utexas.edu/users/dafr/paris_nt.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Greg+Moore">Greg Moore</a>, <em>The RR-charge of an orientifold</em> (<a href="http://www.physics.rutgers.edu/~gmoore/AnnArbor_Feb2010_FINAL.ppt">ppt</a>)</p> </li> </ul> <h3 id="strings_and_sduality"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p,q)</annotation></semantics></math>-Strings and S-duality</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/John+Schwarz">John Schwarz</a>, opening talk at <a href="http://strings2013.sogang.ac.kr/">Strings 2013</a>, (<a href="http://strings2013.sogang.ac.kr/design/default/data/john_schwarz.pdf">pdf</a>)</li> </ul> <h3 id="quantum_anomalies">Quantum anomalies</h3> <p>Discussion of type II <a class="existingWikiWord" href="/nlab/show/quantum+anomalies">quantum anomalies</a> is in</p> <ul id="DFMI"> <li><a class="existingWikiWord" href="/nlab/show/Jacques+Distler">Jacques Distler</a>, <a class="existingWikiWord" href="/nlab/show/Dan+Freed">Dan Freed</a>, <a class="existingWikiWord" href="/nlab/show/Greg+Moore">Greg Moore</a>, <em>Orientifold Précis</em> in: <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a> (eds.) <em><a class="existingWikiWord" href="/schreiber/show/Mathematical+Foundations+of+Quantum+Field+and+Perturbative+String+Theory">Mathematical Foundations of Quantum Field and Perturbative String Theory</a></em> Proceedings of Symposia in Pure Mathematics, AMS (2011) (<a href="http://arxiv.org/abs/0906.0795">arXiv</a>, <a href="http://www.ma.utexas.edu/users/dafr/bilbao.pdf">slides</a>)</li> </ul> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacques+Distler">Jacques Distler</a>, <a class="existingWikiWord" href="/nlab/show/Dan+Freed">Dan Freed</a>, <a class="existingWikiWord" href="/nlab/show/Greg+Moore">Greg Moore</a>, <em>Spin structures and superstrings</em> (<a href="http://arxiv.org/abs/1007.4581">arXiv:1007.4581</a>)</li> </ul> <p>An exposition is at</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Dan+Freed">Dan Freed</a>, <em>Lectures on K-theory and orientifolds</em> (2012) (<a href="http://www.ma.utexas.edu/users/dafr/ESI.pdf">pdf</a>)</li> </ul> <h3 id="classical_solutions__vacua">Classical solutions / vacua</h3> <p>Description of type II backgrounds in terms of <a class="existingWikiWord" href="/nlab/show/generalized+complex+geometry">generalized complex geometry</a>/<a class="existingWikiWord" href="/nlab/show/Courant+Lie+2-algebroids">Courant Lie 2-algebroids</a> is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Mariana+Grana">Mariana Grana</a>, Francesco Orsi, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">N=1</annotation></semantics></math> vacua in Exceptional Generalized Geometry</em> (<a href="http://arxiv.org/abs/1105.4855">arXiv:1105.4855</a>)</li> </ul> <h3 id="holography">Holography</h3> <p>A <a class="existingWikiWord" href="/nlab/show/holographic+principle">holographic</a> description of type II by <a class="existingWikiWord" href="/nlab/show/higher+dimensional+Chern-Simons+theory">higher dimensional Chern-Simons theory</a> is discussed in</p> <ul> <li>Dmitriy Belov, <a class="existingWikiWord" href="/nlab/show/Greg+Moore">Greg Moore</a>, <em>Type II Actions from 11-Dimensional Chern-Simons Theories</em> (<a href="http://arxiv.org/abs/hep-th/0611020">arXiv:hep-th/0611020</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 1, 2023 at 22:47:09. See the <a href="/nlab/history/type+II+string+theory" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/type+II+string+theory" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/2917/#Item_1">Discuss</a><span class="backintime"><a href="/nlab/revision/type+II+string+theory/33" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/type+II+string+theory" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/type+II+string+theory" accesskey="S" class="navlink" id="history" rel="nofollow">History (33 revisions)</a> <a href="/nlab/show/type+II+string+theory/cite" style="color: black">Cite</a> <a href="/nlab/print/type+II+string+theory" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/type+II+string+theory" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>