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float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="string_theory">String theory</h4> <div class="hide"><div> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a></strong></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/books+about+string+theory">books about string theory</a></p> </li> </ul> <h3 id="ingredients">Ingredients</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a>,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/CFT">CFT</a>, <a class="existingWikiWord" href="/nlab/show/perturbation+theory">perturbation theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+QFT">effective background QFT</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>, <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a>, <a class="existingWikiWord" href="/nlab/show/quantum+gravity">quantum gravity</a></li> </ul> </li> </ul> <h3 id="critical_string_models">Critical string models</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a>, <a class="existingWikiWord" href="/nlab/show/differential+string+structure">differential string structure</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dual+heterotic+string+theory">dual heterotic string theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+fivebrane+structure">differential fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/type+IIA+string+theory">type IIA string theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/type+IIB+string+theory">type IIB string theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/F-theory">F-theory</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+field+theory">string field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/duality+in+string+theory">duality in string theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/T-duality">T-duality</a>, <a class="existingWikiWord" href="/nlab/show/mirror+symmetry">mirror symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/S-duality">S-duality</a>, <a class="existingWikiWord" href="/nlab/show/electric-magnetic+duality">electric-magnetic duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/U-duality">U-duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open%2Fclosed+string+duality">open/closed string duality</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/AdS%2FCFT+correspondence">AdS/CFT correspondence</a>, <a class="existingWikiWord" href="/nlab/show/holographic+principle">holographic principle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a>, <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%C5%99ava-Witten+theory">Hořava-Witten theory</a></li> </ul> </li> </ul> <h3 id="extended_objects">Extended objects</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/brane">brane</a></p> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/D0-brane">D0-brane</a>, <a class="existingWikiWord" href="/nlab/show/D2-brane">D2-brane</a>, <a class="existingWikiWord" href="/nlab/show/D4-brane">D4-brane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D1-brane">D1-brane</a>, <a class="existingWikiWord" href="/nlab/show/D3-brane">D3-brane</a>, <a class="existingWikiWord" href="/nlab/show/D5-brane">D5-brane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/RR-field">RR-field</a>, <a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/NS-brane">NS-brane</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>, <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/B2-field">B2-field</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/NS5-brane">NS5-brane</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/B6-field">B6-field</a></li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/M-brane">M-brane</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/C3-field">C3-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ABJM+theory">ABJM theory</a>, <a class="existingWikiWord" href="/nlab/show/BLG+model">BLG model</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/C6-field">C6-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a></p> </li> </ul> </li> </ul> </li> </ul> <h3 id="topological_strings">Topological strings</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a>, <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+M-theory">topological M-theory</a></p> </li> </ul> <h2 id="backgrounds">Backgrounds</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/target+space">target space</a>, <a class="existingWikiWord" href="/nlab/show/background+gauge+field">background gauge field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+smooth+cohomology+in+string+theory">twisted smooth cohomology in string theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/landscape+of+string+theory+vacua">landscape of string theory vacua</a></p> </li> </ul> <h2 id="phenomenology">Phenomenology</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/string+phenomenology">string phenomenology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/moduli+stabilization">moduli stabilization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/G%E2%82%82-MSSM">G₂-MSSM</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/string+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <ul> <li><a href='#in__rational_cft'>In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>d</mi></mrow><annotation encoding="application/x-tex">2d</annotation></semantics></math> rational CFT</a></li> <li><a href='#in__tft'>In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>d</mi></mrow><annotation encoding="application/x-tex">2d</annotation></semantics></math> TFT</a></li> <li><a href='#in_terms_of_geometric_data_of_the_model_background'>In terms of geometric data of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-model background</a></li> <li><a href='#AsBlackBranes'>As black branes</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#various_dimensions'>Various dimensions</a></li> <li><a href='#in_the_wzw_model'>In the WZW model</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#as_fbranes_originating_from_mbranes'>As F-branes originating from M-branes</a></li> <li><a href='#characterization_in_terms_of_dirac_structures'>Characterization in terms of Dirac structures</a></li> <li><a href='#DBraneCharge'>D-brane charge</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#DBraneChargeViaAtiyahHirzebruchSpectralSequence'>Via the Atiyah-Hirzebruch spectral sequence</a></li> </ul> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general_2'>General</a></li> <li><a href='#on_orbifolds'>On orbifolds</a></li> <li><a href='#ReferencesAsGSsigmaModels'>As higher super-GS-WZW type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-models</a></li> <li><a href='#DBraneChargeQuantizationInTopologicalKTheory'>D-brane charge quantization in topological K-theory</a></li> <ul> <li><a href='#origin_and_basics'>Origin and basics</a></li> <li><a href='#twisted_equivariant_and_differential_refinement'>Twisted, equivariant and differential refinement</a></li> <li><a href='#reviews'>Reviews</a></li> <li><a href='#DBraneChargeInKTheoryReferencesConceptualProblems'>Conceptual problems</a></li> <li><a href='#for_orbifolds_in_equivariant_ktheory'>For orbifolds in equivariant K-theory</a></li> </ul> <li><a href='#ReferencesEntropy'>Entropy</a></li> <li><a href='#for_rational_cft'>For rational CFT</a></li> <li><a href='#branes_within_branes'>Branes within branes</a></li> <li><a href='#for_topological_strings'>For topological strings</a></li> <li><a href='#open_string_worldsheet_anomaly_cancellation'>Open string worldsheet Anomaly cancellation</a></li> <li><a href='#relation_to_dirac_structures'>Relation to Dirac structures</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>An abstractly defined <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> is a consistent assignment of <a class="existingWikiWord" href="/nlab/show/state">state</a>-space and correlators to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a>s with certain structure (topological structure, conformal structure, Riemannian structure, etc. see <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a>/<a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a>). In an <em>open-closed QFT</em> the cobordisms are allowed to have <a class="existingWikiWord" href="/nlab/show/boundary+CFT">boundaries</a>.</p> <p>In this abstract formulation of QFT a <strong>D-brane</strong> is a type of data assigned by the QFT to boundaries of cobordisms.</p> <p>For a broader perspective see at <em><a class="existingWikiWord" href="/nlab/show/brane">brane</a></em>.</p> <h3 id="in__rational_cft">In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>d</mi></mrow><annotation encoding="application/x-tex">2d</annotation></semantics></math> rational CFT</h3> <p>A well understood class of examples is this one: among all 2-dimensional <a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a> that case of <em>full rational 2d CFT</em> has been understood completely, using <a class="existingWikiWord" href="/nlab/show/FFRS-formalism">FFRS-formalism</a>. It is then a theorem that full 2-rational CFTs are classified by</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/modular+tensor+category">modular tensor category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> (to be thought of as being the category of representations of the <a class="existingWikiWord" href="/nlab/show/vertex+operator+algebra">vertex operator algebra</a> of the 2d CFT);</p> </li> <li> <p>a special symmetric <a class="existingWikiWord" href="/nlab/show/Frobenius+algebra">Frobenius algebra</a> object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/internalization">internal</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>.</p> </li> </ol> <p>In this formulation a type of <strong>brane</strong> of the theory is precisely an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/module">module</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> (an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/bimodule">bimodule</a> is a <a class="existingWikiWord" href="/nlab/show/bi-brane">bi-brane</a> or <em>defect line</em> ):</p> <p>the 2d cobordisms with <a class="existingWikiWord" href="/nlab/show/boundary+CFT">boundary</a> on which the theory defined by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">A \in \mathcal{C}</annotation></semantics></math> carry as extra structure on their connected boundary pieces a label given by an equivalence class of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-module in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>. The assignment of the CFT to such a cobordism with boundary is obtained by</p> <ul> <li> <p>triangulating the cobordism,</p> </li> <li> <p>labeling all internal edges by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></p> </li> <li> <p>labelling all boundary pieces by the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-module</p> </li> <li> <p>all vertices where three internal edges meet by the multiplication operation</p> </li> <li> <p>and all points where an internal edge hits a boundary by the corresponding <a class="existingWikiWord" href="/nlab/show/action">action</a> morphism</p> </li> <li> <p>and finally evaluating the resulting <a class="existingWikiWord" href="/nlab/show/string+diagram">string diagram</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>.</p> </li> </ul> <p>So in this abstract algebraic formulation of QFT on the worldvolume, a brane is just the datum assigned by the QFT to the boundary of a cobordism. But abstractly defined QFTs may arise from <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> of <a class="existingWikiWord" href="/nlab/show/sigma+model">sigma model</a>s. This gives these boundary data a geometric interpretation in some space. This we discuss in the next section.</p> <p><img src="https://ncatlab.org/nlab/files/BraneAndOpenString.jpg" width="700" /></p> <blockquote> <p>graphics grabbed from <a href="#IbanezUranga12">Ibanez-Uranga 12</a></p> </blockquote> <h3 id="in__tft">In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>d</mi></mrow><annotation encoding="application/x-tex">2d</annotation></semantics></math> TFT</h3> <p>Another case where the branes of a QFT are under good mathematical control is <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a>: the <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category">(infinity,1)-category</a>-version of a 2d <a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a>.</p> <p>Particularly the <a class="existingWikiWord" href="/nlab/show/A-model">A-model</a> and the <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a> are well understood.</p> <ul> <li> <p>the branes of the B-model (“B-branes”) form the the <a class="existingWikiWord" href="/nlab/show/stable+%28infinity%2C1%29-category">stable (infinity,1)-category</a> of <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a>es of <a class="existingWikiWord" href="/nlab/show/quasicoherent+sheaves">quasicoherent sheaves</a> on the target space (often considered just in terms of its <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28infinity%2C1%29-category">homotopy category of an (infinity,1)-category</a>, the <a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a> of quasicoherent sheaves);</p> </li> <li> <p>the branes of the A-model form the <a class="existingWikiWord" href="/nlab/show/Fukaya+category">Fukaya category</a> of the target space.</p> </li> <li> <p>the category of D-branes of the A-model on a symplectic <a class="existingWikiWord" href="/nlab/show/Landau-Ginzburg+model">Landau-Ginzburg model</a>, is a <a class="existingWikiWord" href="/nlab/show/Fukaya-Seidel+category">Fukaya-Seidel category</a>;</p> </li> <li> <p>the category of D-branes of the B-model on a complex Landau-Ginzburg model is a category of <a class="existingWikiWord" href="/nlab/show/matrix+factorization">matrix factorization</a>s.</p> </li> </ul> <p>There is also a mathematical structure called <em><a class="existingWikiWord" href="/nlab/show/string+topology">string topology</a></em> with D-branes. At present this is more “string inspired” than actually derived from string theory, though.</p> <h3 id="in_terms_of_geometric_data_of_the_model_background">In terms of geometric data of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-model background</h3> <p>An abstractly defined QFT (as a consistent assignment of state spaces and propagators to cobordisms as in <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a>) may be obtained by <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> from <em>geometric data</em> :</p> <p>Such a <em><a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> QFT</em> is the <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> of an <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> on a space of maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\Sigma \to X</annotation></semantics></math> from a cobordism (“worldvolume”) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> to some target space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> that may carry further geometric data such as a <a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemannian metric</a>, or other background <a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a>s.</p> <p>One may therefore try to match the geometric data on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> that encodes the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-model with the algebraic data of the <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a> that results after quantization. This gives a geometric interpretation to many of the otherwise purely abstract algebraic properties of the worldvolume QFT.</p> <p>It turns out that if one checks which geometric data corresponds to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-modules in the above discussion, one finds that these tend to come from structures that look at least roughly like <em>submanifolds</em> of the target space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. And typically these submanifolds themselves carry their own background <a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a> data.</p> <p>A well-understood case is the <a class="existingWikiWord" href="/nlab/show/Wess-Zumino-Witten+model">Wess-Zumino-Witten model</a>: for this the target space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a simple <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">X = G</annotation></semantics></math> and the background field is a <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 2-bundle with connection</a> (a <a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe</a>) on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>, representing the background field that is known as the <a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a>.</p> <p>In this case it turns out that branes for the sigma model on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> are given in the simplest case by conjugacy classes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>⊂</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">D \subset G</annotation></semantics></math> inside the group, and that these carry <a class="existingWikiWord" href="/nlab/show/twisted+bundle">twisted vector bundle</a> with the twist given by the Kalb-Ramond background bundle. These vector bundles are known in the <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> literature as <em><a class="existingWikiWord" href="/nlab/show/Chan-Paton+vector+bundles">Chan-Paton vector bundles</a></em> . The geometric intuition is that a QFT with certain <a class="existingWikiWord" href="/nlab/show/boundary+CFT">boundary condition</a> comes from a quantization of spaces of maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">\Sigma \to G</annotation></semantics></math> that are restricted to take the boundary of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> to these submanifolds.</p> <p>More generally, one finds that the geometric data that corresponds to the branes in the algebraically defined 2d QFT is given by cocycles in the twisted <a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>. These may be quite far from having a direct interpretation as submanifolds of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>.</p> <p>The case of rational 2d CFT considered so far is only the best understood of a long sequence of other examples. Here the collection of all <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> – identified with the collection of all internal modules over an internal frobenius algebra, forms an ordinary <a class="existingWikiWord" href="/nlab/show/category">category</a>.</p> <p>More generally, at least for 2-dimensional <a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a>s analogous considerations yield not just categories but <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-categories">stable (∞,1)-categories</a> of boundary condition objects. For instance, for what is called the <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a> 2-d <a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a> the category of <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> is the <a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a> of <a class="existingWikiWord" href="/nlab/show/coherent+sheaves">coherent sheaves</a> on some Calabi-Yau space.</p> <p>Starting with Kontsevich’s <a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a> reformulation of <a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">mirror symmetry</a> the study of (derived) D-brane categories has become a field in its own right in pure mathematics.</p> <p>… lots of further things to say …</p> <h3 id="AsBlackBranes">As black branes</h3> <p>In <a class="existingWikiWord" href="/nlab/show/perturbative+string+theory">perturbative string theory</a>, hence for small <a class="existingWikiWord" href="/nlab/show/string+coupling+constant">string coupling constant</a> the D-branes are incarnated as boundary conditions for the string’s <a class="existingWikiWord" href="/nlab/show/worldsheet">worldsheet</a> <a class="existingWikiWord" href="/nlab/show/2d+CFT">2d CFT</a>, exhibiting submanifolds in <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>. As the string <a class="existingWikiWord" href="/nlab/show/coupling+constant">coupling constant</a> increases and becomes non-perturbative, this picture of <a class="existingWikiWord" href="/nlab/show/perturbative+string+theory">perturbative string theory</a> breaks down, but at low energy (large scales) now <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> becomes a good description, and now the D-branes are incarnated as <a class="existingWikiWord" href="/nlab/show/black+branes">black branes</a>.</p> <p><img src="https://ncatlab.org/nlab/files/BlackBrane.png" /></p> <blockquote> <p>graphics grabbed from <a href="#IbanezUranga12">Ibanez-Uranga 12</a></p> </blockquote> <p>This transition is also the key to understanding <a class="existingWikiWord" href="/nlab/show/black+holes+in+string+theory">black holes in string theory</a>.</p> <div> <p><strong>1/2 <a class="existingWikiWord" href="/nlab/show/BPS+state">BPS</a> <a class="existingWikiWord" href="/nlab/show/black+branes">black branes</a> in <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></strong>: <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a>, <a class="existingWikiWord" href="/nlab/show/F1-brane">F1-brane</a>, <a class="existingWikiWord" href="/nlab/show/NS5-brane">NS5-brane</a>, <a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a>, <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a></p> <p><img width="600" src="http://ncatlab.org/nlab/files/BraneSolutionsInSupergravity.jpg" /></p> <blockquote> <p>(table from <a href="black+brane#BlumenhagenLustTheisen13">Blumenhagen, Lüst & Theisen 2013, Chapter 18.5</a>)</p> </blockquote> </div> <h2 id="examples">Examples</h2> <h3 id="various_dimensions">Various dimensions</h3> <p>In <a class="existingWikiWord" href="/nlab/show/type+IIA+supergravity">type IIA supergravity</a></p> <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>, <a class="existingWikiWord" href="/nlab/show/D6-brane">D6-brane</a>, <a class="existingWikiWord" href="/nlab/show/D8-brane">D8-brane</a>.</li> </ul> <p>In <a class="existingWikiWord" href="/nlab/show/type+IIB+supergravity">type IIB supergravity</a></p> <ul> <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>, <a class="existingWikiWord" href="/nlab/show/D7-brane">D7-brane</a></li> </ul> <h3 id="in_the_wzw_model">In the WZW model</h3> <p>For D-branes in the <a class="existingWikiWord" href="/nlab/show/WZW-model">WZW-model</a> see <em><a href="Wess-Zumino-Witten+model#DBranes">WZW-model – D-branes</a></em>.</p> <h2 id="properties">Properties</h2> <h3 id="as_fbranes_originating_from_mbranes">As F-branes originating from M-branes</h3> <div> <p><strong>from <a class="existingWikiWord" href="/nlab/show/M-branes">M-branes</a> to F-branes: <a class="existingWikiWord" href="/nlab/show/superstrings">superstrings</a>, <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> and <a class="existingWikiWord" href="/nlab/show/NS5-branes">NS5-branes</a></strong></p> <table><thead><tr><th><strong><a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a></strong> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>A</mi> <mn>1</mn></msubsup><mo>×</mo><msubsup><mi>S</mi> <mi>B</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S^1_A \times S^1_B</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/elliptic+fibration">elliptic fibration</a></th><th><a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>A</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S^1_A</annotation></semantics></math></th><th><strong><a class="existingWikiWord" href="/nlab/show/type+IIA+string+theory">type IIA string theory</a></strong></th><th><a class="existingWikiWord" href="/nlab/show/T-duality">T-dual</a> <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>B</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S^1_B</annotation></semantics></math></th><th><strong><a class="existingWikiWord" href="/nlab/show/type+IIB+string+theory">type IIB string theory</a></strong></th><th>geometrize the <a class="existingWikiWord" href="/nlab/show/axio-dilaton">axio-dilaton</a></th><th><strong><a class="existingWikiWord" href="/nlab/show/F-theory">F-theory</a></strong> on <a class="existingWikiWord" href="/nlab/show/elliptic+fibration">elliptically fibered</a>-<a class="existingWikiWord" href="/nlab/show/K3">K3</a> fibration</th><th><a class="existingWikiWord" href="/nlab/show/duality+between+F-theory+and+heterotic+string+theory">duality between F-theory and heterotic string theory</a></th><th><strong><a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a></strong> on <a class="existingWikiWord" href="/nlab/show/elliptic+fibration">elliptic fibration</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a> <a class="existingWikiWord" href="/nlab/show/wrapped+brane">wrapping</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>A</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S_A^1</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/double+dimensional+reduction">double dimensional reduction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+IIA+superstring">type IIA superstring</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+IIB+superstring">type IIB superstring</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;">“</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/heterotic+superstring">heterotic superstring</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a> <a class="existingWikiWord" href="/nlab/show/wrapped+brane">wrapping</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>B</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S_B^1</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D2-brane">D2-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D1-brane">D1-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></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;"><a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a> <a class="existingWikiWord" href="/nlab/show/wrapped+brane">wrapping</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> times around <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>A</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S_A^1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math> times around <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>B</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S_B^1</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/strings">strings</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/D2-branes">D2-branes</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28p%2Cq%29-string">(p,q)-string</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></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;"><a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> <a class="existingWikiWord" href="/nlab/show/wrapped+brane">wrapping</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>A</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S_A^1</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/double+dimensional+reduction">double dimensional reduction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D4-brane">D4-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D5-brane">D5-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></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;"><a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> <a class="existingWikiWord" href="/nlab/show/wrapped+brane">wrapping</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>B</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S_B^1</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/NS5-brane">NS5-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/NS5-brane">NS5-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;">“</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/NS5-brane">NS5-brane</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> <a class="existingWikiWord" href="/nlab/show/wrapped+brane">wrapping</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> times around <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>A</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S_A^1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math> times around <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>B</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S_B^1</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/D4-brane">D4-brane</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/NS5-branes">NS5-branes</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28p%2Cq%295-brane">(p,q)5-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></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;"><a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> <a class="existingWikiWord" href="/nlab/show/wrapped+brane">wrapping</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>A</mi> <mn>1</mn></msubsup><mo>×</mo><msubsup><mi>S</mi> <mi>B</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S_A^1 \times S_B^1</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D3-brane">D3-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></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;"><a class="existingWikiWord" href="/nlab/show/KK-monopole">KK-monopole</a>/<a class="existingWikiWord" href="/nlab/show/ADE+singularity">A-type ADE singularity</a> (degeneration locus of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>A</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S^1_A</annotation></semantics></math>-circle fibration, <a class="existingWikiWord" href="/nlab/show/Sen+limit">Sen limit</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>A</mi> <mn>1</mn></msubsup><mo>×</mo><msubsup><mi>S</mi> <mi>B</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S^1_A \times S^1_B</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/elliptic+fibration">elliptic fibration</a>)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D6-brane">D6-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D7-branes">D7-branes</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;">A-type <a class="existingWikiWord" href="/nlab/show/nodal+curve">nodal curve</a> cycle degeneration locus of <a class="existingWikiWord" href="/nlab/show/elliptic+fibration">elliptic fibration</a> <img src="http://ncatlab.org/nlab/files/ADE2Cycle.jpeg" width="220" alt="ADE 2Cycle" /> (<a href="F-theory#Sen97b">Sen 97, section 2</a>)</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/special+unitary+group">SU</a>-<a class="existingWikiWord" href="/nlab/show/M-theory+lift+of+gauge+enhancement+on+D6-branes">gauge enhancement</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/KK-monopole">KK-monopole</a> <a class="existingWikiWord" href="/nlab/show/orientifold">orientifold</a>/<a class="existingWikiWord" href="/nlab/show/ADE+singularity">D-type ADE singularity</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D6-brane">D6-brane</a> with <a class="existingWikiWord" href="/nlab/show/O6-planes">O6-planes</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D7-branes">D7-branes</a> with <a class="existingWikiWord" href="/nlab/show/O7-planes">O7-planes</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;">D-type <a class="existingWikiWord" href="/nlab/show/nodal+curve">nodal curve</a> cycle degeneration locus of <a class="existingWikiWord" href="/nlab/show/elliptic+fibration">elliptic fibration</a> <img src="http://ncatlab.org/nlab/files/ADE2Cycle.jpeg" width="220" alt="ADE 2Cycle" /> (<a href="F-theory#Sen97b">Sen 97, section 3</a>)</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">SO</a>-<a class="existingWikiWord" href="/nlab/show/M-theory+lift+of+gauge+enhancement+on+D6-branes">gauge enhancement</a></td></tr> <tr><td style="text-align: left;">exceptional <a class="existingWikiWord" href="/nlab/show/ADE-singularity">ADE-singularity</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a href="elliptic+fibration#ClassificationOfSingularFibers">exceptional ADE-singularity of elliptic fibration</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/E6">E6</a>-, <a class="existingWikiWord" href="/nlab/show/E7">E7</a>-, <a class="existingWikiWord" href="/nlab/show/E8">E8</a>-<a class="existingWikiWord" href="/nlab/show/gauge+enhancement">gauge enhancement</a></td></tr> </tbody></table> <p>(e.g. <a href="F-theory#Johnson97">Johnson 97</a>, <a href="F-theory#Blumenhagen10">Blumenhagen 10</a>)</p> </div> <h3 id="characterization_in_terms_of_dirac_structures">Characterization in terms of Dirac structures</h3> <p>D-branes may be identified with <a class="existingWikiWord" href="/nlab/show/Dirac+structures">Dirac structures</a> on a <a class="existingWikiWord" href="/nlab/show/Courant+Lie+2-algebroid">Courant Lie 2-algebroid</a> over spacetime related to the <a class="existingWikiWord" href="/nlab/show/type+II+geometry">type II geometry</a> (<a href="#AsakawaSasaWatamura">Asakawa-Sasa-Watamura</a>). See at <em><a class="existingWikiWord" href="/nlab/show/Dirac+structure">Dirac structure</a></em> for more on this.</p> <h3 id="DBraneCharge">D-brane charge</h3> <p>In analogy to how in <a class="existingWikiWord" href="/nlab/show/electromagnetism">electromagnetism</a> <a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a> is given by a class in <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>, so D-brane charge is given in (<a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted</a>) <a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a>, or, if preferred, in its image under the <a class="existingWikiWord" href="/nlab/show/Chern+character">Chern character</a>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/Chan-Paton+bundle">Chan-Paton bundle</a> carried by a D-brane defines a class in <a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a> on the D-brane <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> and the D-brane charge is the push-forward (<a class="existingWikiWord" href="/nlab/show/Umkehr+map">Umkehr map</a>) of this class to <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>, using a <a class="existingWikiWord" href="/nlab/show/K-orientation">K-orientation</a> of the embedding of the D-brane (a <a class="existingWikiWord" href="/nlab/show/spin%5Ec+structure">spin^c structure</a>).</p> <p>See at <em><a class="existingWikiWord" href="/nlab/show/K-theory+classification+of+D-brane+charge">K-theory classification of D-brane charge</a></em></p> <h4 id="general">General</h4> <p>More in detail this means the following (<a href="#BMRS2">BMRS2</a>).</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a manifold regarded as <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><mi>Q</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">i \colon Q \hookrightarrow X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/submanifold">submanifold</a> regarded as the <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> of a D-brane. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mi>B</mi></msub><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\nabla_B \colon X \to \mathbf{B}^2 U(1)_{conn}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/circle+2-bundle+with+connection">circle 2-bundle with connection</a> which models the <a class="existingWikiWord" href="/nlab/show/background+gauge+field">background</a> <a class="existingWikiWord" href="/nlab/show/B-field">B-field</a>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>χ</mi> <mi>B</mi></msub><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\chi_B \colon X \to \mathbf{B}^2 U(1)</annotation></semantics></math> for the underlying <a class="existingWikiWord" href="/nlab/show/circle+2-group">circle 2-group</a>-<a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>.</p> <p>The corresponding <a class="existingWikiWord" href="/nlab/show/Chan-Paton+bundle">Chan-Paton bundle</a> (a <a class="existingWikiWord" href="/nlab/show/twisted+bundle">twisted</a> <a class="existingWikiWord" href="/nlab/show/line+bundle">line bundle</a> for the case of a single D-brane) is the trivialization <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ξ</mi></mrow><annotation encoding="application/x-tex">\xi</annotation></semantics></math> in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Q</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mi>i</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd></mtd> <mtd><msub><mo>⇙</mo> <mi>ξ</mi></msub></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><msub><mi>χ</mi> <mi>B</mi></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Q</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mi>i</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><msub><mo>⇙</mo> <mi>ξ</mi></msub></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msup><mi>i</mi> <mo>*</mo></msup><msub><mi>χ</mi> <mi>B</mi></msub></mrow></mpadded></msup></mtd> <mtd><msub><mo>⇙</mo> <mi>id</mi></msub></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><msub><mi>χ</mi> <mi>B</mi></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && Q \\ & \swarrow && \searrow^{\mathrlap{i}} \\ \ast && \swArrow_{\xi} && X \\ & \searrow && \swarrow_{\mathrlap{\chi_B}} \\ && \mathbf{B}^2 U(1) } \;\;\;\;\; \simeq \;\;\;\;\; \array{ && Q \\ & \swarrow &\downarrow& \searrow^{\mathrlap{i}} \\ \ast &\swArrow_{\xi}& \downarrow^{\mathrlap{i^\ast \chi_B}} &\swArrow_{id}& X \\ & \searrow &\downarrow& \swarrow_{\mathrlap{\chi_B}} \\ && \mathbf{B}^2 U(1) } \,. </annotation></semantics></math></div> <p>Assuming that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><mi>Q</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">i \colon Q \to X</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/K-orientation">K-oriented</a> in that for instance <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> has a <a class="existingWikiWord" href="/nlab/show/spin-structure">spin-structure</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/spin%5Ec-structure">spin^c-structure</a>, then under the <a class="existingWikiWord" href="/nlab/show/groupoid+convolution+algebra">groupoid convolution algebra</a> <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math> this is incarnated as a <a class="existingWikiWord" href="/nlab/show/Hilbert+bimodule">Hilbert bimodule</a> which defines a class in <a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted</a> <a class="existingWikiWord" href="/nlab/show/operator+K-theory">operator K-theory</a>, realized as the following comoposite in <a class="existingWikiWord" href="/nlab/show/KK-theory">KK-theory</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><mover><mo>→</mo><mrow><mi>Γ</mi><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow></mover><mi>C</mi><mo stretchy="false">(</mo><mi>Q</mi><msub><mo stretchy="false">)</mo> <mrow><msup><mi>i</mi> <mo>*</mo></msup><msub><mi>χ</mi> <mi>B</mi></msub></mrow></msub><mover><mo>→</mo><mrow><msub><mi>i</mi> <mo>!</mo></msub></mrow></mover><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><msub><mi>χ</mi> <mi>B</mi></msub></mrow></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathbb{C} \stackrel{\Gamma(\xi)}{\to} C(Q)_{i^\ast \chi_B} \stackrel{i_!}{\to} C(X)_{\chi_B} \,, </annotation></semantics></math></div> <p>where</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(Q)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(X)</annotation></semantics></math> are the <a class="existingWikiWord" href="/nlab/show/C%2A-algebra">C*-</a><a class="existingWikiWord" href="/nlab/show/algebras+of+functions">algebras of functions</a> (<a class="existingWikiWord" href="/nlab/show/vanishing+at+infinity">vanishing at infinity</a>) on the D-brane and on <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>, respectively;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><msub><mi>χ</mi> <mi>B</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">C(X)_{\chi_B}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/groupoid+convolution+algebra">groupoid convolution algebra</a> of <a class="existingWikiWord" href="/nlab/show/sections">sections</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>χ</mi> <mi>B</mi></msub></mrow><annotation encoding="application/x-tex">\chi_B</annotation></semantics></math> regarded as a <a class="existingWikiWord" href="/nlab/show/centrally+extended+groupoid">centrally extended groupoid</a> over a <a class="existingWikiWord" href="/nlab/show/Cech+groupoid">Cech groupoid</a> <a class="existingWikiWord" href="/nlab/show/resolution">resolution</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> which supports a <a class="existingWikiWord" href="/nlab/show/Cech+cohomology">Cech cocycle</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>χ</mi> <mi>B</mi></msub></mrow><annotation encoding="application/x-tex">\chi_B</annotation></semantics></math>, and similarly for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>Q</mi><msub><mo stretchy="false">)</mo> <mrow><msup><mi>i</mi> <mo>*</mo></msup><mi>χ</mi><mi>B</mi></mrow></msub></mrow><annotation encoding="application/x-tex">C(Q)_{i^\ast \chi B}</annotation></semantics></math> and the pullback/restriction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>i</mi> <mo>*</mo></msup><msub><mi>χ</mi> <mi>B</mi></msub></mrow><annotation encoding="application/x-tex">i^\ast \chi_B</annotation></semantics></math> of the background B-field to the brane;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>!</mo></mrow><annotation encoding="application/x-tex">i!</annotation></semantics></math> is the push-forward (<a class="existingWikiWord" href="/nlab/show/Umkehr+map">Umkehr map</a>) dual to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>i</mi> <mo>*</mo></msup><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><msub><mi>χ</mi> <mi>B</mi></msub></mrow></msub><mo>→</mo><mi>C</mi><mo stretchy="false">(</mo><mi>Q</mi><msub><mo stretchy="false">)</mo> <mrow><msup><mi>i</mi> <mo>*</mo></msup><msub><mi>χ</mi> <mi>B</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">i^\ast \colon C(X)_{\chi_B} \to C(Q)_{i^\ast \chi_B}</annotation></semantics></math>, realizes as a <a class="existingWikiWord" href="/nlab/show/KK-theory">KK-theory</a> class</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>Q</mi></msub><mo>=</mo><mi>i</mi><mo>!</mo><mo>=</mo><mo>∈</mo><mi>KK</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">)</mo><mo>,</mo><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><msub><mi>χ</mi> <mi>B</mi></msub></mrow></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> D_Q = i! = \in KK(C(Q), C(X)_{\chi_B}) \,. </annotation></semantics></math></div></li> </ul> <p>The corresponding <strong>D-brane charge</strong> in KK-theory is the resulting composite (relative <a class="existingWikiWord" href="/nlab/show/index">index</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mo>!</mo></msub><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>D</mi> <mi>Q</mi></msub><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>KK</mi><mo stretchy="false">(</mo><mi>ℂ</mi><mo>,</mo><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><msub><mi>χ</mi> <mi>B</mi></msub></mrow></msub><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>K</mi> <mrow><mo stretchy="false">[</mo><msub><mi>χ</mi> <mi>b</mi></msub><mo stretchy="false">]</mo></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> i_!(\xi) = D_Q(\xi) \in KK(\mathbb{C}, C(X)_{\chi_B}) \simeq K^{[\chi_b]}(X) </annotation></semantics></math></div> <p>in <a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a>. Traditionally only the image of this under the <a class="existingWikiWord" href="/nlab/show/Chern+character">Chern character</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ch</mi><mo lspace="verythinmathspace">:</mo><mi>KK</mi><mo>→</mo><mi>HL</mi></mrow><annotation encoding="application/x-tex"> ch \colon KK \to HL </annotation></semantics></math></div> <p>in real cohomology/<a class="existingWikiWord" href="/nlab/show/cyclic+cohomology">cyclic cohomology</a> is considered, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ch</mi><mo stretchy="false">(</mo><msub><mi>D</mi> <mi>Q</mi></msub><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ch(D_Q(\xi))</annotation></semantics></math>. Moreover, traditonally one thinks of first applying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ch</mi></mrow><annotation encoding="application/x-tex">ch</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ξ</mi></mrow><annotation encoding="application/x-tex">\xi</annotation></semantics></math> and then pushing forward in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>HL</mi></mrow><annotation encoding="application/x-tex">HL</annotation></semantics></math>. By the <a class="existingWikiWord" href="/nlab/show/C%2A-algebra">C*-algebraic</a> <a class="existingWikiWord" href="/nlab/show/Grothendieck-Riemann-Roch+theorem">Grothendieck-Riemann-Roch theorem</a> this gives the <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a> expression</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ch</mi><mo stretchy="false">(</mo><msub><mi>D</mi> <mi>Q</mi></msub><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mrow><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><msub><mi>χ</mi> <mi>B</mi></msub></mrow></msub></mrow></msub><msqrt><mi>Todd</mi></msqrt><mo>∈</mo><mi>HL</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> ch(D_Q(\xi)) \otimes_{C(X)_{\chi_B}} \sqrt{Todd} \in HL \,, </annotation></semantics></math></div> <p>where on the right we have the relative <a class="existingWikiWord" href="/nlab/show/Todd+class">Todd class</a>. This is the form the D-brane charge was originally found in the physics literature and in which it is still often given.</p> <p>(In (<a href="#BMRS2">BMRS2, Section 4</a>) this is discussed for the untwisted case.)</p> <p>For more general discussion see at <em><a href="Freed-Witten+anomaly#Details">Freed-Witten anomaly – Details</a></em> as well as at <em><a href="Poincar%C3%A9+duality+algebra#PropertiesKOrientationAndUmkehrMaps">Poincaré duality algebra – Properties – K-Orientation and Umkehr maps</a></em>.</p> <h4 id="DBraneChargeViaAtiyahHirzebruchSpectralSequence">Via the Atiyah-Hirzebruch spectral sequence</h4> <p>The <a class="existingWikiWord" href="/nlab/show/Atiyah-Hirzeburch+spectral+sequence">Atiyah-Hirzeburch spectral sequence</a> expresses, starting from its <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">E_2</annotation></semantics></math> pages, <a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a> classes on <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> as <a class="existingWikiWord" href="/nlab/show/kernels">kernels</a> of certain differential acting on ordinary cohomology in all even degrees (for type IIA strings) or all odd degrees (for type IIB strings)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>E</mi> <mn>2</mn> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup><mo>=</mo><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mi>KU</mi> <mi>q</mi></msup><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⇒</mo><msup><mi>KU</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E_2^{p,q} = H^p(X, KU^q(\ast)) \Rightarrow KU^\bullet(X) \,. </annotation></semantics></math></div> <p>Discussion of D-brane charge this way is in (<a href="#MaldacenaMooreSeiberg01">Maldacena-Moore-Seiberg 01</a>, <a href="#EvslinSati06">Evslin-Sati 06</a>).</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/boundary+conformal+field+theory">boundary conformal field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fractional+D-brane">fractional D-brane</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/permutation+D-brane">permutation D-brane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/color+branes+and+flavor+branes">color branes and flavor branes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chan-Paton+bundle">Chan-Paton bundle</a>, <a class="existingWikiWord" href="/nlab/show/twisted+bundle">twisted bundle</a>, <a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a>, <a class="existingWikiWord" href="/nlab/show/Chan-Paton+gauge+field">Chan-Paton gauge field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freed-Witten+anomaly+cancellation">Freed-Witten anomaly cancellation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dirac-Born-Infeld+action">Dirac-Born-Infeld action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/black+brane">black brane</a>, <a class="existingWikiWord" href="/nlab/show/black+hole+in+string+theory">black hole in string theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/intersecting+D-brane+model">intersecting D-brane model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-homology">K-homology</a>, <a class="existingWikiWord" href="/nlab/show/KK-theory">KK-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/O-plane">O-plane</a>, <a class="existingWikiWord" href="/nlab/show/KR-theory">KR-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D-brane+geometry">D-brane geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/anti+D-brane">anti D-brane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Myers+effect">Myers effect</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bridgeland+stability+condition">Bridgeland stability condition</a></p> </li> </ul> <div> <p><strong>Table of <a class="existingWikiWord" href="/nlab/show/brane">branes</a> appearing in <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>/<a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a></strong> (for classification see at <em><a class="existingWikiWord" href="/nlab/show/brane+scan">brane scan</a></em>).</p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/brane">brane</a></th><th>in <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></th><th><a class="existingWikiWord" href="/nlab/show/charge">charge</a>d under <a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a></th><th>has <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> theory</th></tr></thead><tbody><tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/black+brane">black brane</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/higher+gauge+field">higher gauge field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SCFT">SCFT</a></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+II+supergravity">type II</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/RR-field">RR-field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">super Yang-Mills theory</a></td></tr> <tr><td style="text-align: left;"><strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>D</mi><mo>=</mo><mn>2</mn><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(D = 2n)</annotation></semantics></math></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+IIA+supergravity">type IIA</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D%28-2%29-brane">D(-2)-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D0-brane">D0-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/BFSS+matrix+model">BFSS matrix model</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D2-brane">D2-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D4-brane">D4-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D%3D5+super+Yang-Mills+theory">D=5 super Yang-Mills theory</a> with <a class="existingWikiWord" href="/nlab/show/Khovanov+homology">Khovanov homology</a> <a class="existingWikiWord" href="/nlab/show/observables">observables</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D6-brane">D6-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D%3D7+super+Yang-Mills+theory">D=7 super Yang-Mills theory</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D8-brane">D8-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>D</mi><mo>=</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(D = 2n+1)</annotation></semantics></math></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+IIB+supergravity">type IIB</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D%28-1%29-brane">D(-1)-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D1-brane">D1-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;">2d <a class="existingWikiWord" href="/nlab/show/CFT">CFT</a> with <a class="existingWikiWord" href="/nlab/show/BH+entropy">BH entropy</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D3-brane">D3-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/N%3D4+D%3D4+super+Yang-Mills+theory">N=4 D=4 super Yang-Mills theory</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D5-brane">D5-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D7-brane">D7-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D9-brane">D9-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28p%2Cq%29-string">(p,q)-string</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/D25-brane">D25-brane</a>)</td><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/bosonic+string+theory">bosonic string theory</a>)</td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/NS-brane">NS-brane</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/supergravity">type I, II, heterotic</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-connection</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/string">string</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/B2-field">B2-field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2d+SCFT">2d SCFT</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/NS5-brane">NS5-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/B6-field">B6-field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/little+string+theory">little string theory</a></td></tr> <tr><td style="text-align: left;"><strong>D-brane for <a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/A-brane">A-brane</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/B-brane">B-brane</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/M-brane">M-brane</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11D SuGra</a>/<a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-connection</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/C3-field">C3-field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/ABJM+theory">ABJM theory</a>, <a class="existingWikiWord" href="/nlab/show/BLG+model">BLG model</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/C6-field">C6-field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-superconformal+QFT">6d (2,0)-superconformal QFT</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M9-brane">M9-brane</a>/<a class="existingWikiWord" href="/nlab/show/O-plane">O9-plane</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/heterotic+string+theory">heterotic string theory</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M-wave">M-wave</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;"><a class="existingWikiWord" href="/nlab/show/topological+M2-brane">topological M2-brane</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/topological+M-theory">topological M-theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/C3-field">C3-field</a> on <a class="existingWikiWord" href="/nlab/show/G%E2%82%82-manifold">G₂-manifold</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/topological+M5-brane">topological M5-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/C6-field">C6-field</a> on <a class="existingWikiWord" href="/nlab/show/G%E2%82%82-manifold">G₂-manifold</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/S-brane">S-brane</a></strong></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;"><a class="existingWikiWord" href="/nlab/show/SM2-brane">SM2-brane</a>,<br /><a class="existingWikiWord" href="/nlab/show/membrane+instanton">membrane instanton</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;"><a class="existingWikiWord" href="/nlab/show/M5-brane+instanton">M5-brane instanton</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;"><a class="existingWikiWord" href="/nlab/show/D3-brane+instanton">D3-brane instanton</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;"><strong><a class="existingWikiWord" href="/nlab/show/solitons">solitons</a></strong> on <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-superconformal+QFT">6d (2,0)-superconformal QFT</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><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/self-dual+higher+gauge+field">self-dual</a> <a class="existingWikiWord" href="/nlab/show/B-field">B-field</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/3-brane+in+6d">3-brane in 6d</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> </tbody></table> </div><div> <p><strong><a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Chern-Simons+theory">∞-Chern-Simons theory</a> from binary and non-degenerate <a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a></strong></p> <table><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0477fc87f3bab6c93955b29dd7c515f68e3108fd_1"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math></th><th><a class="existingWikiWord" href="/nlab/show/symplectic+Lie+n-algebroid">symplectic Lie n-algebroid</a></th><th><a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integrated</a> <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a> = <a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stack">moduli ∞-stack</a> of <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0477fc87f3bab6c93955b29dd7c515f68e3108fd_2"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-d <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></th><th><a class="existingWikiWord" href="/nlab/show/higher+symplectic+geometry">higher symplectic geometry</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0477fc87f3bab6c93955b29dd7c515f68e3108fd_3"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>d <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></th><th><a class="existingWikiWord" href="/nlab/show/dg-manifold">dg-</a><a class="existingWikiWord" href="/nlab/show/Lagrangian+submanifold">Lagrangian submanifold</a>/ <a class="existingWikiWord" href="/nlab/show/real+polarization">real polarization</a> <a class="existingWikiWord" href="/nlab/show/leaf">leaf</a></th><th>= <a class="existingWikiWord" href="/nlab/show/brane">brane</a></th><th><a class="existingWikiWord" href="/nlab/show/n-module">(n+1)-module</a> of <a class="existingWikiWord" href="/nlab/show/quantum+states">quantum states</a> in <a class="existingWikiWord" href="/nlab/show/codimension">codimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0477fc87f3bab6c93955b29dd7c515f68e3108fd_4"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math></th><th>discussed in:</th></tr></thead><tbody><tr><td style="text-align: left;">0</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lagrangian+submanifold">Lagrangian submanifold</a></td><td style="text-align: left;">–</td><td style="text-align: left;">ordinary <a class="existingWikiWord" href="/nlab/show/space+of+states+%28in+geometric+quantization%29">space of states (in geometric quantization)</a></td><td style="text-align: left;"><em><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a></em></td></tr> <tr><td style="text-align: left;">1</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroid">Poisson Lie algebroid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-plectic+geometry">2-plectic geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Poisson+sigma-model">Poisson sigma-model</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coisotropic+submanifold">coisotropic submanifold</a> (of underlying <a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a>)</td><td style="text-align: left;"><a href="Poisson+sigma-model#Branes">brane of Poisson sigma-model</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-module">2-module</a> = <a class="existingWikiWord" href="/nlab/show/category+of+modules">category of modules</a> over <a class="existingWikiWord" href="/nlab/show/strict+deformation+quantization">strict deformation quantiized</a> <a class="existingWikiWord" href="/nlab/show/algebra+of+observables">algebra of observables</a></td><td style="text-align: left;"><em><a class="existingWikiWord" href="/nlab/show/extended+geometric+quantization+of+2d+Chern-Simons+theory">extended geometric quantization of 2d Chern-Simons theory</a></em></td></tr> <tr><td style="text-align: left;">2</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Courant+Lie+2-algebroid">Courant Lie 2-algebroid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+2-groupoid">symplectic 2-groupoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/3-plectic+geometry">3-plectic geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Courant+sigma-model">Courant sigma-model</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Dirac+structure">Dirac structure</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a> in <a class="existingWikiWord" href="/nlab/show/type+II+geometry">type II geometry</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0477fc87f3bab6c93955b29dd7c515f68e3108fd_5"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+Lie+n-algebroid">symplectic Lie n-algebroid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+infinity-groupoid">symplectic n-groupoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/n-plectic+geometry">(n+1)-plectic geometry</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0477fc87f3bab6c93955b29dd7c515f68e3108fd_6"><semantics><mrow><mi>d</mi><mo>=</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">d = n+1</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/AKSZ+sigma-model">AKSZ sigma-model</a></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></tr> </tbody></table> <p>(adapted from <a class="existingWikiWord" href="/nlab/show/Some+title+containing+the+words+%22homotopy%22+and+%22symplectic%22%2C+e.g.+this+one">Ševera 00</a>)</p></div> <h2 id="references">References</h2> <h3 id="general_2">General</h3> <p>The original article is</p> <ul> <li id="Polchinski95"> <p><a class="existingWikiWord" href="/nlab/show/Joseph+Polchinski">Joseph Polchinski</a>, <em>Dirichlet-Branes and Ramond-Ramond Charges</em>, Phys. Rev. Lett. <strong>75</strong> (1995) 4724-4727 [<a href="https://arxiv.org/abs/hep-th/9510017">arXiv:hep-th/9510017</a>, <a href="https://doi.org/10.1103/PhysRevLett.75.4724">doi:10.1103/PhysRevLett.75.4724</a>]</p> <p>from p.7:</p> </li> </ul> <blockquote> <p>although it appears that we have modified the type II theory by adding something new to it, we are now arguing that these objects are actually intrinsic to any nonperturbative formulation of the type II theory; presumably one should think of them as an alternate representation of the black <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-branes</p> </blockquote> <p>General review:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Joseph+Polchinski">Joseph Polchinski</a>, <em>TASI Lectures on D-Branes</em> [<a href="https://arxiv.org/abs/hep-th/9611050">arXiv:hep-th/9611050</a>]</p> </li> <li id="Johnson98"> <p><a class="existingWikiWord" href="/nlab/show/Clifford+Johnson">Clifford Johnson</a>, <em>Études on D-Branes</em>, in: <a class="existingWikiWord" href="/nlab/show/Mike+Duff">Mike Duff</a> et. al. (eds.) <em>Nonperturbative aspects of strings, branes and supersymmetry</em>, Proceedings, Trieste, Italy, March 23-April 3, (1998) [<a href="https://arxiv.org/abs/hep-th/9812196">arXiv:hep-th/9812196</a>, <a href="https://inspirehep.net/literature/481393">spire:481393</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Koji+Hashimoto">Koji Hashimoto</a>, <em>D-Brane – Superstrings and New Perspective of Our World</em>, Springer 2012 (<a href="https://link.springer.com/book/10.1007%2F978-3-642-23574-0">doi:10.1007/978-3-642-23574-0</a>, <a href="http://inspirehep.net/record/1188897">spire:1188897</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pietro+Fre">Pietro Fre</a>, <em>The Branes: Three Viewpoints</em>, In: <em>Gravity, a Geometrical Course</em> Springer 2013 (<a href="http://inspirehep.net/record/1242195">spire:1242195</a>, <a href="https://doi.org/10.1007/978-94-007-5443-0_7">doi:10.1007/978-94-007-5443-0_7</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Constantin+Bachas">Constantin Bachas</a>, <em>D-branes</em>, in: <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Quantum+Gravity">Handbook of Quantum Gravity</a></em>, Springer (2023) [<a href="https://arxiv.org/abs/2311.18456">arXiv:2311.18456</a>, <a href="https://doi.org/10.1007/978-981-19-3079-9">doi:10.1007/978-981-19-3079-9</a>]</p> </li> </ul> <p>A classical text describing how the physics way to think of D-branes leads to seeing that they are objects in derived categories is</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Paul+Aspinwall">Paul Aspinwall</a>, <em>D-Branes on Calabi-Yau Manifolds</em> (<a href="http://arxiv.org/abs/hep-th/0403166">arXiv:hep-th/0403166</a>)</li> </ul> <p>Discussion with an eye towards <a class="existingWikiWord" href="/nlab/show/string+phenomenology">string phenomenology</a> is in</p> <ul> <li id="IbanezUranga12"><a class="existingWikiWord" href="/nlab/show/Luis+Ib%C3%A1%C3%B1ez">Luis Ibáñez</a>, <a class="existingWikiWord" href="/nlab/show/Angel+Uranga">Angel Uranga</a>, <em><a class="existingWikiWord" href="/nlab/show/String+Theory+and+Particle+Physics+--+An+Introduction+to+String+Phenomenology">String Theory and Particle Physics – An Introduction to String Phenomenology</a></em>, Cambridge University Press 2012</li> </ul> <p>This can to a large extent be read as a dictionary from <a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a> terminology to that of D-brane physics.</p> <p>More recent similar material with the emphasis on the <a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a> aspects is</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Richard+Szabo">Richard Szabo</a>, <em><a class="existingWikiWord" href="/nlab/files/Szabo09.pdf" title="D-Branes and Bivariant K-Theory">D-Branes and Bivariant K-Theory</a></em></li> </ul> <p>Comments on the role of D-branes in <a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a> and <a class="existingWikiWord" href="/nlab/show/mathematics">mathematics</a> is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Gregory+Moore">Gregory Moore</a>, <em><a class="existingWikiWord" href="/nlab/show/The+Impact+of+D-Branes+on+Mathematics">The Impact of D-Branes on Mathematics</a></em> (2014)</li> </ul> <h3 id="on_orbifolds">On orbifolds</h3> <p>Review includes</p> <ul> <li> <p>Frederik Roose, <em>Strings and D-branes on orbifolds: from boundary states to geometry</em>, 2001 (<a href="https://fys.kuleuven.be/itf/groups/hep/files/phd/fredrroose.pdf">pdf</a>)</p> </li> <li> <p>Nikolas Prezas, <em>Aspects of branes and orbifolds in string theory</em>, 2002 (<a href="https://dspace.mit.edu/bitstream/handle/1721.1/8486/50759455-MIT.pdf?sequence=2">pdf</a>, <a href="http://hdl.handle.net/1721.1/8486">web</a>)</p> </li> </ul> <p>See also the references at <em><a class="existingWikiWord" href="/nlab/show/orientifold">orientifold</a></em>.</p> <h3 id="ReferencesAsGSsigmaModels">As higher super-GS-WZW type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-models</h3> <p>The <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+sigma+model">Green-Schwarz sigma model</a> with <a class="existingWikiWord" href="/nlab/show/DBI+action">DBI action</a> for D-branes is discussed in</p> <ul> <li id="CGNSW97"> <p><a class="existingWikiWord" href="/nlab/show/Martin+Cederwall">Martin Cederwall</a>, Alexander von Gussich, <a class="existingWikiWord" href="/nlab/show/Bengt+Nilsson">Bengt Nilsson</a>, Per Sundell, Anders Westerberg, <em>The Dirichlet Super-p-Branes in Ten-Dimensional Type IIA and IIB Supergravity</em>, Nucl.Phys. B490 (1997) 179-201 (<a href="http://arxiv.org/abs/hep-th/9611159">arXiv:hep-th/9611159</a>)</p> </li> <li id="APPS97b"> <p><a class="existingWikiWord" href="/nlab/show/Mina+Aganagic">Mina Aganagic</a>, <a class="existingWikiWord" href="/nlab/show/Jaemo+Park">Jaemo Park</a>, Costin Popescu, <a class="existingWikiWord" href="/nlab/show/John+Schwarz">John Schwarz</a>, <em>Dual D-Brane Actions</em>, Nucl. Phys. B496 (1997) 215-230 (<a href="https://arxiv.org/abs/hep-th/9702133">arXiv:hep-th/9702133</a>)</p> </li> </ul> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+action+functionals">Green-Schwarz action functionals</a> for super D-branes and the interpretation of the WZW cocycles for the <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> as cocycles on “<a class="existingWikiWord" href="/nlab/show/extended+super-Minkowski+spacetime">extended super-Minkowski spacetime</a>” is due to</p> <ul> <li id="CAIB99"> <p>C. Chrysso‌malakos, <a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+de+Azc%C3%A1rraga">José de Azcárraga</a>, <a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+M.+Izquierdo">José M. Izquierdo</a>, C. Pérez Bueno, <em>The geometry of branes and extended superspaces</em>, Nucl. Phys. B <strong>567</strong> (2000) 293-330 [<a href="http://arxiv.org/abs/hep-th/9904137">arXiv:hep-th/9904137</a>, <a href="https://doi.org/10.1016/S0550-3213(99)00512-X">doi:10.1016/S0550-3213(99)00512-X</a>]</p> </li> <li id="Sakaguchi00"> <p>Makoto Sakaguchi, <em>IIB-Branes and New Spacetime Superalgebras</em>, JHEP 0004 (2000) 019 (<a href="https://arxiv.org/abs/hep-th/9909143">arXiv:hep-th/9909143</a>)</p> </li> <li id="AzcarragaIzquierdo01"> <p><a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+de+Azc%C3%A1rraga">José de Azcárraga</a>, J. M. Izquierdo, <em>Superalgebra cohomology, the geometry of extended superspaces and superbranes</em> (<a href="http://arxiv.org/abs/hep-th/0105125">arXiv:hep-th/0105125</a>)</p> </li> </ul> <p>See also <em><a class="existingWikiWord" href="/nlab/show/division+algebras+and+supersymmetry">division algebras and supersymmetry</a></em>.</p> <p>A corresponding discussion as <a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Wess-Zumino-Witten+theory">∞-Wess-Zumino-Witten theory</a> and refinement of the brane scan to a “brane bouquet” of <a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebra">super L-∞ algebra</a> <a class="existingWikiWord" href="/nlab/show/infinity-Lie+algebra+cohomology">extensions</a> (hence in <a class="existingWikiWord" href="/nlab/show/infinity-Lie+theory">infinity-Lie theory</a> via <a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Wess-Zumino-Witten+theory">∞-Wess-Zumino-Witten theory</a>) is discussed in</p> <ul> <li id="FiorenzaSatiSchreiber13"> <p><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/The+brane+bouquet">Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields</a></em>, International Journal of Geometric Methods in Modern Physics Volume 12, Issue 02 (2015) 1550018 (<a href="http://arxiv.org/abs/1308.5264">arXiv:1308.5264</a>)</p> </li> <li id="FiorenzaSatiSchreiber15"> <p><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a> <em><a class="existingWikiWord" href="/schreiber/show/The+WZW+term+of+the+M5-brane+and+differential+cohomotopy">The WZW term of the M5-brane and differential cohomotopy</a></em>, J. Math. Phys. 56, 102301 (2015) (<a href="http://arxiv.org/abs/1506.07557">arXiv:1506.07557</a>)</p> </li> <li id="FiorenzaSatiSchreiber16"> <p><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Rational+sphere+valued+supercocycles+in+M-theory+and+type+IIA+string+theory">Rational sphere valued supercocycles in M-theory and type IIA string theory</a></em> (<a href="http://arxiv.org/abs/1606.03206">arXiv:1606.03206</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Vincent+Braunack-Mayer">Vincent Braunack-Mayer</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Gauge+enhancement+of+Super+M-Branes">Gauge enhancement of Super M-Branes</a></em> (<a href="https://arxiv.org/abs/1806.01115">arXiv:1806.01115</a>)</p> </li> </ul> <div> <h3 id="DBraneChargeQuantizationInTopologicalKTheory">D-brane charge quantization in topological K-theory</h3> <p>On the conjectural <a class="existingWikiWord" href="/nlab/show/D-brane+charge+quantization+in+topological+K-theory">D-brane charge quantization in topological K-theory</a>:</p> <h4 id="origin_and_basics">Origin and basics</h4> <p>The idea that <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> have <a class="existingWikiWord" href="/nlab/show/Dirac+charge+quantization">Dirac charge quantization</a> in <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a> originates with the observation that their charge expressed in <a class="existingWikiWord" href="/nlab/show/RR-field">RR-field</a> <a class="existingWikiWord" href="/nlab/show/flux+densities">flux densities</a> resembles the image of a <a class="existingWikiWord" href="/nlab/show/Chern+character">Chern character</a>:</p> <ul> <li id="GreenHarveyMoore97"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Green">Michael Green</a>, <a class="existingWikiWord" href="/nlab/show/Jeffrey+A.+Harvey">Jeffrey A. Harvey</a>, <a class="existingWikiWord" href="/nlab/show/Gregory+Moore">Gregory Moore</a>, <em>I-Brane Inflow and Anomalous Couplings on D-Branes</em>, Class. Quant. Grav. <strong>14</strong> (1997) 47-52 [<a href="https://doi.org/10.1088/0264-9381/14/1/008">doi:10.1088/0264-9381/14/1/008</a>, <a href="https://arxiv.org/abs/hep-th/9605033">arXiv:hep-th/9605033</a>]</p> </li> <li id="MinasianMoore97"> <p><a class="existingWikiWord" href="/nlab/show/Ruben+Minasian">Ruben Minasian</a>, <a class="existingWikiWord" href="/nlab/show/Gregory+Moore">Gregory Moore</a>, <em>K-theory and Ramond-Ramond charge</em>, JHEP 9711:002 (1997) [<a href="https://doi.org/10.1088/1126-6708/1997/11/002">doi:10.1088/1126-6708/1997/11/002</a>, <a href="http://arxiv.org/abs/hep-th/9710230">arXiv:hep-th/9710230</a>]</p> </li> </ul> <p>Further early discussion:</p> <ul> <li id="Witten98"> <p><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>D-Branes And K-Theory</em>, JHEP 9812:019 (1998) [<a href="http://arxiv.org/abs/hep-th/9810188">arXiv:hep-th/9810188</a>, <a href="https://doi.org/10.1088/1126-6708/1998/12/019">doi:10.1088/1126-6708/1998/12/019</a>]</p> </li> <li id="Horava98"> <p><a class="existingWikiWord" href="/nlab/show/Petr+Ho%C5%99ava">Petr Hořava</a>, <em>Type IIA D-Branes, K-Theory, and Matrix Theory</em> (1998). (<a href="https://arxiv.org/abs/hep-th/9812135">hep-th/9812135</a>).</p> </li> <li id="FreedHopkins00"> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <em>On Ramond-Ramond fields and K-theory</em>, JHEP 0005 (2000) 044 [<a href="https://doi.org/10.1088/1126-6708/2000/05/044">doi:10.1088/1126-6708/2000/05/044</a>, <a href="http://arxiv.org/abs/hep-th/0002027">arXiv:hep-th/0002027</a>]</p> </li> </ul> <p>and with emphasis on the full picture of <a class="existingWikiWord" href="/nlab/show/twisted+differential+K-theory">twisted</a> <a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a> in:</p> <ul> <li id="Freed00"><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <em><a class="existingWikiWord" href="/nlab/show/Dirac+charge+quantization+and+generalized+differential+cohomology">Dirac charge quantization and generalized differential cohomology</a></em>, Surveys in Differential Geometry, Int. Press, Somerville, MA, 2000, pp. 129–194 (<a href="http://arxiv.org/abs/hep-th/0011220">arXiv:hep-th/0011220</a>, <a href="https://dx.doi.org/10.4310/SDG.2002.v7.n1.a6">doi:10.4310/SDG.2002.v7.n1.a6</a>, <a href="https://inspirehep.net/literature/537392">spire:537392</a>)</li> </ul> <p id="BasisOfTheConjecture"> Here:</p> <ul> <li><a href="#GreenHarveyMoore97">Green, Harvey & Moore (1997)</a>, <a href="#MinasianMoore97">Minasian & Moore (1997)</a> observe that <a class="existingWikiWord" href="/nlab/show/RR-field">RR-field</a> <a class="existingWikiWord" href="/nlab/show/flux">flux</a> <a class="existingWikiWord" href="/nlab/show/differential+form">form</a>-expressions for <a class="existingWikiWord" href="/nlab/show/D-brane+charge">D-brane charge</a> look like images of K-theory classes under the <a class="existingWikiWord" href="/nlab/show/Chern+character">Chern character</a>;</li> </ul> <div class="float_right_image" style="margin: -40px 0px 20px 10px"> <figure style="margin: 0 0 0 0"> <img src="/nlab/files/AntiBraneAnnihilationAsKGroupEquivRelation.jpg" width="300px" /> <figcaption style="text-align: center">From <a href="https://ncatlab.org/schreiber/show/Equivariant+Stable+Cohomotopy+and+Branes">Sch 18</a></figcaption> </figure> </div> <ul> <li><a href="#Witten98">Witten 98, Section 3</a> adds the observation that the <a class="existingWikiWord" href="/nlab/show/tachyon+condensation">tachyon condensation</a> – which is expected (this is <em><a class="existingWikiWord" href="/nlab/show/Sen%27s+conjecture">Sen's conjecture</a></em> from <a href="Sen's+conjecture#ReferencesForSuperstrings">Sen 98</a>) for <a class="existingWikiWord" href="/nlab/show/open+strings">open strings</a> between <a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a>/<a class="existingWikiWord" href="/nlab/show/anti-D-branes">anti-D-branes</a> – plausibly implements on their <a class="existingWikiWord" href="/nlab/show/Chan-Paton+gauge+field">Chan-Paton</a> <a class="existingWikiWord" href="/nlab/show/vector+bundles">vector bundles</a> the defining <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a> (<a href="topological+K-theory#eq:DefiningEquivalenceRelation">here</a>) of <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a>.</li> </ul> <p>Expression of these D-brane K-theory classes via the <a class="existingWikiWord" href="/nlab/show/Atiyah-Hirzebruch+spectral+sequence">Atiyah-Hirzebruch spectral sequence</a>:</p> <ul> <li id="MaldacenaMooreSeiberg01"> <p><a class="existingWikiWord" href="/nlab/show/Juan+Maldacena">Juan Maldacena</a>, <a class="existingWikiWord" href="/nlab/show/Gregory+Moore">Gregory Moore</a>, <a class="existingWikiWord" href="/nlab/show/Nathan+Seiberg">Nathan Seiberg</a>, <em>D-Brane Instantons and K-Theory Charges</em>, JHEP 0111:062,2001 (<a href="http://arxiv.org/abs/hep-th/0108100">arXiv:hep-th/0108100</a>)</p> </li> <li id="EvslinSati06"> <p><a class="existingWikiWord" href="/nlab/show/Jarah+Evslin">Jarah Evslin</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <em>Can D-Branes Wrap Nonrepresentable Cycles?</em>, JHEP0610:050,2006 (<a href="http://arxiv.org/abs/hep-th/0607045">arXiv:hep-th/0607045</a>)</p> </li> </ul> <p>Specifically for D-branes in <a class="existingWikiWord" href="/nlab/show/WZW+models">WZW models</a> see</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Peter+Bouwknegt">Peter Bouwknegt</a>, <em>A note on equality of algebraic and geometric D-brane charges in WZW models</em> (<a href="http://people.physics.anu.edu.au/~drt105/papers/BR0312259.pdf">pdf</a>)</li> </ul> <p>Understanding the <a class="existingWikiWord" href="/nlab/show/solitonic+brane">solitonic</a> (non-singular) D-branes and their <a class="existingWikiWord" href="/nlab/show/T-duality">T-duality</a> in K-theory:</p> <ul> <li id="BergmanGimonHořava99"><a class="existingWikiWord" href="/nlab/show/Oren+Bergman">Oren Bergman</a>, <a class="existingWikiWord" href="/nlab/show/Eric+G.+Gimon">Eric G. Gimon</a>, <a class="existingWikiWord" href="/nlab/show/Petr+Ho%C5%99ava">Petr Hořava</a>, <em>Brane Transfer Operations and T-Duality of Non-BPS States</em>, JHEP 9904 (1999) 010 [<a href="https://doi.org/10.1088/1126-6708/1999/04/010">doi:10.1088/1126-6708/1999/04/010</a>, <a href="https://arxiv.org/abs/hep-th/9902160">arXiv:hep-th/9902160</a>]</li> </ul> <p>Towards a <a class="existingWikiWord" href="/nlab/show/BFSS+matrix+model">matrix model</a> taking these K-theoretic effects into account (<a class="existingWikiWord" href="/nlab/show/K-matrix+model">K-matrix model</a>):</p> <ul> <li id="AsakawaSugimotoTerashima01"><a class="existingWikiWord" href="/nlab/show/Tsuguhiko+Asakawa">Tsuguhiko Asakawa</a>, <a class="existingWikiWord" href="/nlab/show/Shigeki+Sugimoto">Shigeki Sugimoto</a>, <a class="existingWikiWord" href="/nlab/show/Seiji+Terashima">Seiji Terashima</a>, <em>D-branes, Matrix Theory and K-homology</em>, JHEP 0203 (2002) 034 [<a href="https://doi.org/10.1088/1126-6708/2002/03/034">doi:10.1088/1126-6708/2002/03/034</a>, <a href="https://arxiv.org/abs/hep-th/0108085">arXiv:hep-th/0108085</a>]</li> </ul> <h4 id="twisted_equivariant_and_differential_refinement">Twisted, equivariant and differential refinement</h4> <p>Discussion of charge quantization in <a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a> for the case of non-vanishing <a class="existingWikiWord" href="/nlab/show/B-field">B-field</a>:</p> <ul> <li> <p><a href="#Witten98">Witten 98, Sec. 5.3</a> (for <a class="existingWikiWord" href="/nlab/show/torsion+subgroup">torsion</a> twists)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Peter+Bouwknegt">Peter Bouwknegt</a>, <a class="existingWikiWord" href="/nlab/show/Varghese+Mathai">Varghese Mathai</a>, <em>D-branes, B-fields and twisted K-theory</em>, Int. J. Mod. Phys. A <strong>16</strong> (2001) 693-706 [<a href="https://doi.org/10.1088/1126-6708/2000/03/007">doi:10.1088/1126-6708/2000/03/007</a>, <a href="https://arxiv.org/abs/hep-th/0002023">arXiv:hep-th/0002023</a>]</p> </li> </ul> <p>An elaborate proposal for the correct flavour of <a class="existingWikiWord" href="/nlab/show/equivariant+K-theory">equivariant</a> <a class="existingWikiWord" href="/nlab/show/KR-theory">KR-theory</a> needed for <a class="existingWikiWord" href="/nlab/show/orientifolds">orientifolds</a> is sketched in:</p> <ul> <li id="DistlerFreedMoore09"><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>Discussion of full-blown <a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted</a> <a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential</a> <a class="existingWikiWord" href="/nlab/show/topological+K-theory">K-theory</a> and its relation to <a class="existingWikiWord" href="/nlab/show/D-brane+charge">D-brane charge</a> in <a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a></p> <ul> <li id="GradySati19a"><a class="existingWikiWord" href="/nlab/show/Daniel+Grady">Daniel Grady</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <em>Ramond-Ramond fields and twisted differential K-theory</em>, Advances in Theoretical and Mathematical Physics <strong>26</strong> 5 (2022) [<a href="https://dx.doi.org/10.4310/ATMP.2022.v26.n5.a2">doi:10.4310/ATMP.2022.v26.n5.a2</a>, <a href="https://arxiv.org/abs/1903.08843">arXiv:1903.08843</a>]</li> </ul> <p>Discussion of full-blown <a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted</a> <a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential</a> <a class="existingWikiWord" href="/nlab/show/KO-theory">orthogonal</a> <a class="existingWikiWord" href="/nlab/show/topological+K-theory">K-theory</a> and its relation to <a class="existingWikiWord" href="/nlab/show/D-brane+charge">D-brane charge</a> in <a class="existingWikiWord" href="/nlab/show/type+I+string+theory">type I string theory</a> (on <a class="existingWikiWord" href="/nlab/show/orientifolds">orientifolds</a>):</p> <ul> <li id="GradySati19b"><a class="existingWikiWord" href="/nlab/show/Daniel+Grady">Daniel Grady</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <em>Twisted differential KO-theory</em> (<a href="https://arxiv.org/abs/1905.09085">arXiv:1905.09085</a>)</li> </ul> <h4 id="reviews">Reviews</h4> <ul> <li id="OlsenSzabo00"> <p>Kasper Olsen, <a class="existingWikiWord" href="/nlab/show/Richard+Szabo">Richard Szabo</a>, <em>Brane Descent Relations in K-theory</em>, Nucl.Phys. B566 (2000) 562-598 (<a href="https://arxiv.org/abs/hep-th/9904153">arXiv:hep-th/9904153</a>)</p> </li> <li> <p>Kasper Olsen, <a class="existingWikiWord" href="/nlab/show/Richard+Szabo">Richard Szabo</a>, <em>Constructing D-Branes from K-Theory</em>, Adv. Theor. Math. Phys. 3 (1999) 889-1025 (<a href="https://arxiv.org/abs/hep-th/9907140">arXiv:hep-th/9907140</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/John+Schwarz">John Schwarz</a>, <em>TASI Lectures on Non-BPS D-Brane Systems</em> (<a href="https://arxiv.org/abs/hep-th/9908144">arXiv:hep-th/9908144</a>)</p> </li> <li id="Witten00"> <p><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>Overview Of K-Theory Applied To Strings</em>, Int. J. Mod. Phys. A16:693-706, 2001 (<a href="https://arxiv.org/abs/hep-th/0007175">arXiv:hep-th/0007175</a>)</p> </li> <li id="Moore02"> <p><a class="existingWikiWord" href="/nlab/show/Greg+Moore">Greg Moore</a>, <em>K-Theory from a physical perspective</em>, in: <a class="existingWikiWord" href="/nlab/show/Ulrike+Tillmann">Ulrike Tillmann</a> (ed.) <em>Topology, Geometry and Quantum Field Theory</em>, Proceedings of the 2002 Oxford Symposium in Honour of the 60th Birthday of Graeme Segal, Cambridge University Press (2004) (<a href="http://arxiv.org/abs/hep-th/0304018">arXiv:hep-th/0304018</a>, <a href="https://doi.org/10.1017/CBO9780511526398.011">doi:10.1017/CBO9780511526398.011</a>)</p> </li> <li id="Manjarin04"> <p><a class="existingWikiWord" href="/nlab/show/Juan+Jos%C3%A9+Manjar%C3%ADn">Juan José Manjarín</a>, <em>Topics on D-brane charges with B-fields</em>, Int. J. Geom. Meth. Mod. Phys. 1 (2004) (<a href="http://arxiv.org/abs/hep-th/0405074">arXiv:hep-th/0405074</a>)</p> </li> <li id="Evslin06"> <p><a class="existingWikiWord" href="/nlab/show/Jarah+Evslin">Jarah Evslin</a>, <em>What Does(n’t) K-theory Classify?</em>, Modave Summer School in Mathematical Physics (<a href="https://arxiv.org/abs/hep-th/0610328">arXiv:hep-th/0610328</a>, <a href="https://inspirehep.net/literature/730502">spire:730502</a>)</p> </li> <li id="Fredenhagen08"> <p><a class="existingWikiWord" href="/nlab/show/Stefan+Fredenhagen">Stefan Fredenhagen</a>, <em>Physical Background to the K-Theory Classification of D-Branes: Introduction and References</em> (<a href="https://doi.org/10.1007/978-3-540-74956-1_1">doi:10.1007/978-3-540-74956-1_1</a>), chapter in: <a class="existingWikiWord" href="/nlab/show/Dale+Husemoeller">Dale Husemoeller</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Joachim">Michael Joachim</a>, <a class="existingWikiWord" href="/nlab/show/Branislav+Jur%C4%8Do">Branislav Jurčo</a>, <a class="existingWikiWord" href="/nlab/show/Martin+Schottenloher">Martin Schottenloher</a>, <em><a class="existingWikiWord" href="/nlab/show/Basic+Bundle+Theory+and+K-Cohomology+Invariants">Basic Bundle Theory and K-Cohomology Invariants</a></em>, Lecture Notes in Physics, Springer (2008) 1-9 [<a href="https://doi.org/10.1007/978-3-540-74956-1">doi:10.1007/978-3-540-74956-1</a>, <a href="http://www.mathematik.uni-muenchen.de/~schotten/Texte/978-3-540-74955-4_Book_LNP726corr1.pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fabio+Ruffino">Fabio Ruffino</a>, <em>Topics on topology and superstring theory</em> (<a href="http://arxiv.org/abs/0910.4524">arXiv:0910.4524</a>)</p> </li> <li id="SatiSchreiber24"> <p><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>: <em><a class="existingWikiWord" href="/schreiber/show/Flux+Quantization">Flux Quantization</a></em>, in <em><a class="existingWikiWord" href="/nlab/show/Encyclopedia+of+Mathematical+Physics+2nd+ed">Encyclopedia of Mathematical Physics 2nd ed</a></em>, Elsevier (2024) [<a href="https://arxiv.org/abs/2402.18473">arXiv:2402.18473</a>]</p> </li> </ul> <p>Amplification of <a class="existingWikiWord" href="/nlab/show/torsion+subgroup">torsion</a>-charges implied by charge quantization in Ktheory</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Volker+Braun">Volker Braun</a>, <em>K-Theory Torsion</em> &lbrack;<a href="https://arxiv.org/abs/hep-th/0005103">arXiv:hep-th/0005103</a>&rbrack;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ilka+Brunner">Ilka Brunner</a>, <a class="existingWikiWord" href="/nlab/show/Jacques+Distler">Jacques Distler</a>, <em>Torsion D-Branes in Nongeometrical Phases</em>, Adv. Theor. Math. Phys. <strong>5</strong> (2002) 265-309 [<a href="https://doi.org/10.4310/ATMP.2001.v5.n2.a3">doi:10.4310/ATMP.2001.v5.n2.a3</a>, <a href="https://arxiv.org/abs/hep-th/0102018">arXiv:hep-th/0102018</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ilka+Brunner">Ilka Brunner</a>, <a class="existingWikiWord" href="/nlab/show/Jacques+Distler">Jacques Distler</a>, Rahul Mahajan, <em>Return of the Torsion D-Branes</em>, Adv. Theor. Math. Phys. <strong>5</strong> (2002) 311-352 [<a href="https://doi.org/10.4310/ATMP.2001.v5.n2.a4">doi:10.4310/ATMP.2001.v5.n2.a4</a>, <a href="https://arxiv.org/abs/hep-th/0106262">arXiv:hep-th/0106262</a>]</p> </li> </ul> <p>Review of D-branes charge seen in <a class="existingWikiWord" href="/nlab/show/KK-theory">KK-theory</a>:</p> <ul> <li id="Szabo"><a class="existingWikiWord" href="/nlab/show/Richard+Szabo">Richard Szabo</a>, <em>D-branes and bivariant K-theory</em>, Noncommutative Geometry and Physics 3 1 (2013): 131. (<a href="http://arxiv.org/abs/0809.3029">arXiv:0809.3029</a>)</li> </ul> <p>based on</p> <ul> <li id="ReisSzabo05"> <p>Rui Reis, <a class="existingWikiWord" href="/nlab/show/Richard+Szabo">Richard Szabo</a>, <em>Geometric K-Homology of Flat D-Branes</em> , Commun. Math. Phys. <strong>266</strong> (2006) 71-122 &lbrack;<a href="https://arxiv.org/abs/hep-th/0507043">arXiv:hep-th/0507043</a>&rbrack;</p> </li> <li id="BrodzkiMathaiRosenbergSzabo06"> <p><a class="existingWikiWord" href="/nlab/show/Jacek+Brodzki">Jacek Brodzki</a>, <a class="existingWikiWord" href="/nlab/show/Varghese+Mathai">Varghese Mathai</a>, <a class="existingWikiWord" href="/nlab/show/Jonathan+Rosenberg">Jonathan Rosenberg</a>, <a class="existingWikiWord" href="/nlab/show/Richard+Szabo">Richard Szabo</a>, <em>D-Branes, RR-Fields and Duality on Noncommutative Manifolds</em>, Commun. Math. Phys. <strong>277</strong> (2008) 643-706 [<a href="https://doi.org/10.1007/s00220-007-0396-y">doi:10.1007/s00220-007-0396-y</a>, <a href="http://arxiv.org/abs/hep-th/0607020">arXiv:hep-th/0607020</a>]</p> </li> <li id="BMRS2"> <p><a class="existingWikiWord" href="/nlab/show/Jacek+Brodzki">Jacek Brodzki</a>, <a class="existingWikiWord" href="/nlab/show/Varghese+Mathai">Varghese Mathai</a>, <a class="existingWikiWord" href="/nlab/show/Jonathan+Rosenberg">Jonathan Rosenberg</a>, <a class="existingWikiWord" href="/nlab/show/Richard+Szabo">Richard Szabo</a>, <em>Noncommutative correspondences, duality and D-branes in bivariant K-theory</em>, Adv. Theor. Math. Phys. 13:497-552, 2009 (<a href="http://arxiv.org/abs/0708.2648">arXiv:0708.2648</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jacek+Brodzki">Jacek Brodzki</a>, <a class="existingWikiWord" href="/nlab/show/Varghese+Mathai">Varghese Mathai</a>, <a class="existingWikiWord" href="/nlab/show/Jonathan+Rosenberg">Jonathan Rosenberg</a>, <a class="existingWikiWord" href="/nlab/show/Richard+Szabo">Richard Szabo</a>, <em>D-branes, KK-theory and duality on noncommutative spaces</em>, J. Phys. Conf. Ser. <strong>103</strong> 012004 (2008) [<a href="https://doi.org/10.1088/1742-6596/103/1/012004">doi:10.1088/1742-6596/103/1/012004</a>, <a href="http://arxiv.org/abs/0709.2128">arXiv:0709.2128</a>]</p> </li> </ul> <p>In particular (<a href="#BMRS2">BMRS2</a>) discusses the definition and construction of D-brane charge as a generalized <a class="existingWikiWord" href="/nlab/show/index">index</a> in <a class="existingWikiWord" href="/nlab/show/KK-theory">KK-theory</a>. The discussion there focuses on the untwisted case. Comments on the generalization of this to topologicall non-trivial <a class="existingWikiWord" href="/nlab/show/B-field">B-field</a> and hence <a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a> is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Richard+Szabo">Richard Szabo</a>, <em>D-Branes, Tachyons and K-Homology</em>, Mod. Phys. Lett. A17 (2002) 2297-2316 (<a href="http://arxiv.org/abs/hep-th/0209210">arXiv:hep-th/0209210</a>)</li> </ul> <h4 id="DBraneChargeInKTheoryReferencesConceptualProblems">Conceptual problems</h4> <p id="KTheorySubtleties"> But there remain conceptual issues with the proposal that D-brane charge is in K-theory, as highlighted in</p> <ul> <li id="BDHKMMS01"> <p><a class="existingWikiWord" href="/nlab/show/Jan+de+Boer">Jan de Boer</a>, <a class="existingWikiWord" href="/nlab/show/Robbert+Dijkgraaf">Robbert Dijkgraaf</a>, <a class="existingWikiWord" href="/nlab/show/Kentaro+Hori">Kentaro Hori</a>, <a class="existingWikiWord" href="/nlab/show/Arjan+Keurentjes">Arjan Keurentjes</a>, <a class="existingWikiWord" href="/nlab/show/John+Morgan">John Morgan</a>, <a class="existingWikiWord" href="/nlab/show/David+Morrison">David Morrison</a>, <a class="existingWikiWord" href="/nlab/show/Savdeep+Sethi">Savdeep Sethi</a>, section 4.5.2 and 4.6.5 of <em>Triples, Fluxes, and Strings</em>, Adv. Theor. Math. Phys. <strong>4</strong> (2002) 995-1186 [<a href="https://arxiv.org/abs/hep-th/0103170">arXiv:hep-th/0103170</a>, <a href="https://www.intlpress.com/site/pub/files/_fulltext/journals/atmp/2000/0004/0005/ATMP-2000-0004-0005-a001.pdf">pdf</a>]</p> </li> <li id="Evslin06"> <p><a class="existingWikiWord" href="/nlab/show/Jarah+Evslin">Jarah Evslin</a>, section 8 of: <em>What Does(n’t) K-theory Classify?</em>, Second Modave Summer School in Mathematical Physics [<a href="https://arxiv.org/abs/hep-th/0610328">arXiv:hep-th/0610328</a>]</p> </li> </ul> <p>In particular, actual checks of the proposal that D-brane charge is given by K-theory, via concrete computation in <a class="existingWikiWord" href="/nlab/show/boundary+conformal+field+theory">boundary conformal field theory</a>, have revealed some subtleties:</p> <ul> <li id="FredenhagenQuella05"> <p><a class="existingWikiWord" href="/nlab/show/Stefan+Fredenhagen">Stefan Fredenhagen</a>, <a class="existingWikiWord" href="/nlab/show/Thomas+Quella">Thomas Quella</a>, <em>Generalised permutation branes</em>, JHEP 0511:004 (2005) [<a href="https://arxiv.org/abs/hep-th/0509153">arXiv:hep-th/0509153</a>, <a href="https://iopscience.iop.org/article/10.1088/1126-6708/2005/11/004">doi:10.1088/1126-6708/2005/11/004</a>]</p> <blockquote> <p>It might surprise that despite all the progress that has been made in understanding branes on group manifolds, there are usually not enough D-branes known to explain the whole charge group predicted by (twisted) K-theory. […] it is fair to say that a satisfactory answer is still missing.</p> </blockquote> </li> </ul> <p>The closest available towards an actual check of the argument for K-theory via <a class="existingWikiWord" href="/nlab/show/Sen%27s+conjecture">open superstring tachyon condensation</a> (<a href="#Witten98">Witten 98, Section 3</a>) seems to be</p> <ul> <li id="Erler13"><a class="existingWikiWord" href="/nlab/show/Theodore+Erler">Theodore Erler</a>, <em>Analytic Solution for Tachyon Condensation in Berkovits’ Open Superstring Field Theory</em>, JHEP 1311 (2013) 007 [<a href="https://doi.org/10.1007/JHEP11(2013)007">doi:10.1007/JHEP11(2013)007</a>, <a href="https://arxiv.org/abs/1308.4400">arXiv:1308.4400</a>]</li> </ul> <p>which, however, concludes (on <a href="https://arxiv.org/pdf/1308.4400.pdf#page=33">p. 32</a>) with:</p> <blockquote> <p>It would also be interesting to see if these developments can shed light on the long-speculated relation between string field theory and the K-theoretic description of D-brane charge <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="D-brane#DBraneChargeQuantizationInTopologicalKTheory">75, 76, 77</a>]. We leave these questions for future work.</p> </blockquote> <p>See also</p> <ul> <li id="Erler19"><a class="existingWikiWord" href="/nlab/show/Theodore+Erler">Theodore Erler</a>, <em>Four Lectures on Analytic Solutions in Open String Field Theory</em> (<a href="https://arxiv.org/abs/1912.00521">arXiv:1912.00521</a>, <a href="https://inspirehep.net/literature/1768105">spire:1768105</a>)</li> </ul> <p>which still lists (on <a href="https://arxiv.org/pdf/1912.00521.pdf#page=113">p. 112</a>) among open problems of <a class="existingWikiWord" href="/nlab/show/string+field+theory">string field theory</a>:</p> <blockquote> <p>“Are there topological invariants of the open string star algebra representing D-brane charges?”</p> </blockquote> <h4 id="for_orbifolds_in_equivariant_ktheory">For orbifolds in equivariant K-theory</h4> <p id="ChargeInEquivariantKTheory"> The proposal that D-brane charge on <a class="existingWikiWord" href="/nlab/show/orbifolds">orbifolds</a> is measured in <a class="existingWikiWord" href="/nlab/show/equivariant+K-theory">equivariant K-theory</a> (<a class="existingWikiWord" href="/nlab/show/orbifold+K-theory">orbifold K-theory</a>) goes back to</p> <ul> <li><a href="#Witten98">Witten 98, section 5.1</a></li> </ul> <p>It was pointed out that only a subgroup of equivariant K-theory can be physically relevant in</p> <ul> <li id="BDHKMMS01"><a class="existingWikiWord" href="/nlab/show/Jan+de+Boer">Jan de Boer</a>, <a class="existingWikiWord" href="/nlab/show/Robbert+Dijkgraaf">Robbert Dijkgraaf</a>, <a class="existingWikiWord" href="/nlab/show/Kentaro+Hori">Kentaro Hori</a>, <a class="existingWikiWord" href="/nlab/show/Arjan+Keurentjes">Arjan Keurentjes</a>, <a class="existingWikiWord" href="/nlab/show/John+Morgan">John Morgan</a>, <a class="existingWikiWord" href="/nlab/show/David+Morrison">David Morrison</a>, <a class="existingWikiWord" href="/nlab/show/Savdeep+Sethi">Savdeep Sethi</a>, around (137) of: <em>Triples, Fluxes, and Strings</em>, Adv.Theor.Math.Phys. 4 (2002) 995-1186 (<a href="https://arxiv.org/abs/hep-th/0103170">arXiv:hep-th/0103170</a>)</li> </ul> <p>Further discussion of <a class="existingWikiWord" href="/nlab/show/equivariant+K-theory">equivariant K-theory</a> for D-branes on <a class="existingWikiWord" href="/nlab/show/orbifolds">orbifolds</a> includes the following:</p> <ul> <li> <p>Hugo García-Compeán, <em>D-branes in orbifold singularities and equivariant K-theory</em>, Nucl.Phys. B557 (1999) 480-504 (<a href="https://arxiv.org/abs/hep-th/9812226">arXiv:hep-th/9812226</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Matthias+Gaberdiel">Matthias Gaberdiel</a>, <a class="existingWikiWord" href="/nlab/show/Bogdan+Stefanski">Bogdan Stefanski</a>, <em>Dirichlet Branes on Orbifolds</em>, Nucl.Phys.B578:58-84, 2000 (<a href="https://arxiv.org/abs/hep-th/9910109">arXiv:hep-th/9910109</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Igor+Kriz">Igor Kriz</a>, Leopoldo A. Pando Zayas, Norma Quiroz, <em>Comments on D-branes on Orbifolds and K-theory</em>, Int. J. Mod. Phys. A <strong>23</strong> (2008) 933-974 [<a href="https://arxiv.org/abs/hep-th/0703122">arXiv:hep-th/0703122</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Richard+Szabo">Richard Szabo</a>, <a class="existingWikiWord" href="/nlab/show/Alessandro+Valentino">Alessandro Valentino</a>, <em>Ramond-Ramond Fields, Fractional Branes and Orbifold Differential K-Theory</em>, Commun.Math.Phys.294:647-702, 2010 (<a href="https://arxiv.org/abs/0710.2773">arXiv:0710.2773</a>)</p> </li> </ul> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/real+K-theory">real K-theory</a> for D-branes on <a class="existingWikiWord" href="/nlab/show/orientifolds">orientifolds</a> includes the following:</p> <p>The original observation that <a class="existingWikiWord" href="/nlab/show/D-brane+charge">D-brane charge</a> for <a class="existingWikiWord" href="/nlab/show/orientifolds">orientifolds</a> should be in <a class="existingWikiWord" href="/nlab/show/KR-theory">KR-theory</a> is due to</p> <ul> <li><a href="#Witten98">Witten 98, section 5</a></li> </ul> <p>and was then re-amplified in</p> <ul> <li id="Gukov99"> <p><a class="existingWikiWord" href="/nlab/show/Sergei+Gukov">Sergei Gukov</a>, <em>K-Theory, Reality, and Orientifolds</em>, Commun.Math.Phys. 210 (2000) 621-639 (<a href="https://arxiv.org/abs/hep-th/9901042">arXiv:hep-th/9901042</a>)</p> </li> <li id="BergmanGimonSugimoto01"> <p><a class="existingWikiWord" href="/nlab/show/Oren+Bergman">Oren Bergman</a>, E. Gimon, <a class="existingWikiWord" href="/nlab/show/Shigeki+Sugimoto">Shigeki Sugimoto</a>, <em>Orientifolds, RR Torsion, and K-theory</em>, JHEP 0105:047, 2001 (<a href="https://arxiv.org/abs/hep-th/0103183">arXiv:hep-th/0103183</a>)</p> </li> </ul> <p>With further developments in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Varghese+Mathai">Varghese Mathai</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Murray">Michael Murray</a>, <a class="existingWikiWord" href="/nlab/show/Daniel+Stevenson">Daniel Stevenson</a>, <em>Type I D-branes in an H-flux and twisted KO-theory</em>, JHEP 0311 (2003) 053 (<a href="https://arxiv.org/abs/hep-th/0310164">arXiv:hep-th/0310164</a>)</li> </ul> <p>Discussion of orbi-orienti-folds using <a class="existingWikiWord" href="/nlab/show/equivariant+K-theory">equivariant</a> <a class="existingWikiWord" href="/nlab/show/KO-theory">KO-theory</a> is in</p> <ul> <li> <p>N. Quiroz, <a class="existingWikiWord" href="/nlab/show/Bogdan+Stefanski">Bogdan Stefanski</a>, <em>Dirichlet Branes on Orientifolds</em>, Phys.Rev. D66 (2002) 026002 (<a href="https://arxiv.org/abs/hep-th/0110041">arXiv:hep-th/0110041</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Volker+Braun">Volker Braun</a>, <a class="existingWikiWord" href="/nlab/show/Bogdan+Stefanski">Bogdan Stefanski</a>, <em>Orientifolds and K-theory</em> (<a href="https://arxiv.org/abs/hep-th/0206158">arXiv:hep-th/0206158</a>)</p> </li> <li> <p>H. Garcia-Compean, W. Herrera-Suarez, B. A. Itza-Ortiz, O. Loaiza-Brito, <em>D-Branes in Orientifolds and Orbifolds and Kasparov KK-Theory</em>, JHEP 0812:007, 2008 (<a href="https://arxiv.org/abs/0809.4238">arXiv:0809.4238</a>)</p> </li> </ul> <p>Discussion of the alleged K-theory classification of D-brane charge in relation to the <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a> <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">C-field</a> is in</p> <ul> <li id="DMW00"><a class="existingWikiWord" href="/nlab/show/Duiliu-Emanuel+Diaconescu">Duiliu-Emanuel Diaconescu</a>, <a class="existingWikiWord" href="/nlab/show/Gregory+Moore">Gregory Moore</a>, <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">E_8</annotation></semantics></math> Gauge Theory, and a Derivation of K-Theory from M-Theory</em>, Adv.Theor.Math.Phys.6:1031-1134,2003 (<a href="http://arxiv.org/abs/hep-th/0005090">arXiv:hep-th/0005090</a>), summarised in <em>A Derivation of K-Theory from M-Theory</em> (<a href="http://arxiv.org/abs/hep-th/0005091">arXiv:hep-th/0005091</a>)</li> </ul> <p>See also</p> <ul> <li id="GarciaUranga05">Inaki Garcia-Etxebarria, <a class="existingWikiWord" href="/nlab/show/Angel+Uranga">Angel Uranga</a>, <em>From F/M-theory to K-theory and back</em>, JHEP 0602:008,2006 (<a href="https://arxiv.org/abs/hep-th/0510073">arXiv:hep-th/0510073</a>)</li> </ul> <p>More complete discussion of <a class="existingWikiWord" href="/nlab/show/double+dimensional+reduction">double dimensional reduction</a> of the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a> in 11d to the expected <a class="existingWikiWord" href="/nlab/show/B-field">B-field</a> and <a class="existingWikiWord" href="/nlab/show/RR-field">RR-field</a> <a class="existingWikiWord" href="/nlab/show/flux">flux</a> forms in 10d:</p> <ul> <li id="MathaiSati03"> <p><a class="existingWikiWord" href="/nlab/show/Varghese+Mathai">Varghese Mathai</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <em>Some Relations between Twisted K-theory and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">E_8</annotation></semantics></math> Gauge Theory</em>, JHEP0403:016,2004 (<a href="http://arxiv.org/abs/hep-th/0312033">arXiv:hep-th/0312033</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Rational+sphere+valued+supercocycles+in+M-theory+and+type+IIA+string+theory">Rational sphere valued supercocycles in M-theory and type IIA string theory</a></em>, Journal of Geometry and Physics, Volume 114, Pages 91-108 April 2017 (<a href="http://arxiv.org/abs/1606.03206">arXiv:1606.03206</a>, <a href="http://dx.doi.org/10.1016/j.geomphys.2016.11.024">doi:10.1016/j.geomphys.2016.11.024</a>)</p> </li> </ul> </div> <h3 id="ReferencesEntropy">Entropy</h3> <p>The <a class="existingWikiWord" href="/nlab/show/entropy">entropy</a> of D-branes scales with the <em>square</em> of their number:</p> <p>Around equation (3) in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Steven+Gubser">Steven Gubser</a>, <a class="existingWikiWord" href="/nlab/show/Igor+Klebanov">Igor Klebanov</a> <a class="existingWikiWord" href="/nlab/show/Arkady+Tseytlin">Arkady Tseytlin</a>, <em>Coupling Constant Dependence in the Thermodynamics of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">N = 4</annotation></semantics></math> Supersymmetric Yang-Mills Theory</em>, Nucl. Phys. B534:202-222, 1998 (<a href="https://arxiv.org/abs/hep-th/9805156">arXiv:hep-th/9805156</a>)</li> </ul> <p>Around (3) in:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Igor+Klebanov">Igor Klebanov</a>, <em>From Threebranes to Large <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math> Gauge Theories</em> (<a href="https://arxiv.org/abs/hep-th/9901018">arXiv:hep-th/9901018</a>)</li> </ul> <p>Around equation (5.5) in:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Elias+Kiritsis">Elias Kiritsis</a>, T.R. Taylor, <em>Thermodynamics of D-brane Probes</em> (<a href="https://arxiv.org/abs/hep-th/9906048">arXiv:hep-th/9906048</a>)</li> </ul> <p>See also:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Mark+van+Raamsdonk">Mark van Raamsdonk</a>, <em><a href="http://www.physics.mcgill.ca/seminars/PSC_van-raamsdonk.html">The Amazing Matrix in String Theory</a></em> (2006):</p> <blockquote> <p>Most of the dramatic progress in string theory over the past decade has in some way involved the degrees of freedom of D-branes, solitonic objects in string theory whose low energy physics is described by quantum field theories living on the branes. Essential to many of the applications of D-brane physics is the fact that the number of degrees of freedom of a collection of branes scales not as the number of branes, but as the square of the number of branes.</p> </blockquote> </li> </ul> <p>Around (1.1) in:</p> <ul> <li>Qianqian Du, Michael Strickland, Ubaid Tantary, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒩</mi><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">\mathcal{N} = 4</annotation></semantics></math> supersymmetric Yang-Mills thermodynamics to order <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>λ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\lambda^2</annotation></semantics></math> (<a href="https://arxiv.org/abs/2105.02101">arXiv:2105.02101</a>)</li> </ul> <h3 id="for_rational_cft">For rational CFT</h3> <p>For exhaustive details on D-branes in 2-dimensional rational <a class="existingWikiWord" href="/nlab/show/CFT">CFT</a> see the references given at</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/FFRS-formalism">FFRS-formalism</a></li> </ul> <h3 id="branes_within_branes">Branes within branes</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/Michael+Douglas">Michael Douglas</a>, <em>Branes within Branes</em> (<a href="http://arxiv.org/abs/hep-th/9512077">arXiv:hep-th/9512077</a>)</li> </ul> <h3 id="for_topological_strings">For topological strings</h3> <p>A discussion of topological D-branes in the context of <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a> is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Anton+Kapustin">Anton Kapustin</a>, <em>Topological Field Theory, Higher Categories, and Their Applications</em> (<a href="http://arxiv.org/abs/1004.2307">arXiv:1004.2307</a>)</li> </ul> <h3 id="open_string_worldsheet_anomaly_cancellation">Open string worldsheet Anomaly cancellation</h3> <p>The need for <a class="existingWikiWord" href="/nlab/show/twisted+spin%5Ec+structures">twisted spin^c structures</a> as <a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a>-cancellaton condition on the <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> of D-branes was first discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>Anomalies in String Theory with D-Branes</em> (<a href="http://arxiv.org/abs/hep-th/9907189">arXiv:hep-th/9907189</a>)</li> </ul> <p>More details are in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Anton+Kapustin">Anton Kapustin</a>, <em>D-branes in a topologically nontrivial B-field</em> , Adv. Theor. Math. Phys. <p>4, no. 1, pp. 127–154 (2000), (<a href="http://arxiv.org/abs/hep-th/9909089">arXiv:hep-th/9909089</a>)</p> </li> </ul> <p>A clean review is provided in</p> <ul> <li>Kim Laine, <em>Geometric and topological aspects of Type IIB D-branes</em> (<a href="http://arxiv.org/abs/0912.0460">arXiv:0912.0460</a>)</li> </ul> <p>For more see at <em><a class="existingWikiWord" href="/nlab/show/Freed-Witten+anomaly+cancellation">Freed-Witten anomaly cancellation</a></em>.</p> <h3 id="relation_to_dirac_structures">Relation to Dirac structures</h3> <ul> <li id="AsakawaSasaWatamura">Tsuguhiko Asakawa, Shuhei Sasa, Satoshi Watamura, <em>D-branes in Generalized Geometry and Dirac-Born-Infeld Action</em> (<a href="http://arxiv.org/abs/1206.6964">arXiv:1206.6964</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on May 27, 2024 at 05:59:49. 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