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href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+net">local net</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/boson">boson</a>, <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/FQFT">functorial QFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem </li> <li> <a class="existingWikiWord" href="/nlab/show/extended+topological+quantum+field+theory">extended topological quantum field theory</a> </li> </ul> </li> </ul> </li> <li> Tools <ul> <li> <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-function+theory">geometric ∞-function theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+%28in+particle+phyiscs%29">models</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/fields+and+quanta+-+table">fields and quanta</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/GUT">Grand 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href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein mechanism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/integrable+systems">integrable systems</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/holonomic+quantum+fields">holonomic quantum fields</a> </li> </ul> </li> <li> Types of quantum field thories <ul> <li> <a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory and elliptic cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/CFT">CFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a>, <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>, <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixing</a> </li> <li> examples <ul> <li><a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a>, <a class="existingWikiWord" href="/nlab/show/QED">QED</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/electric+charge">electric charge</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>, <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spinors+in+Yang-Mills+theory">spinors in Yang-Mills theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a></li> <li><a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a></li> <li><a class="existingWikiWord" href="/nlab/show/RR+field">RR field</a></li> <li><a class="existingWikiWord" href="/nlab/show/first-order+formulation+of+gravity">first-order formulation of gravity</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/general+covariance">general covariance</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gravity+as+a+BF-theory">gravity as a BF-theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/particle">particle</a>, <a class="existingWikiWord" href="/nlab/show/relativistic+particle">relativistic particle</a>, <a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental particle</a>, <a class="existingWikiWord" href="/nlab/show/spinning+particle">spinning particle</a>, <a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>, <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/membrane">membrane</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/AKSZ+theory">AKSZ theory</a> </li> </ul> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/string+theory">String Theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+theory+results+applied+elsewhere">string theory results applied elsewhere</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/number+theory+and+physics">number theory and physics</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Riemann+hypothesis+and+physics">Riemann hypothesis and physics</a></li> </ul> </li> </ul> <div> <a href="/nlab/edit/physicscontents">Edit this sidebar</a> </div></div></div> <h4 id="chernweil_theory"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Chern-Weil theory</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a> <a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Chern-Simons+theory">∞-Chern-Simons theory</a> <a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Wess-Zumino-Witten+theory">∞-Wess-Zumino-Witten theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory+introduction">∞-Chern-Weil theory introduction</a> </li> </ul> <h2 id="ingredients">Ingredients</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> <ul> <li><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+an+%28%E2%88%9E%2C1%29-topos">differential cohomology</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+theory">∞-Lie theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a>, <a class="existingWikiWord" href="/nlab/show/Weil+algebra">Weil algebra</a>, <a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a></li> </ul> </li> </ul> <h2 id="connection">Connection</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid+valued+differential+forms">∞-Lie algebroid valued differential forms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-connection+on+a+principal+%E2%88%9E-bundle">∞-connection on a principal ∞-bundle</a> </li> </ul> <h2 id="curvature">Curvature</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Bianchi+identity">Bianchi identity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/curvature+characteristic+form">curvature characteristic form</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/covariant+derivative">covariant derivative</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/secondary+characteristic+class">secondary characteristic class</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Chern-Simons+form">Chern-Simons form</a> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Chern-Weil+homomorphism">Chern-Weil homomorphism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Chern-Gauss-Bonnet+theorem">Chern-Gauss-Bonnet theorem</a> </li> </ul> <div> <a href="/nlab/edit/infinity-Chern-Weil+theory+-+contents">Edit this sidebar</a> </div></div></div> <h4 id="differential_cohomology">Differential cohomology</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a> <h2 id="ingredients">Ingredients</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> </li> </ul> <h2 id="connections_on_bundles">Connections on bundles</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a bundle</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/parallel+transport">parallel transport</a>, <a class="existingWikiWord" href="/nlab/show/holonomy">holonomy</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/curvature+characteristic+form">curvature characteristic form</a>, <a class="existingWikiWord" href="/nlab/show/Chern+character">Chern character</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Chern-Weil+homomorphism">Chern-Weil homomorphism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/secondary+characteristic+class">secondary characteristic class</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/differential+characteristic+class">differential characteristic class</a> </li> </ul> </li> </ul> <h2 id="higher_abelian_differential_cohomology">Higher abelian differential cohomology</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/differential+function+complex">differential function complex</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/differential+orientation">differential orientation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/differential+Thom+class">differential Thom class</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/differential+characters">differential characters</a>, </li> <li> <a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-bundle with connection</a>, </li> <li> <a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe with connection</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/differential+elliptic+cohomology">differential elliptic cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a> </li> </ul> <h2 id="higher_nonabelian_differential_cohomology">Higher nonabelian differential cohomology</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/connection+on+a+2-bundle">connection on a 2-bundle</a>, <a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">connection on an ∞-bundle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory+in+Smooth%E2%88%9EGrpd">Chern-Weil theory in Smooth∞Grpd</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Simons+theory">∞-Chern-Simons theory</a> </li> </ul> <h2 id="fiber_integration">Fiber integration</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/higher+holonomy">higher holonomy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/fiber+integration+in+differential+cohomology">fiber integration in differential cohomology</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/fiber+integration+in+ordinary+differential+cohomology">fiber integration in ordinary differential cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/fiber+integration+in+differential+K-theory">fiber integration in differential K-theory</a> </li> </ul> </li> </ul> <h2 id="application_to_gauge_theory">Application to gauge theory</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a>/<a class="existingWikiWord" href="/nlab/show/B-field">B-field</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/RR-field">RR-field</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">supergravity</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a> </li> </ul> <div> <a href="/nlab/edit/differential+cohomology+-+contents">Edit this sidebar</a> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <ul> <li><a href='#for_monopole_charges_in_electromagnetism'>For monopole charges in electromagnetism</a></li> <li><a href='#for_monopole_charges_in_nonabelian_yangmills_theory'>For monopole charges in non-abelian Yang-Mills theory</a></li> <li><a href='#for_rrfield_dbrane_charges_in_ktheory'>For RR-field D-brane charges in K-theory</a></li> <li><a href='#for_the_cfield_in_jtwisted_cohomotopy'>For the C-field in J-twisted Cohomotopy</a></li> </ul> <li><a href='#related_entries'>Related entries</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#for_the_electromagnetic_field'>For the electromagnetic field</a></li> <li><a href='#for_the_weak_nuclear_force_field'>For the weak nuclear force field</a></li> <li><a href='#ReferencesDBraneChargeQuantizationInOrdinaryCohomology'>D-brane charge quantization in ordinary cohomology</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='#for_the_cfield_in_ftheory'>For the C-field in F-theory</a></li> <li><a href='#for_the_cfield_in_mtheory'>For the C-field in M-theory</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <h3 id="for_monopole_charges_in_electromagnetism">For monopole charges in electromagnetism</h3> If the field of <a class="existingWikiWord" href="/nlab/show/electromagnetism">electromagnetism</a> serves as a <a class="existingWikiWord" href="/nlab/show/background+gauge+field">background gauge field</a> for electrically charged <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum</a> <a class="existingWikiWord" href="/nlab/show/particles">particles</a> it is subject to a quantization condition: Outside the locus of any <a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a> – for instance a magnetic <a class="existingWikiWord" href="/nlab/show/monopole">monopole</a> <a class="existingWikiWord" href="/nlab/show/topological+defect">topological defect</a> – the <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a> must be a <a class="existingWikiWord" href="/nlab/show/circle+bundle+with+connection">connection on a principal U(1) bundle</a> whose <a class="existingWikiWord" href="/nlab/show/first+Chern+class">first Chern class</a> is the discrete measure for the units of magnetic charge. Equivalently this means that the <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a> is a <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> in <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a> of degree 2. In the underlying topological sector (“<a class="existingWikiWord" href="/nlab/show/monopole">monopole</a>”/“<a class="existingWikiWord" href="/nlab/show/instantons">instantons</a>”-sector) this is <a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a> in degree-2, whose <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> is equivalently the <a class="existingWikiWord" href="/nlab/show/infinite+complex+projective+space">infinite complex projective space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>ℂ</mi><msup><mi>P</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">B U(1) \simeq \mathbb{C}P^\infty</annotation></semantics></math>: <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> <center> <img src="https://ncatlab.org/nlab/files/DiracChargeQuantizationII.jpg" width="640" /> </center> This goes back to an insight due to <a href="#Dirac31">Dirac 31</a>. See <a href="#Heras18">Heras 18</a> for traditional elementary review, but see <a href="#Alvarez85">Alvarez 85</a>, <a href="#Frankel">Frankel</a> and <a href="#MangiarottiSardanashvily00">Mangiarotti-Sardanashvily 00</a> for exposition of the modern picture in terms of <a class="existingWikiWord" href="/nlab/show/fiber+bundles+in+physics">fiber bundles in physics</a>. See <a href="#Freed00">Freed 00, Section 2</a> for review in terms of <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a> with outlook to generalization to <a class="existingWikiWord" href="/nlab/show/higher+gauge+fields">higher gauge fields</a> in <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> (more on which in the references <a href="#ReferencesForBFieldAndRRFields">below</a>). A closely related phenomenon is the magnetic flux quantization in <a class="existingWikiWord" href="/nlab/show/type+II+superconductors">type II superconductors</a>, see <a href="superconductivity#MagneticFluxQuantizationInTypeIISuperconductors">there</a>. On the locus of the magnetic charge itself the situation is more complex. There the <a class="existingWikiWord" href="/nlab/show/magnetic+current">magnetic current</a> is given by a cocycle in <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a> of degree 3 (with compact support) and now the electromagnetic field is a connection on a <a class="existingWikiWord" href="/nlab/show/twisted+bundle">twisted bundle</a> (<a href="#Freed00">Freed 00, Section 2</a>). <h3 id="for_monopole_charges_in_nonabelian_yangmills_theory">For monopole charges in non-abelian Yang-Mills theory</h3> A similar charge quantization condition govers <a class="existingWikiWord" href="/nlab/show/monopoles">monopoles</a> in <a class="existingWikiWord" href="/nlab/show/SU%282%29">SU(2)</a>-<a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a>, see at <a class="existingWikiWord" href="/nlab/show/moduli+space+of+monopoles">moduli space of monopoles</a>. Here the <a class="existingWikiWord" href="/nlab/show/Atiyah-Hitchin+charge+quantization">Atiyah-Hitchin charge quantization</a> (<a href="#AtiyahHitchin88">Atiyah-Hitchin 88, Theorem 2.10</a>) says that the <a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a> of <a class="existingWikiWord" href="/nlab/show/monopoles">monopoles</a> is the <a class="existingWikiWord" href="/nlab/show/holomorphic+function">complex</a>-<a class="existingWikiWord" href="/nlab/show/rational+function">rational</a> 2-<a class="existingWikiWord" href="/nlab/show/Cohomotopy">Cohomotopy</a> of an asymptotic 2-sphere enclosing the monopoles: <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> <center> <img src="https://ncatlab.org/nlab/files/AtiyahHitchinChargeQuantizationII.jpg" width="640" /> </center> <h3 id="for_rrfield_dbrane_charges_in_ktheory">For RR-field D-brane charges in K-theory</h3> See at <a class="existingWikiWord" href="/nlab/show/D-brane+charge+quantization+in+K-theory">D-brane charge quantization in K-theory</a> <h3 id="for_the_cfield_in_jtwisted_cohomotopy">For the C-field in J-twisted Cohomotopy</h3> See at <a href="supergravity+C-field#ShiftedCFieldFluxQuantizationCondition">supergravity C-field – shifted flux quantization condition</a> <h2 id="related_entries">Related entries</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/flux">flux</a>, <a class="existingWikiWord" href="/nlab/show/flux+tube">flux tube</a>, <a class="existingWikiWord" href="/nlab/show/flux+compactification">flux compactification</a>, <a class="existingWikiWord" href="/nlab/show/uncertainty+of+fluxes">uncertainty of fluxes</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+flux+quantization">geometry of physics – flux quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/fiber+bundles+in+physics">fiber bundles in physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/non-perturbative+quantum+field+theory">non-perturbative quantum field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/monopole">monopole</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Dirac+monopole">Dirac monopole</a>, <a class="existingWikiWord" href="/nlab/show/Yang+monopole">Yang monopole</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/moduli+space+of+monopoles">moduli space of monopoles</a> </li> </ul> <h2 id="references">References</h2> See also the references at <a class="existingWikiWord" href="/nlab/show/fiber+bundles+in+physics">fiber bundles in physics</a>. <h3 id="for_the_electromagnetic_field">For the electromagnetic field</h3> The original argument for charge quantization of the <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a>: <ul> <li id="Dirac31"><a class="existingWikiWord" href="/nlab/show/P.A.M.+Dirac">P.A.M. Dirac</a>, Quantized Singularities in the Electromagnetic Field, Proceedings of the Royal Society A 133 (1931) 60-72 [<a href="https://doi.org/10.1098/rspa.1931.0130">doi:10.1098/rspa.1931.0130</a>]</li> </ul> Further early discussion <ul> <li> <a class="existingWikiWord" href="/nlab/show/Julian+Schwinger">Julian Schwinger</a>, Magnetic Charge and Quantum Field Theory, Phys. Rev. 144 (1966) 1087 [<a href="https://doi.org/10.1103/PhysRev.144.1087">doi:10.1103/PhysRev.144.1087</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Daniel+Zwanziger">Daniel Zwanziger</a>, Quantum Field Theory of Particles with Both Electric and Magnetic Charges, Phys. Rev. 176 (1968) 1489 [<a href="https://doi.org/10.1103/PhysRev.176.1489">doi:10.1103/PhysRev.176.1489</a>] </li> </ul> Understanding of charge quantization in terms of <a class="existingWikiWord" href="/nlab/show/%C4%8Cech+cohomology">Čech cohomology</a>: <ul> <li id="Alvarez85a"> <a class="existingWikiWord" href="/nlab/show/Orlando+Alvarez">Orlando Alvarez</a>, Cohomology and Field Theory, talk at: Symposium on Anomalies, Geometry, Topology, Argonne IL (28-30 March 1985) [<a href="https://inspirehep.net/conferences/965785">inspire:965785</a>, <a href="https://lib-extopc.kek.jp/preprints/PDF/1985/8507/8507262.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Alvarez-CohomologyAndFieldTheory.pdf" title="pdf">pdf</a>] </li> <li id="Alvarez85"> <a class="existingWikiWord" href="/nlab/show/Orlando+Alvarez">Orlando Alvarez</a>, Topological quantization and cohomology, Comm. Math. Phys. 100 2 (1985) 279-309 [<a href="https://projecteuclid.org/euclid.cmp/1103943448">euclid:cmp/1103943448</a>] </li> </ul> reviewed in: <ul> <li>Guillermo A. González, Cohomologa de Cĕch y Cuantización Topológica de Parámetros Físicos, Revista de la Academia Colombiana de Ciencias Exactas, Físicas y Naturales 47 182 (2023) 51-71 [<a href="https://doi.org/10.18257/raccefyn.1782">doi:10.18257/raccefyn.1782</a>, <a class="existingWikiWord" href="/nlab/files/Gonzales-TopologicalQuantization.pdf" title="pdf">pdf</a>]</li> </ul> General review and further discussion: <ul> <li id="Brylinski93"> <a class="existingWikiWord" href="/nlab/show/Jean-Luc+Brylinski">Jean-Luc Brylinski</a>, §7.1 in: Loop spaces, characteristic classes and geometric quantization, Birkhäuser (1993) [<a href="https://link.springer.com/book/10.1007/978-0-8176-4731-5">doi:10.1007/978-0-8176-4731-5</a>] </li> <li id="Frankel"> <a class="existingWikiWord" href="/nlab/show/Theodore+Frankel">Theodore Frankel</a>, section 16.4e of: <a class="existingWikiWord" href="/nlab/show/The+Geometry+of+Physics+-+An+Introduction">The Geometry of Physics - An Introduction</a>, Cambridge University Press (1997, 2004, 2012) [<a href="https://doi.org/10.1017/CBO9781139061377">doi:10.1017/CBO9781139061377</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Luis+Alvarez-Gaum%C3%A9">Luis Alvarez-Gaumé</a>, Frederic Zamora, Section II in: Duality in Quantum Field Theory (and String Theory), AIP Conference Proceedings 423 (1998) 46–83 [<a href="https://arxiv.org/abs/hep-th/9709180">arXiv:hep-th/9709180</a>, <a href="https://doi.org/10.1063/1.55087">doi:10.1063/1.55087</a>] </li> <li id="Freed00"> <a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, Section 2 of: <a class="existingWikiWord" href="/nlab/show/Dirac+charge+quantization+and+generalized+differential+cohomology">Dirac charge quantization and generalized differential cohomology</a>, Surveys in Differential Geometry 7, Int. Press (2000) 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> <li id="MangiarottiSardanashvily00"> L. Mangiarotti, <a class="existingWikiWord" href="/nlab/show/Gennadi+Sardanashvily">Gennadi Sardanashvily</a>, Connections in Classical and Quantum Field Theory, World Scientific, 2000 (<a href="https://doi.org/10.1142/2524">doi:10.1142/2524</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Yakov+Shnir">Yakov Shnir</a>, Section 2.1 of: Magnetic Monopoles, Springer 2005 (<a href="https://www.springer.com/gp/book/9783540252771">ISBN:978-3-540-29082-7</a>) </li> <li> <a href="#FreedMooreSegal07">Freed, Moore & Segal 2007, p. 7</a> </li> <li id="Heras18"> Ricardo Heras, Dirac quantisation condition: a comprehensive review, Contemp. Phys. 59, 331 (2018) (<a href="https://arxiv.org/abs/1810.13403">arXiv:1810.13403</a>) </li> </ul> See also <ul> <li>Wikipedia, <a href="https://en.wikipedia.org/wiki/Magnetic_monopole#Dirac%27s_quantization">Dirac’s quantization</a></li> </ul> Arguments for electric charge quantization from <a class="existingWikiWord" href="/nlab/show/anomaly+cancellation">anomaly cancellation</a> in the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a>: <ul> <li>K. S. Babu and Rabindra N. Mohapatra, Quantization of electric charge from anomaly constraints and a Majorana neutrino, Phys. Rev. D 41 (1990) 271 [<a href="https://doi.org/10.1103/PhysRevD.41.271">doi:10.1103/PhysRevD.41.271</a>]</li> </ul> Consideration of combined magnetic and electric charge quantization in <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> (cf. <a class="existingWikiWord" href="/nlab/show/duality-symmetric+electromagnetism">duality-symmetric electromagnetism</a>): <ul> <li id="FreedMooreSegal07"> <a class="existingWikiWord" href="/nlab/show/Daniel+S.+Freed">Daniel S. Freed</a>, <a class="existingWikiWord" href="/nlab/show/Gregory+W.+Moore">Gregory W. Moore</a>, <a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a>, The Uncertainty of Fluxes, Commun. Math. Phys. 271 (2007) 247-274 [<a href="https://arxiv.org/abs/hep-th/0605198">arXiv:hep-th/0605198</a>, <a href="https://doi.org/10.1007/s00220-006-0181-3">doi:10.1007/s00220-006-0181-3</a>] </li> <li id="BBSS15"> <a class="existingWikiWord" href="/nlab/show/Christian+Becker">Christian Becker</a>, <a class="existingWikiWord" href="/nlab/show/Marco+Benini">Marco Benini</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Schenkel">Alexander Schenkel</a>, <a class="existingWikiWord" href="/nlab/show/Richard+Szabo">Richard Szabo</a>, Rem. 2.3 in: Abelian duality on globally hyperbolic spacetimes, Commun. Math. Phys. 349 (2017) 361-392 [<a href="https://arxiv.org/abs/1511.00316">arXiv:1511.00316</a>, <a href="https://doi.org/10.1007/s00220-016-2669-9">doi:10.1007/s00220-016-2669-9</a>] </li> <li id="LazaroiuShahbazi22"> <a class="existingWikiWord" href="/nlab/show/Calin+Lazaroiu">Calin Lazaroiu</a>, <a class="existingWikiWord" href="/nlab/show/Carlos+S.+Shahbazi">Carlos S. Shahbazi</a>, around Def. 1.16 in: The duality covariant geometry and DSZ quantization of abelian gauge theory, Advances in Theoretical and Mathematical Physics 26 (2022) 2213–2312 [<a href="https://arxiv.org/abs/2101.07236">arXiv:2101.07236</a>, <a href="https://dx.doi.org/10.4310/ATMP.2022.v26.n7.a5">doi:10.4310/ATMP.2022.v26.n7.a5</a>] </li> <li id="LazaroiuShahbazi23"> <a class="existingWikiWord" href="/nlab/show/Calin+Lazaroiu">Calin Lazaroiu</a>, <a class="existingWikiWord" href="/nlab/show/Carlos+S.+Shahbazi">Carlos S. Shahbazi</a>, around (3) in: The geometry and DSZ quantization of four-dimensional supergravity, Letters in Mathematical Physics 113 4 (2023) [<a href="https://arxiv.org/abs/2101.07778">arXiv:2101.07778</a>, <a href="https://doi.org/10.1007/s11005-022-01626-y">doi:10.1007/s11005-022-01626-y</a>] </li> </ul> <h3 id="for_the_weak_nuclear_force_field">For the weak nuclear force field</h3> Discussion of the <a class="existingWikiWord" href="/nlab/show/moduli+space+of+monopoles">moduli space of monopoles</a> for <a class="existingWikiWord" href="/nlab/show/SU%282%29">SU(2)</a>-<a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a> (<a class="existingWikiWord" href="/nlab/show/weak+nuclear+force">weak nuclear force</a>): <ul> <li id="AtiyahHitchin88"> <a class="existingWikiWord" href="/nlab/show/Michael+Atiyah">Michael Atiyah</a>, <a class="existingWikiWord" href="/nlab/show/Nigel+Hitchin">Nigel Hitchin</a>, The geometry and dynamics of magnetic monopoles M. B. Porter Lectures. Princeton University Press, Princeton, NJ, 1988 (<a href="https://www.jstor.org/stable/j.ctt7zv206">jstor:j.ctt7zv206</a>) </li> <li id="Segal79"> <a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a>, The topology of spaces of rational functions, Acta Math. Volume 143 (1979), 39-72 (<a href="https://projecteuclid.org/euclid.acta/1485890033">euclid:1485890033</a>) </li> </ul> <h3 id="ReferencesDBraneChargeQuantizationInOrdinaryCohomology">D-brane charge quantization in ordinary cohomology</h3> Discussion of charge quantization of <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> in (<a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative</a>) <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+Figueroa-O%E2%80%99Farrill">José Figueroa-O’Farrill</a>, <a class="existingWikiWord" href="/nlab/show/Sonia+Stanciu">Sonia Stanciu</a>, D-brane charge, flux quantisation and relative (co)homology, JHEP 0101 (2001) 006 [<a href="https://arxiv.org/abs/hep-th/0008038">arXiv:hep-th/0008038</a>, <a href="https://doi.org/10.1088/1126-6708/2001/01/006">doi:10.1088/1126-6708/2001/01/006</a>] </li> <li> Kazuya Yonekura, Constraints on the topology of Type IIB string theory [<a href="https://arxiv.org/abs/2403.04127">arXiv:2403.04127</a>, <a href="https://inspirehep.net/literature/2767009">InSpire:2767009</a>] </li> </ul> <div> <h3 id="DBraneChargeQuantizationInTopologicalKTheory">D-brane charge quantization in topological K-theory</h3> 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>: <h4 id="origin_and_basics">Origin and basics</h4> 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>: <ul> <li id="GreenHarveyMoore97"> <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>, I-Brane Inflow and Anomalous Couplings on D-Branes, Class. Quant. Grav. 14 (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>] </li> <li id="MinasianMoore97"> <a class="existingWikiWord" href="/nlab/show/Ruben+Minasian">Ruben Minasian</a>, <a class="existingWikiWord" href="/nlab/show/Gregory+Moore">Gregory Moore</a>, K-theory and Ramond-Ramond charge, 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>] </li> </ul> Further early discussion: <ul> <li id="Witten98"> <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, D-Branes And K-Theory, 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>] </li> <li id="Horava98"> <a class="existingWikiWord" href="/nlab/show/Petr+Ho%C5%99ava">Petr Hořava</a>, Type IIA D-Branes, K-Theory, and Matrix Theory (1998). (<a href="https://arxiv.org/abs/hep-th/9812135">hep-th/9812135</a>). </li> <li id="FreedHopkins00"> <a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, On Ramond-Ramond fields and K-theory, 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>] </li> </ul> 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: <ul> <li id="Freed00"><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <a class="existingWikiWord" href="/nlab/show/Dirac+charge+quantization+and+generalized+differential+cohomology">Dirac charge quantization and generalized differential cohomology</a>, 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> Here: <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 <a class="existingWikiWord" href="/nlab/show/Sen%27s+conjecture">Sen's conjecture</a> 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> 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>: <ul> <li id="MaldacenaMooreSeiberg01"> <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>, D-Brane Instantons and K-Theory Charges, JHEP 0111:062,2001 (<a href="http://arxiv.org/abs/hep-th/0108100">arXiv:hep-th/0108100</a>) </li> <li id="EvslinSati06"> <a class="existingWikiWord" href="/nlab/show/Jarah+Evslin">Jarah Evslin</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, Can D-Branes Wrap Nonrepresentable Cycles?, JHEP0610:050,2006 (<a href="http://arxiv.org/abs/hep-th/0607045">arXiv:hep-th/0607045</a>) </li> </ul> Specifically for D-branes in <a class="existingWikiWord" href="/nlab/show/WZW+models">WZW models</a> see <ul> <li><a class="existingWikiWord" href="/nlab/show/Peter+Bouwknegt">Peter Bouwknegt</a>, A note on equality of algebraic and geometric D-brane charges in WZW models (<a href="http://people.physics.anu.edu.au/~drt105/papers/BR0312259.pdf">pdf</a>)</li> </ul> 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: <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>, Brane Transfer Operations and T-Duality of Non-BPS States, 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> 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>): <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>, D-branes, Matrix Theory and K-homology, 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> 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>: <ul> <li> <a href="#Witten98">Witten 98, Sec. 5.3</a> (for <a class="existingWikiWord" href="/nlab/show/torsion+subgroup">torsion</a> twists) </li> <li> <a class="existingWikiWord" href="/nlab/show/Peter+Bouwknegt">Peter Bouwknegt</a>, <a class="existingWikiWord" href="/nlab/show/Varghese+Mathai">Varghese Mathai</a>, D-branes, B-fields and twisted K-theory, Int. J. Mod. Phys. A 16 (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>] </li> </ul> 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: <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>, Orientifold Précis in: <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a> (eds.) <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> 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> 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> <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>, Ramond-Ramond fields and twisted differential K-theory, Advances in Theoretical and Mathematical Physics 26 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> 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>): <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>, Twisted differential KO-theory (<a href="https://arxiv.org/abs/1905.09085">arXiv:1905.09085</a>)</li> </ul> <h4 id="reviews">Reviews</h4> <ul> <li id="OlsenSzabo00"> Kasper Olsen, <a class="existingWikiWord" href="/nlab/show/Richard+Szabo">Richard Szabo</a>, Brane Descent Relations in K-theory, Nucl.Phys. B566 (2000) 562-598 (<a href="https://arxiv.org/abs/hep-th/9904153">arXiv:hep-th/9904153</a>) </li> <li> Kasper Olsen, <a class="existingWikiWord" href="/nlab/show/Richard+Szabo">Richard Szabo</a>, Constructing D-Branes from K-Theory, Adv. Theor. Math. Phys. 3 (1999) 889-1025 (<a href="https://arxiv.org/abs/hep-th/9907140">arXiv:hep-th/9907140</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/John+Schwarz">John Schwarz</a>, TASI Lectures on Non-BPS D-Brane Systems (<a href="https://arxiv.org/abs/hep-th/9908144">arXiv:hep-th/9908144</a>) </li> <li id="Witten00"> <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, Overview Of K-Theory Applied To Strings, Int. J. Mod. Phys. A16:693-706, 2001 (<a href="https://arxiv.org/abs/hep-th/0007175">arXiv:hep-th/0007175</a>) </li> <li id="Moore02"> <a class="existingWikiWord" href="/nlab/show/Greg+Moore">Greg Moore</a>, K-Theory from a physical perspective, in: <a class="existingWikiWord" href="/nlab/show/Ulrike+Tillmann">Ulrike Tillmann</a> (ed.) Topology, Geometry and Quantum Field Theory, 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>) </li> <li id="Manjarin04"> <a class="existingWikiWord" href="/nlab/show/Juan+Jos%C3%A9+Manjar%C3%ADn">Juan José Manjarín</a>, Topics on D-brane charges with B-fields, Int. J. Geom. Meth. Mod. Phys. 1 (2004) (<a href="http://arxiv.org/abs/hep-th/0405074">arXiv:hep-th/0405074</a>) </li> <li id="Evslin06"> <a class="existingWikiWord" href="/nlab/show/Jarah+Evslin">Jarah Evslin</a>, What Does(n’t) K-theory Classify?, 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>) </li> <li id="Fredenhagen08"> <a class="existingWikiWord" href="/nlab/show/Stefan+Fredenhagen">Stefan Fredenhagen</a>, Physical Background to the K-Theory Classification of D-Branes: Introduction and References (<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>, <a class="existingWikiWord" href="/nlab/show/Basic+Bundle+Theory+and+K-Cohomology+Invariants">Basic Bundle Theory and K-Cohomology Invariants</a>, 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>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Fabio+Ruffino">Fabio Ruffino</a>, Topics on topology and superstring theory (<a href="http://arxiv.org/abs/0910.4524">arXiv:0910.4524</a>) </li> <li id="SatiSchreiber24"> <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>: <a class="existingWikiWord" href="/schreiber/show/Flux+Quantization">Flux Quantization</a>, in <a class="existingWikiWord" href="/nlab/show/Encyclopedia+of+Mathematical+Physics+2nd+ed">Encyclopedia of Mathematical Physics 2nd ed</a>, Elsevier (2024) [<a href="https://arxiv.org/abs/2402.18473">arXiv:2402.18473</a>] </li> </ul> Amplification of <a class="existingWikiWord" href="/nlab/show/torsion+subgroup">torsion</a>-charges implied by charge quantization in Ktheory <ul> <li> <a class="existingWikiWord" href="/nlab/show/Volker+Braun">Volker Braun</a>, K-Theory Torsion &lbrack;<a href="https://arxiv.org/abs/hep-th/0005103">arXiv:hep-th/0005103</a>&rbrack; </li> <li> <a class="existingWikiWord" href="/nlab/show/Ilka+Brunner">Ilka Brunner</a>, <a class="existingWikiWord" href="/nlab/show/Jacques+Distler">Jacques Distler</a>, Torsion D-Branes in Nongeometrical Phases, Adv. Theor. Math. Phys. 5 (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>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Ilka+Brunner">Ilka Brunner</a>, <a class="existingWikiWord" href="/nlab/show/Jacques+Distler">Jacques Distler</a>, Rahul Mahajan, Return of the Torsion D-Branes, Adv. Theor. Math. Phys. 5 (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>] </li> </ul> Review of D-branes charge seen in <a class="existingWikiWord" href="/nlab/show/KK-theory">KK-theory</a>: <ul> <li id="Szabo"><a class="existingWikiWord" href="/nlab/show/Richard+Szabo">Richard Szabo</a>, D-branes and bivariant K-theory, Noncommutative Geometry and Physics 3 1 (2013): 131. (<a href="http://arxiv.org/abs/0809.3029">arXiv:0809.3029</a>)</li> </ul> based on <ul> <li id="ReisSzabo05"> Rui Reis, <a class="existingWikiWord" href="/nlab/show/Richard+Szabo">Richard Szabo</a>, Geometric K-Homology of Flat D-Branes , Commun. Math. Phys. 266 (2006) 71-122 &lbrack;<a href="https://arxiv.org/abs/hep-th/0507043">arXiv:hep-th/0507043</a>&rbrack; </li> <li id="BrodzkiMathaiRosenbergSzabo06"> <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>, D-Branes, RR-Fields and Duality on Noncommutative Manifolds, Commun. Math. Phys. 277 (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>] </li> <li id="BMRS2"> <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>, Noncommutative correspondences, duality and D-branes in bivariant K-theory, Adv. Theor. Math. Phys. 13:497-552, 2009 (<a href="http://arxiv.org/abs/0708.2648">arXiv:0708.2648</a>) </li> <li> <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>, D-branes, KK-theory and duality on noncommutative spaces, J. Phys. Conf. Ser. 103 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>] </li> </ul> 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 <ul> <li><a class="existingWikiWord" href="/nlab/show/Richard+Szabo">Richard Szabo</a>, D-Branes, Tachyons and K-Homology, 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> But there remain conceptual issues with the proposal that D-brane charge is in K-theory, as highlighted in <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>, section 4.5.2 and 4.6.5 of Triples, Fluxes, and Strings, Adv. Theor. Math. Phys. 4 (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>] </li> <li id="Evslin06"> <a class="existingWikiWord" href="/nlab/show/Jarah+Evslin">Jarah Evslin</a>, section 8 of: What Does(n’t) K-theory Classify?, Second Modave Summer School in Mathematical Physics [<a href="https://arxiv.org/abs/hep-th/0610328">arXiv:hep-th/0610328</a>] </li> </ul> 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: <ul> <li id="FredenhagenQuella05"> <a class="existingWikiWord" href="/nlab/show/Stefan+Fredenhagen">Stefan Fredenhagen</a>, <a class="existingWikiWord" href="/nlab/show/Thomas+Quella">Thomas Quella</a>, Generalised permutation branes, 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>] <blockquote> 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. </blockquote> </li> </ul> 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 <ul> <li id="Erler13"><a class="existingWikiWord" href="/nlab/show/Theodore+Erler">Theodore Erler</a>, Analytic Solution for Tachyon Condensation in Berkovits’ Open Superstring Field Theory, 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> which, however, concludes (on <a href="https://arxiv.org/pdf/1308.4400.pdf#page=33">p. 32</a>) with: <blockquote> 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. </blockquote> See also <ul> <li id="Erler19"><a class="existingWikiWord" href="/nlab/show/Theodore+Erler">Theodore Erler</a>, Four Lectures on Analytic Solutions in Open String Field Theory (<a href="https://arxiv.org/abs/1912.00521">arXiv:1912.00521</a>, <a href="https://inspirehep.net/literature/1768105">spire:1768105</a>)</li> </ul> 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>: <blockquote> “Are there topological invariants of the open string star algebra representing D-brane charges?” </blockquote> <h4 id="for_orbifolds_in_equivariant_ktheory">For orbifolds in equivariant K-theory</h4> 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 <ul> <li><a href="#Witten98">Witten 98, section 5.1</a></li> </ul> It was pointed out that only a subgroup of equivariant K-theory can be physically relevant in <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: Triples, Fluxes, and Strings, Adv.Theor.Math.Phys. 4 (2002) 995-1186 (<a href="https://arxiv.org/abs/hep-th/0103170">arXiv:hep-th/0103170</a>)</li> </ul> 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: <ul> <li> Hugo García-Compeán, D-branes in orbifold singularities and equivariant K-theory, Nucl.Phys. B557 (1999) 480-504 (<a href="https://arxiv.org/abs/hep-th/9812226">arXiv:hep-th/9812226</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Matthias+Gaberdiel">Matthias Gaberdiel</a>, <a class="existingWikiWord" href="/nlab/show/Bogdan+Stefanski">Bogdan Stefanski</a>, Dirichlet Branes on Orbifolds, Nucl.Phys.B578:58-84, 2000 (<a href="https://arxiv.org/abs/hep-th/9910109">arXiv:hep-th/9910109</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Igor+Kriz">Igor Kriz</a>, Leopoldo A. Pando Zayas, Norma Quiroz, Comments on D-branes on Orbifolds and K-theory, Int. J. Mod. Phys. A 23 (2008) 933-974 [<a href="https://arxiv.org/abs/hep-th/0703122">arXiv:hep-th/0703122</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Richard+Szabo">Richard Szabo</a>, <a class="existingWikiWord" href="/nlab/show/Alessandro+Valentino">Alessandro Valentino</a>, Ramond-Ramond Fields, Fractional Branes and Orbifold Differential K-Theory, Commun.Math.Phys.294:647-702, 2010 (<a href="https://arxiv.org/abs/0710.2773">arXiv:0710.2773</a>) </li> </ul> 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: 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 <ul> <li><a href="#Witten98">Witten 98, section 5</a></li> </ul> and was then re-amplified in <ul> <li id="Gukov99"> <a class="existingWikiWord" href="/nlab/show/Sergei+Gukov">Sergei Gukov</a>, K-Theory, Reality, and Orientifolds, Commun.Math.Phys. 210 (2000) 621-639 (<a href="https://arxiv.org/abs/hep-th/9901042">arXiv:hep-th/9901042</a>) </li> <li id="BergmanGimonSugimoto01"> <a class="existingWikiWord" href="/nlab/show/Oren+Bergman">Oren Bergman</a>, E. Gimon, <a class="existingWikiWord" href="/nlab/show/Shigeki+Sugimoto">Shigeki Sugimoto</a>, Orientifolds, RR Torsion, and K-theory, JHEP 0105:047, 2001 (<a href="https://arxiv.org/abs/hep-th/0103183">arXiv:hep-th/0103183</a>) </li> </ul> With further developments in <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>, Type I D-branes in an H-flux and twisted KO-theory, JHEP 0311 (2003) 053 (<a href="https://arxiv.org/abs/hep-th/0310164">arXiv:hep-th/0310164</a>)</li> </ul> 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 <ul> <li> N. Quiroz, <a class="existingWikiWord" href="/nlab/show/Bogdan+Stefanski">Bogdan Stefanski</a>, Dirichlet Branes on Orientifolds, Phys.Rev. D66 (2002) 026002 (<a href="https://arxiv.org/abs/hep-th/0110041">arXiv:hep-th/0110041</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Volker+Braun">Volker Braun</a>, <a class="existingWikiWord" href="/nlab/show/Bogdan+Stefanski">Bogdan Stefanski</a>, Orientifolds and K-theory (<a href="https://arxiv.org/abs/hep-th/0206158">arXiv:hep-th/0206158</a>) </li> <li> H. Garcia-Compean, W. Herrera-Suarez, B. A. Itza-Ortiz, O. Loaiza-Brito, D-Branes in Orientifolds and Orbifolds and Kasparov KK-Theory, JHEP 0812:007, 2008 (<a href="https://arxiv.org/abs/0809.4238">arXiv:0809.4238</a>) </li> </ul> 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 <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>, <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, Adv.Theor.Math.Phys.6:1031-1134,2003 (<a href="http://arxiv.org/abs/hep-th/0005090">arXiv:hep-th/0005090</a>), summarised in A Derivation of K-Theory from M-Theory (<a href="http://arxiv.org/abs/hep-th/0005091">arXiv:hep-th/0005091</a>)</li> </ul> See also <ul> <li id="GarciaUranga05">Inaki Garcia-Etxebarria, <a class="existingWikiWord" href="/nlab/show/Angel+Uranga">Angel Uranga</a>, From F/M-theory to K-theory and back, JHEP 0602:008,2006 (<a href="https://arxiv.org/abs/hep-th/0510073">arXiv:hep-th/0510073</a>)</li> </ul> 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: <ul> <li id="MathaiSati03"> <a class="existingWikiWord" href="/nlab/show/Varghese+Mathai">Varghese Mathai</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, 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, JHEP0403:016,2004 (<a href="http://arxiv.org/abs/hep-th/0312033">arXiv:hep-th/0312033</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <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>, 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>) </li> </ul> </div> <h3 id="for_the_cfield_in_ftheory">For the C-field in F-theory</h3> Discussion of <a class="existingWikiWord" href="/nlab/show/shifted+C-field+flux+quantization">shifted C-field flux quantization</a> of the <a class="existingWikiWord" href="/nlab/show/C-field">C-field</a> in <a class="existingWikiWord" href="/nlab/show/F-theory">F-theory</a>: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Andres+Collinucci">Andres Collinucci</a>, <a class="existingWikiWord" href="/nlab/show/Raffaele+Savelli">Raffaele Savelli</a>, On Flux Quantization in F-Theory, J. High Energ. Phys. 2012 15 (2012) [<a href="https://arxiv.org/abs/1011.6388">arXiv:1011.6388</a>, <a href="https://doi.org/10.1007/JHEP02(2012)015">doi:10.1007/JHEP02(2012)015</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Andres+Collinucci">Andres Collinucci</a>, <a class="existingWikiWord" href="/nlab/show/Raffaele+Savelli">Raffaele Savelli</a>, On Flux Quantization in F-Theory II: Unitary and Symplectic Gauge Groups, J. High Energ. Phys. 2012 94 (2012) [<a href="https://arxiv.org/abs/1203.4542">arXiv:1203.4542</a>, <a href="https://doi.org/10.1007/JHEP08(2012)094">doi:10.1007/JHEP08(2012)094</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Raffaele+Savelli">Raffaele Savelli</a>, Flux Quantization in F-theory and Freed-Witten anomaly, talk at StringMath (2012) [<a href="https://www.hcm.uni-bonn.de/fileadmin/stringmath2012/parallel_talks/savelli.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Savelli-FTheoryFluxQuantization.pdf" title="pdf">pdf</a>] </li> </ul> <h3 id="for_the_cfield_in_mtheory">For the C-field in M-theory</h3> Discussion of <a class="existingWikiWord" href="/nlab/show/shifted+C-field+flux+quantization">shifted C-field flux quantization</a> of the <a class="existingWikiWord" href="/nlab/show/C-field">C-field</a> in <a class="existingWikiWord" href="/nlab/show/D%3D11+supergravity">D=11 supergravity</a>/<a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a>: <ul> <li id="DFM"> E. Diaconescu, <a class="existingWikiWord" href="/nlab/show/Dan+Freed">Dan Freed</a>, <a class="existingWikiWord" href="/nlab/show/Greg+Moore">Greg Moore</a>, The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>-theory 3-form 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, chapter in <a class="existingWikiWord" href="/nlab/show/Haynes+Miller">Haynes Miller</a>, <a class="existingWikiWord" href="/nlab/show/Douglas+Ravenel">Douglas Ravenel</a> (eds.) Elliptic Cohomology Geometry, Applications, and Higher Chromatic Analogues, Cambridge University Press 2007 (<a href="http://arxiv.org/abs/hep-th/0312069">arXiv:hep-th/0312069</a>, <a href="https:
//doi.org/10.1017/CBO9780511721489">doi:10.1017/CBO9780511721489</a>) </li> <li id="FreedMoore04"> <a class="existingWikiWord" href="/nlab/show/Dan+Freed">Dan Freed</a>, <a class="existingWikiWord" href="/nlab/show/Greg+Moore">Greg Moore</a>, Setting the <a class="existingWikiWord" href="/nlab/show/quantum+integrand">quantum integrand</a> of M-theory, Communications in Mathematical Physics, Volume 263, Number 1, 89-132, (<a href="http://arxiv.org/abs/hep-th/0409135">arXiv:hep-th/0409135</a>, <a href="https://doi.org/10.1007/s00220-005-1482-7">doi:10.1007/s00220-005-1482-7</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Greg+Moore">Greg Moore</a>, Anomalies, Gauss laws, and Page charges in M-theory (<a href="http://arxiv.org/abs/hep-th/0409158">arXiv:hep-th/0409158</a>) </li> <li id="FSS12"> <a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/schreiber/show/The+moduli+3-stack+of+the+C-field">The moduli 3-stack of the C-field</a>, Communications in Mathematical Physics, Volume 333, Issue 1 (2015), Page 117-151, (<a href="http://arxiv.org/abs/1202.2455">arXiv:1202.2455</a>, <a href="http://link.springer.com/article/10.1007%2Fs00220-014-2228-1">DOI 10.1007/s00220-014-2228-1</a>) </li> <li id="FiSaSc"> <a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/schreiber/show/7d+Chern-Simons+theory+and+the+5-brane">M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory</a> Advances in Theoretical and Mathematical Physics, Volume 18, Number 2 (2014) p. 229?321 (<a href="http://arxiv.org/abs/1201.5277">arXiv:1201.5277</a>, <a href="https://dx.doi.org/10.4310/ATMP.2014.v18.n2.a1">doi:10.4310/ATMP.2014.v18.n2.a1</a>) </li> </ul> Discussion in <a class="existingWikiWord" href="/nlab/show/twisted+Cohomotopy">twisted Cohomotopy</a> (“<a class="existingWikiWord" href="/nlab/show/Hypothesis+H">Hypothesis H</a>”): <ul> <li> <a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/schreiber/show/Twisted+Cohomotopy+implies+M-theory+anomaly+cancellation">Twisted Cohomotopy implies M-theory anomaly cancellation</a> (<a href="https://arxiv.org/abs/1904.10207">arXiv:1904.10207</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/schreiber/show/Twisted+Cohomotopy+implies+M5+WZ+term+level+quantization">Twisted Cohomotopy implies M5 WZ term level quantization</a> (<a href="https://arxiv.org/abs/1906.07417">arXiv:1906.07417</a>) </li> </ul> and in <a class="existingWikiWord" href="/nlab/show/equivariant+Cohomotopy">equivariant Cohomotopy</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/schreiber/show/Equivariant+Cohomotopy+implies+orientifold+tadpole+cancellation">Equivariant Cohomotopy implies orientifold tadpole cancellation</a> (<a href="https://arxiv.org/abs/1909.12277">arXiv:1909.12277</a>)</li> </ul> </body></html> </div> <div class="revisedby"> Last revised on June 6, 2024 at 21:19:22. 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