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href="/nlab/show/moment+of+inertia">moment of inertia</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/dynamics+on+Lie+groups">dynamics on Lie groups</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/rigid+body+dynamics">rigid body dynamics</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Lagrangian+mechanics">Lagrangian mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a>, <a class="existingWikiWord" href="/nlab/show/state">state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange equations</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-symplectic+manifold">n-symplectic manifold</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/smooth+Lorentzian+manifold">smooth Lorentzian manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/special+relativity">special relativity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/dilaton+gravity">dilaton gravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a> </li> </ul> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/classical+field+theory">Classical field theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/classical+physics">classical physics</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+mechanics">classical mechanics</a></li> <li><a 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class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantization">Quantization</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/path+integral">path integral quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/semiclassical+approximation">semiclassical approximation</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">Quantum Field Theory</a> <ul> <li> Axiomatizations <ul> <li> <a class="existingWikiWord" href="/nlab/show/AQFT">algebraic QFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+net">local net</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/boson">boson</a>, <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/FQFT">functorial QFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem </li> <li> <a class="existingWikiWord" href="/nlab/show/extended+topological+quantum+field+theory">extended topological quantum field theory</a> </li> </ul> </li> </ul> </li> <li> Tools <ul> <li> <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-function+theory">geometric ∞-function theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+%28in+particle+phyiscs%29">models</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/fields+and+quanta+-+table">fields and quanta</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/GUT">Grand Unified Theories</a>, <a class="existingWikiWord" href="/nlab/show/MSSM">MSSM</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/on-shell+recursion">on-shell recursion</a>, <a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></li> </ul> </li> </ul> </li> <li> Structural phenomena <ul> <li> <a class="existingWikiWord" href="/nlab/show/universality+class">universality class</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/instanton">instanton</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spontaneously+broken+symmetry">spontaneously broken symmetry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein mechanism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/integrable+systems">integrable systems</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/holonomic+quantum+fields">holonomic quantum fields</a> </li> </ul> </li> <li> Types of quantum field thories <ul> <li> <a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory and elliptic cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/CFT">CFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a>, <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>, <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixing</a> </li> <li> examples <ul> <li><a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a>, <a class="existingWikiWord" href="/nlab/show/QED">QED</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/electric+charge">electric charge</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>, <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spinors+in+Yang-Mills+theory">spinors in Yang-Mills theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a></li> <li><a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a></li> <li><a class="existingWikiWord" href="/nlab/show/RR+field">RR field</a></li> <li><a class="existingWikiWord" href="/nlab/show/first-order+formulation+of+gravity">first-order formulation of gravity</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/general+covariance">general covariance</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gravity+as+a+BF-theory">gravity as a BF-theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/particle">particle</a>, <a class="existingWikiWord" href="/nlab/show/relativistic+particle">relativistic particle</a>, <a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental particle</a>, <a class="existingWikiWord" href="/nlab/show/spinning+particle">spinning particle</a>, <a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>, <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/membrane">membrane</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/AKSZ+theory">AKSZ theory</a> </li> </ul> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/string+theory">String Theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+theory+results+applied+elsewhere">string theory results applied elsewhere</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/number+theory+and+physics">number theory and physics</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Riemann+hypothesis+and+physics">Riemann hypothesis and physics</a></li> </ul> </li> </ul> <div> <a href="/nlab/edit/physicscontents">Edit this sidebar</a> </div></div></div> <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> <li><a href='#mathematical_model_from_formal_physical_input'>Mathematical model from (formal) physical input</a></li> <li><a href='#over_dbranes'>Over D-branes</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> The Kalb-Ramond field or B-field is the <a class="existingWikiWord" href="/nlab/show/higher+U%281%29-gauge+field">higher U(1)-gauge field</a> that generalizes the <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a> from point <a class="existingWikiWord" href="/nlab/show/particles">particles</a> to <a class="existingWikiWord" href="/nlab/show/string+theory">strings</a>. Its dual incarnation in <a class="existingWikiWord" href="/nlab/show/KK-compactifications">KK-compactifications</a> of <a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a> to 4d is a candidate for the hypothetical <a class="existingWikiWord" href="/nlab/show/axion">axion</a> field (<a href="#SvrcekWitten06">Svrcek-Witten 06, p. 15</a>). Recall that the <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a> is modeled as a <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> in degree 2 <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a> and that this mathematical model is fixed by the fact that charged particles that trace out 1-dimensional trajectories couple to the electromagnetic field by an <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> that sends each trajectory to the holonomy of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection</a> on it. When replacing particles with 1-dimensional trajectories by <a class="existingWikiWord" href="/nlab/show/string+theory">strings</a> with 2-dimensional trajectories, one accordingly expects that they may couple to a higher degree background field given by a degree 3 <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a> cocycle. In <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> this situation arises and the corresponding background field appears, where it is called the Kalb-Ramond field . Often it is also simply called the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-field , after the standard symbol used for the 2-forms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>B</mi> <mi>i</mi></msub><mo>∈</mo><msup><mi>Ω</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(B_i \in \Omega^2(U_i))</annotation></semantics></math> on patches <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math> of a <a class="existingWikiWord" href="/nlab/show/cover">cover</a> of spacetime when the differential cocycle is expressed in a <a class="existingWikiWord" href="/nlab/show/Cech+cohomology">Cech cohomology</a> realization of <a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a>. This is the analog of the local 1-forms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>i</mi></msub><mo>∈</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A_i \in \Omega^1(U_i))</annotation></semantics></math> in a Cech cocycle presentation of a <a class="existingWikiWord" href="/nlab/show/line+bundle">line bundle</a> with <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection</a> encoding the <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a>. The <a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a> of the Kalb-Ramond field is a <a class="existingWikiWord" href="/nlab/show/differential+forms">3-form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>∈</mo><mi>Ω</mi></mrow><annotation encoding="application/x-tex">H \in \Omega</annotation></semantics></math>. On each patch <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math> it is given by <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>H</mi><msub><mo stretchy="false">|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow></msub><mo>=</mo><mi>d</mi><msub><mi>B</mi> <mi>i</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H|_{U_i} = d B_i \,. </annotation></semantics></math></div> And just as a degree 2 <a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cocycle</a> is equivalently encoded in a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">with connection</a>, the degree 3 differential cocycle is equivalently encoded in <ul> <li> a degree 3 <a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cocycle</a>; </li> <li> a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}U(1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a> with connection; </li> <li> a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe</a> with connection. </li> </ul> The study of <a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe</a>s was largely motivated and driven by the desire to understand the Kalb-Ramond field. The next higher degree analog of the electromagnetic field is the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a>. <h2 id="mathematical_model_from_formal_physical_input">Mathematical model from (formal) physical input</h2> The derivation of the fact that the Kalb-Ramond field that is locally given by a 2-form is globally really a degree 3 cocycle in the <a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a> model for <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a> proceeds in in entire analogy with the corresponding discussion of the <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a>: <ul> <li> classical background The <a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a> 3-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>∈</mo><msup><mi>Ω</mi> <mn>3</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H \in \Omega^3(X)</annotation></semantics></math> is required to be closed, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><msub><mi>H</mi> <mn>3</mn></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">d H_3 = 0</annotation></semantics></math>. </li> <li> quantum coupling The gauge interaction with the quantum string is required to yield a well-defined surface holonomy in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math> from locally integrating the 2-forms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mi>i</mi></msub><mo>∈</mo><msup><mi>Ω</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B_i \in \Omega^2(U_2)</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><msub><mi>B</mi> <mi>i</mi></msub><mo>=</mo><mi>H</mi><msub><mo stretchy="false">|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">d B_i = H|_{U_i}</annotation></semantics></math> over its 2-dimensional trajectory. <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>hol</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="false"><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mi>f</mi></msub><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><mstyle displaystyle="false"><msub><mo>∫</mo> <mi>f</mi></msub><msup><mi>Σ</mi> <mo>*</mo></msup><msub><mi>B</mi> <mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mstyle displaystyle="false"><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>e</mi><mo>⊂</mo><mi>f</mi></mrow></msub><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><mstyle displaystyle="false"><msub><mo>∫</mo> <mi>e</mi></msub><msup><mi>Σ</mi> <mo>*</mo></msup><msub><mi>A</mi> <mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mi>ρ</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mstyle displaystyle="false"><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>v</mi><mo>⊂</mo><mi>e</mi><mo>⊂</mo><mi>f</mi></mrow></msub><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><msub><mi>λ</mi> <mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mi>ρ</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mi>ρ</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mstyle></mstyle></mstyle></mstyle></mstyle></mrow><annotation encoding="application/x-tex"> hol(\Sigma) = \textstyle{\prod}_{f} \exp(i \textstyle{\int}_f \Sigma^* B_{\rho(f)}) \textstyle{\prod}_{e \subset f} \exp(i \textstyle{\int}_{e} \Sigma^* A_{\rho(f) \rho(e)}) \textstyle{\prod}_{v \subset e \subset f} \exp(i \lambda_{\rho(f) \rho(e) \rho(v)}) \,. </annotation></semantics></math></div> That this is well defined requires that <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>λ</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo>−</mo><msub><mi>λ</mi> <mrow><mi>i</mi><mi>j</mi><mi>l</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi> <mrow><mi>i</mi><mi>k</mi><mi>l</mi></mrow></msub><mo>−</mo><msub><mi>λ</mi> <mrow><mi>j</mi><mi>k</mi><mi>l</mi></mrow></msub><mo>=</mo><mn>0</mn><mspace width="thickmathspace"></mspace><mi>mod</mi><mspace width="thinmathspace"></mspace><mn>2</mn><mi>π</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \lambda_{i j k} - \lambda_{i j l} + \lambda_{i k l} - \lambda_{j k l} = 0 \;mod \, 2\pi \,, </annotation></semantics></math></div> which says that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>B</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>A</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><msub><mi>λ</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(B_i, A_{i j}, \lambda_{i j k})</annotation></semantics></math> is indeed a degree 3 <a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cocycle</a>. </li> </ul> <h2 id="over_dbranes">Over D-branes</h2> The restriction of the Kalb-Ramond field in the 10-dimensional <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> to a <a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a> is a twist (as in <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a>) of the <a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a> on the D-brane: its 3-class is <a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a> for the <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a>/<a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a> on the D-brane. See also <a class="existingWikiWord" href="/nlab/show/Freed-Witten+anomaly+cancellation">Freed-Witten anomaly cancellation</a> or the discussion in (<a href="#Moore">Moore</a>). <h2 id="related_concepts">Related concepts</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/discrete+torsion">discrete torsion</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/h1-meson">h1-meson</a>, <a class="existingWikiWord" href="/nlab/show/b1-meson">b1-meson</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a>, <a class="existingWikiWord" href="/nlab/show/basic+bundle+gerbe">basic bundle gerbe</a> </li> </ul> <div> 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> (for classification see at <a class="existingWikiWord" href="/nlab/show/brane+scan">brane scan</a>). <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;"><a class="existingWikiWord" href="/nlab/show/black+brane">black brane</a></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;"><a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a></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;"><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></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;"><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></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;"><a class="existingWikiWord" href="/nlab/show/NS-brane">NS-brane</a></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;">D-brane for <a class="existingWikiWord" href="/nlab/show/topological+string">topological string</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/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;"><a class="existingWikiWord" href="/nlab/show/M-brane">M-brane</a></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;"><a class="existingWikiWord" href="/nlab/show/S-brane">S-brane</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/SM2-brane">SM2-brane</a>, <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;"><a class="existingWikiWord" href="/nlab/show/solitons">solitons</a> 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> <h2 id="References">References</h2> The name goes back to: <ul> <li id="KalbRamond">M. Kalb, <a class="existingWikiWord" href="/nlab/show/Pierre+Ramond">Pierre Ramond</a>, Classical direct interstring action, Phys. Rev. D. 9 (1974) 2273-2284 [<a href="https://doi.org/10.1103/PhysRevD.9.2273">doi:10.1103/PhysRevD.9.2273</a>]</li> </ul> The interpretation as a 4d <a class="existingWikiWord" href="/nlab/show/axion">axion</a>: <ul> <li id="SvrcekWitten06">Peter Svrcek, <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, Axions In String Theory, JHEP 0606:051 (2006) [<a href="http://arxiv.org/abs/hep-th/0605206">arXiv:hep-th/0605206</a>]</li> </ul> The interpretation of the B-field as a 3-<a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> in <a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a> is due to <ul> <li><a class="existingWikiWord" href="/nlab/show/Krzysztof+Gaw%C4%99dzki">Krzysztof Gawędzki</a>, Topological Actions in two-dimensional Quantum Field Theories, in: Nonperturbative quantum field theory, Nato Science Series B 185, Springer (1986) [<a href="http://inspirehep.net/record/257658">spire:257658</a>, <a href="https://doi.org/10.1007/978-1-4613-0729-7_5">doi:10.1007/978-1-4613-0729-7_5</a>, <a class="existingWikiWord" href="/nlab/files/Gawedzki-TopologicalActions.pdf" title="pdf">pdf</a>]</li> </ul> picked up in <ul> <li id="FreedWitten99"><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, Anomalies in String Theory with D-Branes, Asian J. Math. 3 4 (1999) 819-852 [<a href="http://arxiv.org/abs/hep-th/9907189">arXiv:hep-th/9907189</a>, <a href="https://inspirehep.net/literature/504509">InSpire:504509</a>]</li> </ul> The equivalent formulation in terms of <a class="existingWikiWord" href="/nlab/show/connections+on+bundle+gerbes">connections on bundle gerbes</a> originates with <ul> <li id="GawędzkiReis02"> <a class="existingWikiWord" href="/nlab/show/Krzysztof+Gaw%C4%99dzki">Krzysztof Gawędzki</a>, <a class="existingWikiWord" href="/nlab/show/Nuno+Reis">Nuno Reis</a>, WZW branes and gerbes, Rev. Math. Phys. 14 (2002) 1281-1334 [<a href="https://arxiv.org/abs/hep-th/0205233">arXiv:hep-th/0205233</a>, <a href="https://doi.org/10.1142/S0129055X02001557">doi:10.1142/S0129055X02001557</a>] </li> <li id="CareyJohnsonMurray02"> <a class="existingWikiWord" href="/nlab/show/Alan+Carey">Alan Carey</a>, Stuart Johnson, <a class="existingWikiWord" href="/nlab/show/Michael+Murray">Michael Murray</a>, Holonomy on D-Branes, Journal of Geometry and Physics 52 2 (2004) 186-216 [<a href="http://arxiv.org/abs/hep-th/0204199">arXiv:hep-th/0204199</a>, <a href="https://doi.org/10.1016/j.geomphys.2004.02.008">doi:10.1016/j.geomphys.2004.02.008</a>] </li> </ul> See also: <ul> <li id="BonoraRuffinoSavelli08"> <a class="existingWikiWord" href="/nlab/show/Loriano+Bonora">Loriano Bonora</a>, <a class="existingWikiWord" href="/nlab/show/Fabio+Ferrari+Ruffino">Fabio Ferrari Ruffino</a>, <a class="existingWikiWord" href="/nlab/show/Raffaele+Savelli">Raffaele Savelli</a>, Classifying A-field and B-field configurations in the presence of D-branes, JHEP 0812:078 (2008) [<a href="https://arxiv.org/abs/0810.4291">arXiv:0810.4291</a>, <a href="https://doi.org/10.1088/1126-6708/2008/12/078">doi:10.1088/1126-6708/2008/12/078</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Fabio+Ferrari+Ruffino">Fabio Ferrari Ruffino</a>, Classifying A-field and B-field configurations in the presence of D-branes - Part II: Stacks of D-branes, Nuclear Physics, Section B 858 (2012) 377-404 [<a href="https://arxiv.org/abs/1104.2798">arXiv:1104.2798</a>, <a href="https://doi.org/10.1016/j.nuclphysb.2012.01.013">doi:10.1016/j.nuclphysb.2012.01.013</a>] </li> </ul> A more refined discussion of the <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a> of the Kalb-Ramond field and the <a class="existingWikiWord" href="/nlab/show/RR-fields">RR-fields</a> that it interacts with: <ul> <li><a class="existingWikiWord" href="/nlab/show/Dan+Freed">Dan 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 7 (2002) 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> In fact, in full generality the Kalb-Ramond field on an <a class="existingWikiWord" href="/nlab/show/orientifold">orientifold</a> background is not a plain bundle gerbe, but a <a class="existingWikiWord" href="/nlab/show/Jandl+gerbe">Jandl gerbe</a>, a connection on a nonabelian <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>AUT</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">AUT(U(1))</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+2-bundles">principal 2-bundles</a> for the <a class="existingWikiWord" href="/nlab/show/automorphism+2-group">automorphism 2-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>AUT</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">AUT(U)(1))</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>: for the bosonic string this is discussed in <ul> <li id="SSW05"><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/nlab/show/Christoph+Schweigert">Christoph Schweigert</a>, <a class="existingWikiWord" href="/nlab/show/Konrad+Waldorf">Konrad Waldorf</a>, Unoriented WZW models and Holonomy of Bundle Gerbes, Communications in Mathematical Physics 274 1 (2007) 31-64 [<a href="http://arxiv.org/abs/hep-th/0512283">arXiv:hep-th/0512283</a>, <a href="https://doi.org/10.1007/s00220-007-0271-x">doi:10.1007/s00220-007-0271-x</a>]</li> </ul> and for the refinement to the <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a> in <ul> <li> <a class="existingWikiWord" href="/nlab/show/Jacques+Distler">Jacques Distler</a>, <a class="existingWikiWord" href="/nlab/show/Dan+Freed">Dan Freed</a>, <a class="existingWikiWord" href="/nlab/show/Greg+Moore">Greg Moore</a>, Orientifold Precis, 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 83, AMS (2011) [<a href="http://arxiv.org/abs/0906.0795">arXiv:0906.0795</a>] </li> <li id="DistlerFreedMooreII"> <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>, Spin structures and superstrings, Surveys in Differential Geometry, Volume 15 (2010) (<a href="http://arxiv.org/abs/1007.4581">arXiv:1007.4581</a>, <a href="http://dx.doi.org/10.4310/SDG.2010.v15.n1.a4">doi:10.4310/SDG.2010.v15.n1.a4</a>) </li> </ul> See at <a class="existingWikiWord" href="/nlab/show/orientifold">orientifold</a> for more on this; also at <a class="existingWikiWord" href="/nlab/show/discrete+torsion">discrete torsion</a>. The role of the KR field in <a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a> (see <a class="existingWikiWord" href="/nlab/show/K-theory+classification+of+D-brane+charge">K-theory classification of D-brane charge</a>) is discussed a bit also in <ul> <li id="Moore"><a class="existingWikiWord" href="/nlab/show/Greg+Moore">Greg Moore</a>, K-theory from a physical perspective [<a href="http://arxiv.org/abs/hep-th/0304018">arXiv:hep-th/0304018</a>]</li> </ul> In relation to <a class="existingWikiWord" href="/nlab/show/first-order+formulation+of+gravity">Einstein-Cartan theory</a>: <ul> <li> Richa Kapoor, A review of Einstein Cartan Theory to describe superstrings with intrinsic torsion (<a href="https://arxiv.org/abs/2009.07211">arXiv:2009.07211</a>) </li> <li> Tanmoy Paul, Antisymmetric tensor fields in modified gravity: a summary (<a href="https://arxiv.org/abs/2009.07732">arXiv:2009.07732</a>) </li> </ul> In the context of <a class="existingWikiWord" href="/nlab/show/cosmology">cosmology</a> with the Kalb-Ramond field as a <a class="existingWikiWord" href="/nlab/show/dark+matter">dark matter</a>-candidate (cf, <a class="existingWikiWord" href="/nlab/show/axion">axion</a> and <a class="existingWikiWord" href="/nlab/show/fuzzy+dark+matter">fuzzy dark matter</a>): <ul> <li>Christian Capanelli, Leah Jenks, Edward W. Kolb, Evan McDonough, Cosmological Implications of Kalb-Ramond-Like-Particles [<a href="https://arxiv.org/abs/2309.02485">arXiv:2309.02485</a>]</li> </ul> See also: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Peter+D.+Jarvis">Peter D. Jarvis</a>, <a class="existingWikiWord" href="/nlab/show/Jean+Thierry-Mieg">Jean Thierry-Mieg</a>, Antisymmetric tensor fields: actions, symmetries and first order Duffin-Kemmer-Petiau formulations [<a href="https://arxiv.org/abs/2311.01675">arXiv:2311.01675</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Jean+Thierry-Mieg">Jean Thierry-Mieg</a>, <a class="existingWikiWord" href="/nlab/show/Peter+D.+Jarvis">Peter D. Jarvis</a>, Conformal invariance of antisymmetric tensor field theories in any even dimension [<a href="https://arxiv.org/abs/2311.01701">arXiv:2311.01701</a>] </li> </ul> </body></html> </div> <div class="revisedby"> Last revised on April 2, 2024 at 06:38:40. 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