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style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="differential_cohomology">Differential cohomology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a></strong></p> <h2 id="ingredients">Ingredients</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a></p> </li> </ul> <h2 id="connections_on_bundles">Connections on bundles</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a bundle</a></p> <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> <p><a class="existingWikiWord" href="/nlab/show/curvature">curvature</a></p> <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> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+homomorphism">Chern-Weil homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/secondary+characteristic+class">secondary characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+characteristic+class">differential characteristic class</a></p> </li> </ul> </li> </ul> <h2 id="higher_abelian_differential_cohomology">Higher abelian differential cohomology</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+function+complex">differential function complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+orientation">differential orientation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+Thom+class">differential Thom class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+characters">differential characters</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-bundle with connection</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe with connection</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+elliptic+cohomology">differential elliptic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a></p> </li> </ul> <h2 id="higher_nonabelian_differential_cohomology">Higher nonabelian differential cohomology</h2> <ul> <li> <p><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></p> </li> <li> <p><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></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory+in+Smooth%E2%88%9EGrpd">Chern-Weil theory in Smooth∞Grpd</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Simons+theory">∞-Chern-Simons theory</a></p> </li> </ul> <h2 id="fiber_integration">Fiber integration</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+holonomy">higher holonomy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration+in+differential+cohomology">fiber integration in differential cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration+in+ordinary+differential+cohomology">fiber integration in ordinary differential cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration+in+differential+K-theory">fiber integration in differential K-theory</a></p> </li> </ul> </li> </ul> <h2 id="application_to_gauge_theory">Application to gauge theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a>/<a class="existingWikiWord" href="/nlab/show/B-field">B-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/RR-field">RR-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">supergravity</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/differential+cohomology+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#SimonsSullivanModel'>The Simons-Sullivan model</a></li> <ul> <li><a href='#idea_2'>Idea</a></li> <li><a href='#details'>Details</a></li> </ul> <li><a href='#the_bunkeschick_model'>The Bunke-Schick model</a></li> <ul> <li><a href='#idea_3'>Idea</a></li> <li><a href='#properties'>Properties</a></li> </ul> <li><a href='#the_hopkinssinger_model'>The Hopkins-Singer model</a></li> <li><a href='#more_models_in_smooth_spectra'>More models in smooth spectra</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#relation_to_index_theory'>Relation to index theory</a></li> <li><a href='#in_string_theory'>In string theory</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p><em>Differential K-theory</em> is the refinement of the <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a> theory <a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a> to <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a>.</p> <p>In as far as we can think of cocycles in <a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a> as represented by <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a>s or <a class="existingWikiWord" href="/nlab/show/vectorial+bundle">vectorial bundle</a>s, cocycles in differential K-theory may be represented by <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a>s <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">with connection</a>.</p> <p>There are various different models that differ in the concrete realization of these cocycles and in their extra properties.</p> <h2 id="SimonsSullivanModel">The Simons-Sullivan model</h2> <p>This section discusses the model presented in (<a href="#SimonsSullivan">SimonsSullivan</a>).</p> <p>More details will eventually be at</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Simons-Sullivan+structured+bundle">Simons-Sullivan structured bundle</a>.</li> </ul> <h3 id="idea_2">Idea</h3> <p>In the Simons-Sullivan model cocycles in differential K-theory are represented by ordinary <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a>s <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">with connection</a>. The crucial ingredient is that two connections on a vector bundle are taken to be the same representative of a differential K-cocycle if they are related by a <a class="existingWikiWord" href="/nlab/show/concordance">concordance</a> such that the corresponding <a class="existingWikiWord" href="/nlab/show/Chern-Simons+form">Chern-Simons form</a> is exact.</p> <h3 id="details">Details</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">V \to X</annotation></semantics></math> be a complex <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a> with <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> 2-form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>=</mo><msub><mi>F</mi> <mo>∇</mo></msub><mo>∈</mo><msup><mi>Ω</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>End</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> F = F_\nabla \in \Omega^2(X,End(V)) \,. </annotation></semantics></math></div> <p><strong>Definition</strong></p> <p>The <strong><a class="existingWikiWord" href="/nlab/show/Chern+character">Chern character</a></strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math> is the inhomogenous <a class="existingWikiWord" href="/nlab/show/curvature+characteristic+form">curvature characteristic form</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ch</mi><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>j</mi><mo>∈</mo><mi>ℕ</mi></mrow></munder><msub><mi>k</mi> <mi>j</mi></msub><mi>tr</mi><mo stretchy="false">(</mo><msub><mi>F</mi> <mo>∇</mo></msub><mo>∧</mo><mi>⋯</mi><mo>∧</mo><msub><mi>F</mi> <mo>∇</mo></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><msup><mi>Ω</mi> <mrow><mn>2</mn><mo>•</mo></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> ch(\nabla) := \sum_{j \in \mathbb{N}} k_j tr( F_\nabla \wedge \cdots \wedge F_\nabla) \;\; \in \Omega^{2 \bullet}(X) \,, </annotation></semantics></math></div> <p>where on the right we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math> wedge factors of the <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> .</p> <p><strong>Definition</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mo>∇</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(V,\nabla)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>V</mi><mo>′</mo><mo>,</mo><mo>∇</mo><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(V',\nabla')</annotation></semantics></math> be two complex vector bundles with connection.</p> <p>A <a class="existingWikiWord" href="/nlab/show/Chern-Simons+form">Chern-Simons form</a> for this pair is a differential form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>CS</mi><mo stretchy="false">(</mo><mo>∇</mo><mo>,</mo><mo>∇</mo><mo>′</mo><mo stretchy="false">)</mo><mo>+</mo><mi>d</mi><mi>ω</mi><mo>∈</mo><msup><mi>Ω</mi> <mrow><mn>2</mn><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> CS(\nabla,\nabla') + d \omega \in \Omega^{2 \bullet + 1}(X) </annotation></semantics></math></div> <p>obtained from the <a class="existingWikiWord" href="/nlab/show/concordance">concordance</a> bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>V</mi><mo stretchy="false">¯</mo></mover><mo>→</mo><mi>X</mi><mo>×</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\bar V \to X \times [0,1]</annotation></semantics></math> given by <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X \times [0,1] \to X</annotation></semantics></math> equipped with a connection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>∇</mo><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">\bar \nabla</annotation></semantics></math> such that …, by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>CS</mi><mo stretchy="false">(</mo><mo>∇</mo><mo>,</mo><mo>∇</mo><mo>′</mo><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo> <mn>0</mn> <mn>1</mn></msubsup><msubsup><mi>ψ</mi> <mi>t</mi> <mo>*</mo></msubsup><mo stretchy="false">(</mo><msub><mi>ι</mi> <mrow><mo>∂</mo><mo stretchy="false">/</mo><mo>∂</mo><mi>t</mi></mrow></msub><mi>ch</mi><mo stretchy="false">(</mo><mover><mo>∇</mo><mo stretchy="false">¯</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><mi>d</mi><mo stretchy="false">(</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> CS(\nabla,\nabla') = \int_0^1 \psi_t^* (\iota_{\partial/\partial t} ch(\bar \nabla)) + d (...) \,. </annotation></semantics></math></div> <p><strong>Proposition</strong> This is indeed well defined in that it is independent of the chosen <a class="existingWikiWord" href="/nlab/show/concordance">concordance</a>, up to an exact term.</p> <p><strong>Definition</strong></p> <p>A <strong>structured bundle</strong> in the sense of the Simons-Sullivan model is a complex vector bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> equipped with the equivalence class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo>∇</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\nabla]</annotation></semantics></math> of a connection under the equivalence relation that identifies two connections <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo><mo>′</mo></mrow><annotation encoding="application/x-tex">\nabla'</annotation></semantics></math> if their Chern-Simons form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CS</mi><mo stretchy="false">(</mo><mo>∇</mo><mo>,</mo><mo>∇</mo><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CS(\nabla,\nabla')</annotation></semantics></math> is exact.</p> <p>Two structured bundles are isomorphic if there is a vector bundle isomorphism under which the two equivalence classes of connections are identified.</p> <p><strong>Definition</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Struc</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Struc(X)</annotation></semantics></math> be the set of <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> classes of structured bundles on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>Under <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> and <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> of vector bundles, this becomes a commutatve <a class="existingWikiWord" href="/nlab/show/rig">rig</a>.</p> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mi>K</mi><mo stretchy="false">(</mo><mi>Struct</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \hat K(X) := K(Struct(X)) </annotation></semantics></math></div> <p>be the additive <a class="existingWikiWord" href="/nlab/show/group+completion">group completion</a> of this rig as usual in <a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a>.</p> <p>So as an additive group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\hat K(X)</annotation></semantics></math> is the quotient of the <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> induced by <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> on pairs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>W</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(V,W)</annotation></semantics></math> of isomorphism classes in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Struc</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Struc(X)</annotation></semantics></math>, modulo the sub-monoid consisting of pairs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(V,V)</annotation></semantics></math>.</p> <p>Hence the pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(V,0)</annotation></semantics></math> is the additive inverse to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,V)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>W</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(V,W)</annotation></semantics></math> may be written as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>−</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">V - W</annotation></semantics></math>.</p> <p><strong>Theorem</strong></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\hat K(X)</annotation></semantics></math> is indeed a <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a> refinement of ordinary K-theory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(X)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> (i.e. of the 0th <a class="existingWikiWord" href="/nlab/show/cohomology+group">cohomology group</a> of <a class="existingWikiWord" href="/nlab/show/K-theory+spectrum">K-cohomology</a>).</p> <p>Moreover…</p> <h2 id="the_bunkeschick_model">The Bunke-Schick model</h2> <h3 id="idea_3">Idea</h3> <p><a class="existingWikiWord" href="/nlab/show/Uli+Bunke">Uli Bunke</a> and <a class="existingWikiWord" href="/nlab/show/Thomas+Schick">Thomas Schick</a> developed in a series of articles a differential-geometric cocycle model of differential K-theory where cocycles are given by smooth families of <a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</a>s.</p> <p>See the reference <a href="BunkeSchickReferences">below</a>.</p> <h3 id="properties">Properties</h3> <p>The restriction of the cocycles in the Bunke-Schick model to those whose “auxialiary form” <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> vanishes reproduces the Simons-Sullivan model above.</p> <h2 id="the_hopkinssinger_model">The Hopkins-Singer model</h2> <p>See at</p> <ul> <li> <p><em><a class="existingWikiWord" href="/nlab/show/differential+function+complex">differential function complex</a></em></p> </li> <li> <p><em><a href="differential+cohomology+diagram#HSDifferentialKU">differential cohomology diagram – Hopkins-Singer differential KU</a></em></p> </li> </ul> <h2 id="more_models_in_smooth_spectra">More models in smooth spectra</h2> <p>See at <em><a href="differential+cohomology+diagram#DifferentialKTheory">Differential cohomology diagram – Differential K-theory</a></em>.</p> <h2 id="examples">Examples</h2> <ul> <li>In <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a> gauge fields are modeled by cocycles in <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a>. The field modeled by differential K-theory is the <a class="existingWikiWord" href="/nlab/show/RR-field">RR-field</a>. A kind of <a class="existingWikiWord" href="/nlab/show/integral+transform">integral transform</a> acting on differential K-theory classes is <a class="existingWikiWord" href="/nlab/show/T-duality">T-duality</a>.</li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/flat+K-theory">flat K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a>, <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a></p> </li> <li> <p><strong>differential K-theory</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration+in+differential+K-theory">fiber integration in differential K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+K-theory+of+smooth+manifolds">algebraic K-theory of smooth manifolds</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+differential+K-theory">twisted differential K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+differential+K-theory">equivariant differential K-theory</a>, <a class="existingWikiWord" href="/nlab/show/orbifold+differential+K-theory">orbifold differential K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+algebraic+K-theory">differential algebraic K-theory</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="general">General</h3> <p>An early sketch of a general definition, motivated by the description of <a class="existingWikiWord" href="/nlab/show/D-brane+charge">D-brane charge</a> in <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a>, is in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <em><a class="existingWikiWord" href="/nlab/show/Dirac+charge+quantization+and+generalized+differential+cohomology">Dirac charge quantization and generalized differential cohomology</a></em>, Surveys in Differential Geometry, Int. Press, Somerville, MA, 2000, pp. 129–194 (<a href="http://arxiv.org/abs/hep-th/0011220">arXiv:hep-th/0011220</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <em>On Ramond-Ramond fields and K-theory</em>, JHEP (2000) 44, 14 [<a href="http://arxiv.org/abs/hep-th/0002027">arXiv:hep-th/0002027</a>, <a href="https://doi.org/10.1088/1126-6708/2000/05/044">doi:10.1088/1126-6708/2000/05/044</a>]</p> </li> </ul> <p>Then the general construction of <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a> theories via <a class="existingWikiWord" href="/nlab/show/differential+function+complexes">differential function complexes</a> of</p> <ul> <li id="HopkinsSinger05"><a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <a class="existingWikiWord" href="/nlab/show/Isadore+Singer">Isadore Singer</a>, <em><a class="existingWikiWord" href="/nlab/show/Quadratic+Functions+in+Geometry%2C+Topology%2C+and+M-Theory">Quadratic Functions in Geometry, Topology, and M-Theory</a></em>, J. Differential Geom. Volume 70, Number 3 (2005), 329-452 (<a href="http://arxiv.org/abs/math.AT/0211216">arXiv:math.AT/0211216</a>)</li> </ul> <p>(motivated in turn by <a class="existingWikiWord" href="/nlab/show/7d+Chern-Simons+theory">7d Chern-Simons theory</a> and the <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> <a class="existingWikiWord" href="/nlab/show/partition+function">partition function</a>)</p> <p>provides in particular a model for differential K-theory.</p> <p>For more historical remarks see section 1.6 of</p> <ul> <li id="FreedLott10"><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <a class="existingWikiWord" href="/nlab/show/John+Lott">John Lott</a>, <em>An index theorem in differential K-theory</em>, Geometry and Topology 14 (2010) (<a href="http://math.berkeley.edu/~lott/gt-2010-14-021p.pdf">pdf</a>)</li> </ul> <p>A discussion of more models and their relation in the context of <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+type+theory">cohesive homotopy type theory</a> and the <a class="existingWikiWord" href="/nlab/show/differential+cohomology+hexagon">differential cohomology hexagon</a> then appears in</p> <ul> <li id="BunkeNikolausVoelkl13"><a class="existingWikiWord" href="/nlab/show/Ulrich+Bunke">Ulrich Bunke</a>, <a class="existingWikiWord" href="/nlab/show/Thomas+Nikolaus">Thomas Nikolaus</a>, <a class="existingWikiWord" href="/nlab/show/Michael+V%C3%B6lkl">Michael Völkl</a>, Section 6 of: <em>Differential cohomology theories as sheaves of spectra</em> (<a href="http://arxiv.org/abs/1311.3188">arXiv:1311.3188</a>)</li> </ul> <p>Survey:</p> <ul> <li id="BunkeSchick10"><a class="existingWikiWord" href="/nlab/show/Ulrich+Bunke">Ulrich Bunke</a>, <a class="existingWikiWord" href="/nlab/show/Thomas+Schick">Thomas Schick</a>, <em>Differential K-theory. A survey</em> (<a href="http://arxiv.org/abs/1011.6663">arXiv:1011.6663</a>).</li> </ul> <p>Cocycle models via <a class="existingWikiWord" href="/nlab/show/vector+bundles">vector bundles</a> <a class="existingWikiWord" href="/nlab/show/bundle+with+connection">bundle with connection</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Max+Karoubi">Max Karoubi</a>, <em>Homologie cyclique et K-théorie</em>, Astérisque, no. 149 (1987) (<a href="http://www.numdam.org/item/AST_1987__149__1_0">numdam:AST_1987__149__1_0</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/John+Lott">John Lott</a>, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>R</mi></mstyle><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>Z</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{R}/\mathbf{Z}</annotation></semantics></math> Index theory</em>, Communications in Analysis and Geometry <strong>2</strong> 2 (1994) 279-311 (<a href="https://math.berkeley.edu/~lott/rztheory.pdf">pdf</a>)</p> </li> <li id="SimonsSullivan"> <p><a class="existingWikiWord" href="/nlab/show/James+Simons">James Simons</a>, <a class="existingWikiWord" href="/nlab/show/Dennis+Sullivan">Dennis Sullivan</a>, <em>Structured vector bundles define differential K-theory</em> (<a href="http://arxiv.org/abs/0810.4935">arXiv:0810.4935</a>)</p> </li> </ul> <p>The basic article for the Bunke-Schick model is</p> <ul> <li id="BunkeSchick09"><a class="existingWikiWord" href="/nlab/show/Ulrich+Bunke">Ulrich Bunke</a>, <a class="existingWikiWord" href="/nlab/show/Thomas+Schick">Thomas Schick</a>, <em>Smooth K-Theory</em>, Astérisque 328 (2009), 45-135 [<a href="http://arxiv.org/abs/0707.0046">arXiv:0707.0046</a>, <a href="http://www.numdam.org/item/?id=AST_2009__328__45_0">numdam:AST_2009__328__45_0</a>]</li> </ul> <p>A survey talk is</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ulrich+Bunke">Ulrich Bunke</a>, <em>Differential cohomology in geometry and analysis</em> (<a href="http://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/Bunke/Vortrag-Erlangen.pdf">pdf slides</a>)</li> </ul> <p>On differential <a class="existingWikiWord" href="/nlab/show/KO-theory">KO-theory</a>:</p> <ul> <li id="GradySati18"> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Grady">Daniel Grady</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <em>Differential KO-theory: Constructions, computations, and applications</em>, Advances in Mathematics Volume 384, 25 June 2021, 107671 (<a href="https://arxiv.org/abs/1809.07059">arXiv:1809.07059</a>, <a href="https://doi.org/10.1016/j.aim.2021.107671">doi:10.1016/j.aim.2021.107671</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kiyonori+Gomi">Kiyonori Gomi</a>, <a class="existingWikiWord" href="/nlab/show/Mayuko+Yamashita">Mayuko Yamashita</a>, <em>Differential KO-theory via gradations and mass terms</em> [<a href="https://arxiv.org/abs/2111.01377">arXiv:2111.01377</a>]</p> </li> </ul> <p>Of <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>:</p> <ul> <li id="GradySati19"><a class="existingWikiWord" href="/nlab/show/Daniel+Grady">Daniel Grady</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <em>Twisted differential KO-theory</em> (<a href="https://arxiv.org/abs/1905.09085">arXiv:1905.09085</a>)</li> </ul> <p>The equivalence of these models with the respective special case of the general construction in <a href="#HopkinsSinger05">Hopkins & Singer 05</a> in terms of <a class="existingWikiWord" href="/nlab/show/differential+function+complexes">differential function complexes</a> is in</p> <ul> <li>Kevin Klonoff, <em>An Index Theorem in Differential K-Theory</em> PdD thesis (2008) (<a href="http://www.lib.utexas.edu/etd/d/2008/klonoffk16802/klonoffk16802.pdf">pdf</a>)</li> </ul> <p>(assuming the existence of a universal connection, which is not strictly proven) and</p> <ul> <li>Michael L. Ortiz, <em>Differential Equivariant K-Theory</em> (<a href="http://arxiv.org/abs/0905.0476">arXiv:0905.0476</a>)</li> </ul> <p>(not needing that assumption).</p> <p>A construction of <a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a> theory in terms of explicit geometric cocycles is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ulrich+Bunke">Ulrich Bunke</a>, <a class="existingWikiWord" href="/nlab/show/Thomas+Schick">Thomas Schick</a>, Ingo Schroeder, Moritz Wiethaup <em>Landweber exact formal group laws and smooth cohomology theories</em> (<a href="http://arxiv.org/abs/0711.1134">arXiv:0711.1134</a>)</li> </ul> <p>By tensoring this with the suitable ring, this also gives a model for differential K-theory, as well as for <a class="existingWikiWord" href="/nlab/show/differential+elliptic+cohomology">differential elliptic cohomology</a>.</p> <p>A variant of this definition with the advantage that there is a natural morphism to <a class="existingWikiWord" href="/nlab/show/Cheeger-Simons+differential+characters">Cheeger-Simons differential characters</a> refining the total <a class="existingWikiWord" href="/nlab/show/Chern+class">Chern class</a> is (as opposed to the <a class="existingWikiWord" href="/nlab/show/Chern+character">Chern character</a>) is presented in</p> <ul> <li>Alain Berthomieu, <em>A version of smooth K-theory adapted to the total Chern class</em> (<a href="http://arxiv.org/PS_cache/arxiv/pdf/0806/0806.4728v1.pdf">pdf</a>)</li> </ul> <p>Discussion for the <a class="existingWikiWord" href="/nlab/show/odd+Chern+character">odd Chern character</a> is in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Thomas+Tradler">Thomas Tradler</a>, <a class="existingWikiWord" href="/nlab/show/Scott+Wilson">Scott Wilson</a>, <a class="existingWikiWord" href="/nlab/show/Mahmoud+Zeinalian">Mahmoud Zeinalian</a>, <em>An Elementary Differential Extension of Odd K-theory</em>, J. of K-theory, K-theory and its Applications to Algebra, Geometry, Analysis</p> <p>and Topology, (<a href="http://arxiv.org/abs/1211.4477">arXiv:1211.4477</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Scott+Wilson">Scott Wilson</a>, <em>A loop group extension of the odd Chern character</em> (<a href="http://arxiv.org/abs/1311.6393">arXiv:1311.6393</a>)</p> </li> </ul> <p>A model in terms of <a class="existingWikiWord" href="/nlab/show/super+vector+bundles">super vector bundles</a> is given in</p> <ul> <li>Jae Min Lee, Byungdo Park, <em>A Superbundle Description of Differential K-Theory</em>, Axioms 2023, 12(1), 82, (<a href="https://doi.org/10.3390/axioms12010082">doi:10.3390/axioms12010082</a>).</li> </ul> <h3 id="relation_to_index_theory">Relation to index theory</h3> <p>Relation to <a class="existingWikiWord" href="/nlab/show/index+theory">index theory</a>:</p> <ul> <li> <p>Kevin Klonoff, <em>An Index Theorem in Differential K-Theory</em> PdD thesis (2008) (<a href="http://www.lib.utexas.edu/etd/d/2008/klonoffk16802/klonoffk16802.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <a class="existingWikiWord" href="/nlab/show/John+Lott">John Lott</a>, <em>An index theorem in differential K-theory</em>, Geometry and Topology 14 (2010) (<a href="http://math.berkeley.edu/~lott/gt-2010-14-021p.pdf">pdf</a>)</p> </li> </ul> <p>See also the references at <em><a class="existingWikiWord" href="/nlab/show/fiber+integration+in+differential+K-theory">fiber integration in differential K-theory</a></em>.</p> <h3 id="in_string_theory">In string theory</h3> <p>A survey of the role of differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>-theory in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> and <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <em>K-Theory in Quantum Field Theory</em>, Current developments in Mathematics (2001) International Press Somerville (<a href="http://arxiv.org/abs/math-ph/0206031">arXiv:math-ph/0206031</a>)</li> </ul> <p>The operation of <a class="existingWikiWord" href="/nlab/show/T-duality">T-duality</a> on hypothetical <a class="existingWikiWord" href="/nlab/show/twisted+differential+K-theory">twisted differential K-theory</a> is discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Alexander+Kahle">Alexander Kahle</a>, <a class="existingWikiWord" href="/nlab/show/Alessandro+Valentino">Alessandro Valentino</a>, <em><a class="existingWikiWord" href="/nlab/show/T-Duality+and+Differential+K-Theory">T-Duality and Differential K-Theory</a></em></li> </ul> <p>Discussion of <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> (see also <a href="D-brane#ReferencesKTheoryDescription">there</a>):</p> <ul> <li id="GradySati19a"><a class="existingWikiWord" href="/nlab/show/Daniel+Grady">Daniel Grady</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <em>Ramond-Ramond fields and twisted differential K-theory</em>, <p>Advances in Theoretical and Mathematical Physics <strong>26</strong> (2022) 5 [<a href="https://arxiv.org/abs/1903.08843">arXiv:1903.08843</a>, <a href="https://dx.doi.org/10.4310/ATMP.2022.v26.n5.a2">doi:10.4310/ATMP.2022.v26.n5.a2</a>]</p> </li> </ul> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted</a> <a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential</a> <a class="existingWikiWord" href="/nlab/show/KO-theory">orthogonal</a> <a class="existingWikiWord" href="/nlab/show/topological+K-theory">K-theory</a> and its relation to <a class="existingWikiWord" href="/nlab/show/D-brane+charge">D-brane charge</a> in <a class="existingWikiWord" href="/nlab/show/type+I+string+theory">type I string theory</a> (on <a class="existingWikiWord" href="/nlab/show/orientifolds">orientifolds</a>):</p> <ul> <li id="GradySati19b"><a class="existingWikiWord" href="/nlab/show/Daniel+Grady">Daniel Grady</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <em>Twisted differential KO-theory</em> (<a href="https://arxiv.org/abs/1905.09085">arXiv:1905.09085</a>)</li> </ul> <p>See also:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Thomas+Tradler">Thomas Tradler</a>, <a class="existingWikiWord" href="/nlab/show/Scott+Wilson">Scott Wilson</a>, <a class="existingWikiWord" href="/nlab/show/Mahmoud+Zeinalian">Mahmoud Zeinalian</a>, <em>Loop Differential K-theory</em> (<a href="http://arxiv.org/abs/1201.4593">arXiv:1201.4593</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 1, 2023 at 11:37:14. 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