Differential K-theory is the refinement of the generalized (Eilenberg-Steenrod) cohomology theory K-theory to differential cohomology.
In as far as we can think of cocycles in K-theory as represented by vector bundles or vectorial bundles, cocycles in differential K-theory may be represented by vector bundles with connection.
There are various different models that differ in the concrete realization of these cocycles and in their extra properties.
This section discusses the model presented in (SimonsSullivan).
More details will eventually be at
In the Simons-Sullivan model cocycles in differential K-theory are represented by ordinary vector bundles with connection. 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 concordance such that the corresponding Chern-Simons form is exact.
Let $V \to X$ be a complex vector bundle with connection $\nabla$ and curvature 2-form
Definition
The Chern character of $\nabla$ is the inhomogenous curvature characteristic form
where on the right we have $j$ wedge factors of the curvature .
Definition
Let $(V,\nabla)$ and $(V',\nabla')$ be two complex vector bundles with connection.
A Chern-Simons form for this pair is a differential form
obtained from the concordance bundle $\bar V \to X \times [0,1]$ given by pullback along $X \times [0,1] \to X$ equipped with a connection $\bar \nabla$ such that …, by
Proposition This is indeed well defined in that it is independent of the chosen concordance, up to an exact term.
Definition
A structured bundle in the sense of the Simons-Sullivan model is a complex vector bundle $V$ equipped with the equivalence class $[\nabla]$ of a connection under the equivalence relation that identifies two connections $\nabla$ and $\nabla'$ if their Chern-Simons form $CS(\nabla,\nabla')$ is exact.
Two structured bundles are isomorphic if there is a vector bundle isomorphism under which the two equivalence classes of connections are identified.
Definition
Let $Struc(X)$ be the set of isomorphism classes of structured bundles on $X$.
Under direct sum and tensor product of vector bundles, this becomes a commutatve rig.
Let
be the additive group completion of this rig as usual in K-theory.
So as an additive group $\hat K(X)$ is the quotient of the monoid induced by direct sum on pairs $(V,W)$ of isomorphism classes in $Struc(X)$, modulo the sub-monoid consisting of pairs $(V,V)$.
Hence the pair $(V,0)$ is the additive inverse to $(0,V)$ and $(V,W)$ may be written as $V - W$.
Theorem
$\hat K(X)$ is indeed a differential cohomology refinement of ordinary K-theory $K(X)$ of $X$ (i.e. of the 0th cohomology group of K-cohomology).
Moreover…
Uli Bunke and Thomas Schick developed in a series of articles a differential-geometric cocycle model of differential K-theory where cocycles are given by smooth families of Dirac operators.
See the reference below.
The restriction of the cocycles in the Bunke-Schick model to those whose “auxialiary form” $\omega$ vanishes reproduces the Simons-Sullivan model above.
See at
See at Differential cohomology diagram – Differential K-theory.
differential K-theory
equivariant differential K-theory, orbifold differential K-theory
An early sketch of a general definition, motivated by the description of D-brane charge in string theory, is in
Daniel Freed, Dirac charge quantization and generalized differential cohomology, Surveys in Differential Geometry, Int. Press, Somerville, MA, 2000, pp. 129–194 (arXiv:hep-th/0011220)
Daniel Freed, Michael Hopkins, On Ramond-Ramond fields and K-theory, JHEP (2000) 44, 14 [arXiv:hep-th/0002027, doi:10.1088/1126-6708/2000/05/044]
Then the general construction of differential cohomology theories via differential function complexes of
(motivated in turn by 7d Chern-Simons theory and the M5-brane partition function)
provides in particular a model for differential K-theory.
For more historical remarks see section 1.6 of
A discussion of more models and their relation in the context of cohesive homotopy type theory and the differential cohomology hexagon then appears in
Survey:
Cocycle models via vector bundles bundle with connection:
Max Karoubi, Homologie cyclique et K-théorie, Astérisque, no. 149 (1987) (numdam:AST_1987__149__1_0)
John Lott, $\mathbf{R}/\mathbf{Z}$ Index theory, Communications in Analysis and Geometry 2 2 (1994) 279-311 (pdf)
James Simons, Dennis Sullivan, Structured vector bundles define differential K-theory (arXiv:0810.4935)
The basic article for the Bunke-Schick model is
A survey talk is
On differential KO-theory:
Daniel Grady, Hisham Sati, Differential KO-theory: Constructions, computations, and applications, Advances in Mathematics Volume 384, 25 June 2021, 107671 (arXiv:1809.07059, doi:10.1016/j.aim.2021.107671)
Kiyonori Gomi, Mayuko Yamashita, Differential KO-theory via gradations and mass terms [arXiv:2111.01377]
Of twisted differential orthogonal K-theory:
The equivalence of these models with the respective special case of the general construction in Hopkins & Singer 05 in terms of differential function complexes is in
(assuming the existence of a universal connection, which is not strictly proven) and
(not needing that assumption).
A construction of differential cobordism cohomology theory in terms of explicit geometric cocycles is in
By tensoring this with the suitable ring, this also gives a model for differential K-theory, as well as for differential elliptic cohomology.
A variant of this definition with the advantage that there is a natural morphism to Cheeger-Simons differential characters refining the total Chern class is (as opposed to the Chern character) is presented in
Discussion for the odd Chern character is in
Thomas Tradler, Scott Wilson, Mahmoud Zeinalian, An Elementary Differential Extension of Odd K-theory, J. of K-theory, K-theory and its Applications to Algebra, Geometry, Analysis
and Topology, (arXiv:1211.4477)
Scott Wilson, A loop group extension of the odd Chern character (arXiv:1311.6393)
A model in terms of super vector bundles is given in
Relation to index theory:
Kevin Klonoff, An Index Theorem in Differential K-Theory PdD thesis (2008) (pdf)
Daniel Freed, John Lott, An index theorem in differential K-theory, Geometry and Topology 14 (2010) (pdf)
See also the references at fiber integration in differential K-theory.
A survey of the role of differential $K$-theory in quantum field theory and string theory is in
The operation of T-duality on hypothetical twisted differential K-theory is discussed in
Discussion of twisted differential K-theory and its relation to D-brane charge in type II string theory (see also there):
Advances in Theoretical and Mathematical Physics 26 (2022) 5 [arXiv:1903.08843, doi:10.4310/ATMP.2022.v26.n5.a2]
Discussion of twisted differential orthogonal K-theory and its relation to D-brane charge in type I string theory (on orientifolds):
See also:
Last revised on August 1, 2023 at 11:37:14. See the history of this page for a list of all contributions to it.