nLab differential K-theory

Idea

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.

The Simons-Sullivan model

This section discusses the model presented in (SimonsSullivan).

More details will eventually be at

• Simons-Sullivan structured bundle.

Idea

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.

Details

Let $V \to X$ be a complex vector bundle with connection $\nabla$ and curvature 2-form

$$F = F_\nabla \in \Omega^2(X,End(V)) \,.$$

Definition

The Chern character of $\nabla$ is the inhomogenous curvature characteristic form

$$ch(\nabla) := \sum_{j \in \mathbb{N}} k_j tr( F_\nabla \wedge \cdots \wedge F_\nabla) \;\; \in \Omega^{2 \bullet}(X) \,,$$

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

$$CS(\nabla,\nabla') + d \omega \in \Omega^{2 \bullet + 1}(X)$$

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

$$CS(\nabla,\nabla') = \int_0^1 \psi_t^* (\iota_{\partial/\partial t} ch(\bar \nabla)) + d (...) \,.$$

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

$$\hat K(X) := K(Struct(X))$$

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…

The Bunke-Schick model

Idea

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.

Properties

The restriction of the cocycles in the Bunke-Schick model to those whose “auxialiary form” $\omega$ vanishes reproduces the Simons-Sullivan model above.

The Hopkins-Singer model

See at

• differential function complex

• differential cohomology diagram – Hopkins-Singer differential KU

More models in smooth spectra

See at Differential cohomology diagram – Differential K-theory.

Examples

• In gauge theory gauge fields are modeled by cocycles in differential cohomology. The field modeled by differential K-theory is the RR-field. A kind of integral transform acting on differential K-theory classes is T-duality.

Related concepts

• flat K-theory

• K-theory, topological K-theory

• twisted K-theory

• differential K-theory

  • fiber integration in differential K-theory

  • algebraic K-theory of smooth manifolds

• twisted differential K-theory

• equivariant differential K-theory, orbifold differential K-theory

• differential algebraic K-theory

References

General

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

• Michael Hopkins, Isadore Singer, Quadratic Functions in Geometry, Topology, and M-Theory, J. Differential Geom. Volume 70, Number 3 (2005), 329-452 (arXiv:math.AT/0211216)

(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

• Daniel Freed, John Lott, An index theorem in differential K-theory, Geometry and Topology 14 (2010) (pdf)

A discussion of more models and their relation in the context of cohesive homotopy type theory and the differential cohomology hexagon then appears in

• Ulrich Bunke, Thomas Nikolaus, Michael Völkl, Section 6 of: Differential cohomology theories as sheaves of spectra (arXiv:1311.3188)

Survey:

• Ulrich Bunke, Thomas Schick, Differential K-theory. A survey (arXiv:1011.6663).

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

• Ulrich Bunke, Thomas Schick, Smooth K-Theory, Astérisque 328 (2009), 45-135 [arXiv:0707.0046, numdam:AST_2009__328__45_0]

A survey talk is

• Ulrich Bunke, Differential cohomology in geometry and analysis (pdf slides)

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:

• Daniel Grady, Hisham Sati, Twisted differential KO-theory (arXiv:1905.09085)

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

• Kevin Klonoff, An Index Theorem in Differential K-Theory PdD thesis (2008) (pdf)

(assuming the existence of a universal connection, which is not strictly proven) and

• Michael L. Ortiz, Differential Equivariant K-Theory (arXiv:0905.0476)

(not needing that assumption).

A construction of differential cobordism cohomology theory in terms of explicit geometric cocycles is in

• Ulrich Bunke, Thomas Schick, Ingo Schroeder, Moritz Wiethaup Landweber exact formal group laws and smooth cohomology theories (arXiv:0711.1134)

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

• Alain Berthomieu, A version of smooth K-theory adapted to the total Chern class (pdf)

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

• Jae Min Lee, Byungdo Park, A Superbundle Description of Differential K-Theory, Axioms 2023, 12(1), 82, (doi:10.3390/axioms12010082).

Relation to index theory

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.

In string theory

A survey of the role of differential $K$-theory in quantum field theory and string theory is in

• Daniel Freed, K-Theory in Quantum Field Theory, Current developments in Mathematics (2001) International Press Somerville (arXiv:math-ph/0206031)

The operation of T-duality on hypothetical twisted differential K-theory is discussed in

• Alexander Kahle, Alessandro Valentino, T-Duality and Differential K-Theory

Discussion of twisted differential K-theory and its relation to D-brane charge in type II string theory (see also there):

• Daniel Grady, Hisham Sati, Ramond-Ramond fields and twisted differential K-theory,

  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):

• Daniel Grady, Hisham Sati, Twisted differential KO-theory (arXiv:1905.09085)

See also:

• Thomas Tradler, Scott Wilson, Mahmoud Zeinalian, Loop Differential K-theory (arXiv:1201.4593)