# nLab differential K-theory

Contents

### Context

#### Differential cohomology

differential cohomology

# Contents

## 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

### 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.

See at

## Examples

### History

An early sketch of a definition, motivated by the description of D-brane charge in string theory, is in

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

### General

A review is in

The Simons-Sullivan model is due to

The basic article for the Bunke-Schick model is

A survey talk is

Differential KO-theory is studied in

Discussion of twisted differential orthogonal K-theory in

Discussion of differential versions of equivariant K-theory, in the generality of orbifolds (orbifold differential K-theory):

The equivalence of these models with the respective special case of the general construction in

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

(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

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

Discussion for the odd Chern character is in

### 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)

### In string 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):

Discussion of twisted differential orthogonal K-theory and its relation to D-brane charge in type I string theory (on orientifolds):