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differential K-theory

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

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

Definition

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

ch():= jk jtr(F F )Ω 2(X), 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 jj wedge factors of the curvature .

Definition

Let (V,)(V,\nabla) and (V,)(V',\nabla') be two complex vector bundles with connection.

A Chern-Simons form for this pair is a differential form

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

obtained from the concordance bundle V¯X×[0,1]\bar V \to X \times [0,1] given by pullback along X×[0,1]XX \times [0,1] \to X equipped with a connection ¯\bar \nabla such that …, by

CS(,)= 0 1ψ t *(ι /tch(¯))+d(...). 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 VV 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(,)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)Struc(X) be the set of isomorphism classes of structured bundles on XX.

Under direct sum and tensor product of vector bundles, this becomes a commutatve rig.

Let

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

be the additive group completion of this rig as usual in K-theory.

So as an additive group K^(X)\hat K(X) is the quotient of the monoid induced by direct sum on pairs (V,W)(V,W) of isomorphism classes in Struc(X)Struc(X), modulo the sub-monoid consisting of pairs (V,V)(V,V).

Hence the pair (V,0)(V,0) is the additive inverse to (0,V)(0,V) and (V,W)(V,W) may be written as VWV - W.

Theorem

K^(X)\hat K(X) is indeed a differential cohomology refinement of ordinary K-theory K(X)K(X) of XX (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” ω\omega vanishes reproduces the Simons-Sullivan model above.

The Hopkins-Singer model

See at

More models in smooth spectra

See at Differential cohomology diagram – Differential K-theory.

Examples

References

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

The equivariant cohomology version of this is in

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

Another model 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)

See also the references at fiber integration in differential K-theory.

In the context of QFT / string theory

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

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

Further

See also

Revised on December 23, 2015 04:19:21 by Urs Schreiber (195.37.209.180)