nLab
Simons-Sullivan structured bundle

Contents

Idea

What Simons-Sullivan 08 call a structured bundle is a certain equivalence class of vector bundles with connection, such that the corresponding Grothendieck group is a model for differential K-theory.

The equivalence relation divided out is relation 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…

Refereces

Revised on November 25, 2016 11:50:16 by Urs Schreiber (89.204.139.40)