Simons-Sullivan structured bundle

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.

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…

- James Simons, Dennis Sullivan,
*Structured vector bundles define differential K-theory*(arXiv:0810.4935)

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