nLab Simons-Sullivan structured bundle

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 $V \to X$ 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…

Refereces

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