Connections on bundles
Higher abelian differential cohomology
Higher nonabelian differential cohomology
Application to gauge theory
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 be a complex vector bundle with connection and curvature 2-form
The Chern character of is the inhomogenous curvature characteristic form
where on the right we have wedge factors of the curvature .
Let and be two complex vector bundles with connection.
A Chern-Simons form for this pair is a differential form
obtained from the concordance bundle given by pullback along equipped with a connection such that …, by
Proposition This is indeed well defined in that it is independent of the chosen concordance, up to an exact term.
A structured bundle in the sense of the Simons-Sullivan model is a complex vector bundle equipped with the equivalence class of a connection under the equivalence relation that identifies two connections and if their Chern-Simons form is exact.
Two structured bundles are isomorphic if there is a vector bundle isomorphism under which the two equivalence classes of connections are identified.
Let be the set of isomorphism classes of structured bundles on .
Under direct sum and tensor product of vector bundles, this becomes a commutatve rig.
be the additive group completion of this rig as usual in K-theory.
So as an additive group is the quotient of the monoid induced by direct sum on pairs of isomorphism classes in , modulo the sub-monoid consisting of pairs .
Hence the pair is the additive inverse to and may be written as .
is indeed a differential cohomology refinement of ordinary K-theory of (i.e. of the 0th cohomology group of K-cohomology).