There are various different models that differ in the concrete realization of these cocycles and in their extra properties.
This section discusses the model presented in (SimonsSullivan).
More details will eventually be at
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.
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
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 .
Hence the pair is the additive inverse to and may be written as .
See the reference below.
The restriction of the cocycles in the Bunke-Schick model to those whose “auxialiary form” vanishes reproduces the Simons-Sullivan model above.
Daniel Freed, Dirac charge quantization and generalized differential cohomology, Surveys in Differential Geometry, Int. Press, Somerville, MA, 2000, pp. 129–194 (arXiv:hep-th/0011220)
provides in particular a model for differential K-theory.
For more historical remarks see section 1.6 of
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
(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
Discussion for the odd Chern character is in
Thomas Tradler, Scott Wilson, Mahmoud Zeinalian, An Elementary Differential Extension of Odd K-theory, J. of K-theory, K-theory and its Applications to Algebra, Geometry, Analysis and Topology, (arXiv:1211.4477)
Another model is in
Relation to index theory:
Kevin Klonoff, An Index Theorem in Differential K-Theory PdD thesis (2008) (pdf)
See also the references at fiber integration in differential K-theory.