The notion of differential character as introduced by CheegerSimons is one geometric model for the differential cohomology-refinement of ordinary integral cohomology – i.e. of the cohomology theory represented by the Eilenberg-MacLane spectrum $K(-,\mathbb{Z})$.
Accordingly, Cheeger-Simons differential characters model connections on circle n-group-principal ∞-bundles ( $U(1)$-$(n-1)$-gerbes) and as such are equivalent to other models for these structures, such as Deligne cohomology. For $n=1$ these are ordinary connections on ordinary circle group-principal bundles.
The definition of CS-differential characters encodes rather directly the higher dimensional notion of parallel transport of such higher connections: a CS-character is a rule that assigns values in the circle group $U(1)$ (whence “character”) to $n$-dimensional surfaces $\Sigma_n \to X$ in a manifold $X$, such that whenever $\Sigma_n = \partial \Sigma_{n+1}$ is the boundary of a $\phi : \Sigma_{n+1} \to X$, this assignment coincides with the integral $\int_{\Sigma_{n+1}} \phi^* F$ of a smooth curvature $(n+1)$-form $F \in \Omega^{n+1}_{cl}(X)$.
Since Cheeger-Simons characters enocde information beyond the curvature characteristic form which represents the underlying characteristic class in de Rham cohomology, they are frequently called secondary characteristic classes, in particular if the curvature characteristic form vanishes so that the corresponding Chern-Simons form becomes exact.
The original article is
building on
Further developments are in
James Simons, Dennis Sullivan, Axiomatic characterization of ordinary differential cohomology, J Topology (2008) 1 (1): 45-56 (arXiv:math/0701077, doi:10.1112/jtopol/jtm006 web pdf)
Mark Brightwell, Paul Turner, Relative differential characters (arXiv:math.AT/0408333)
For a review in the broader context of differential cohomology see also
Last revised on September 14, 2020 at 12:32:30. See the history of this page for a list of all contributions to it.