nLab Cheeger-Simons differential character




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(,)K(-,\mathbb{Z}).

Accordingly, Cheeger-Simons differential characters model connections on circle n-group-principal ∞-bundles ( U(1)U(1)-(n1)(n-1)-gerbes) and as such are equivalent to other models for these structures, such as Deligne cohomology. For n=1n=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)U(1) (whence “character”) to nn-dimensional surfaces Σ nX\Sigma_n \to X in a manifold XX, such that whenever Σ n=Σ n+1\Sigma_n = \partial \Sigma_{n+1} is the boundary of a ϕ:Σ n+1X\phi : \Sigma_{n+1} \to X, this assignment coincides with the integral Σ n+1ϕ *F\int_{\Sigma_{n+1}} \phi^* F of a smooth curvature (n+1)(n+1)-form FΩ cl n+1(X)F \in \Omega^{n+1}_{cl}(X).

As secondary characteristic classes

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

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Further developments are in

See also:

For a review in the broader context of differential cohomology see also

Last revised on December 9, 2022 at 07:47:51. See the history of this page for a list of all contributions to it.