nLab
Cheeger-Simons differential character

Contents

Idea

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.

Relatet concepts

References

The original article is

  • Jeff Cheeger, James Simons, Differential characters and geometric invariants , LNM 1167, pages 50–80. Springer Verlag, (1985) (pdf)

building on

Further developments are in

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

Revised on October 29, 2013 13:52:59 by Urs Schreiber (82.169.114.243)