# nLab Cheeger-Simons differential character

### Context

#### Differential cohomology

differential cohomology

## Ingredients

• cohomology

• differential geometry

• ## Connections on bundles

• connection on a bundle

• curvature

• Chern-Weil theory

• ## Higher abelian differential cohomology

• differential function complex

• differential orientation

• ordinary differential cohomology

• differential K-theory

• differential elliptic cohomology

• differential cobordism cohomology

• ## Higher nonabelian differential cohomology

• Chern-Weil theory in Smooth∞Grpd

• ∞-Chern-Simons theory

• ## Fiber integration

• higher holonomy

• fiber integration in differential cohomology

• ## Application to gauge theory

• gauge theory

• gauge field

• quantum anomaly

• # 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 $K(-,\mathbb{Z})$.

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 $F \in \Omega^{n+1}_{cl}(X)$.

## 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.

## 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