nLab Cheeger-Simons differential character

Idea

Differential characters [Cheeger & Simons 1985] are one geometric model for ordinary differential cohomology, hence for the differential cohomology-refinement of integral ordinary cohomology – i.e. of the cohomology theory represented by the Eilenberg-MacLane spectrum $K(-,\mathbb{Z})$.

Accordingly, Cheeger-Simons differential characters model circle n-bundles with connection ($U(1)$-$(n-1)$-bundle gerbes) and as such are equivalent to other models for these structures, notably to Deligne cohomology. For $n=1$ these are equivalently 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 Cheeger-Simons-character is a rule that assigns values in the circle group $U(1)$ (whence “character”) to $n$-dimensional smooth manifolds $\Sigma_n \to X$ in a smooth manifold $X$, such that whenever $\Sigma_n = \partial \Sigma_{n+1}$ is the boundary of a $\phi \colon \Sigma_{n+1} \to X$, this assignment coincides with the integral $\int_{\Sigma_{n+1}} \phi^* F$ of the pullback of a 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 closed.

Related concepts

• ordinary differential cohomology

• Chern-Simons form

• circle n-bundle with connection

• Chern-Weil homomorphism

• secondary characteristic class

• Cheeger-Simons homomorphism

References

The original article:

• Jeff Cheeger, James Simons: Differential characters and geometric invariants, in: Geometry and Topology, Lecture Notes in Mathematics, 1167 Springer (1985) 50-80 [pdf, doi:10.1007/BFb0075212]

building on

• Shiing-shen Chern, James Simons: Characteristic forms and geometric invariants, The Annals of Mathematics, Second Series 99 1 (1974) 48-69 [doi:10.2307/1971013, jstor:1971013]

Monograph:

• Christian Bär, Christian Becker: Differential Chafacters, Lecture Notes in Mathematics 2112 [doi:10.1007/978-3-319-07034-6]

See in particular:

• Christian Bär, Christian Becker, Differential Characters and Geometric Chains, in: Differential Characters, Lecture Notes in Mathematics 2112, Springer (2014) [arXiv:1303.6457, doi:10.1007/978-3-319-07034-6_1]

Review in the broader context of differential cohomology:

• Mike Hopkins, Isadore Singer: Cheeger-Simons Cohomology, section 3 in: Quadratic Functions in Geometry, Topology, and M-Theory, J. Differential Geom. 70 3 (2005) 329-452 [arXiv:math.AT/0211216, doi:10.4310/jdg/1143642908, euclid:1143642908]

Further discussion:

• Mark Brightwell, Paul Turner: Relative differential characters, Communications in Analysis and Geometry, 14 2 (2006) 269-282 [doi:10.4310/CAG.2006.v14.n2.a3, arXiv:math.AT/0408333]

  (relative cohomology)

• James Simons, Dennis Sullivan: Axiomatic characterization of ordinary differential cohomology, J Topology 1 1 (2008) 45-56 [arXiv:math/0701077, doi:10.1112/jtopol/jtm006m web pdf]

  (the differential cohomology hexagon)

• Christian Becker, Marco Benini, Alexander Schenkel, Richard J. Szabo: Cheeger-Simons differential characters with compact support and Pontryagin duality, Communications in Analysis and Geometry 27 7 (2019) 1473-1522 [arXiv:1511.00324, doi:10.4310/CAG.2019.v27.n7.a2]

  (Pontryagin duality)