Schreiber Cocycles for differential characteristic classes

• Domenico Fiorenza, Urs Schreiber, Jim Stasheff,

  Cech cocycles for differential characteristic classes – An $\infty$-Lie theoretic construction

  (arXiv:1011.4735).

Abstract

We define for every L-∞-algebra $\mathfrak{g}$ a smooth ∞-group $G$ integrating it, and define $G$-principal ∞-bundles with connection. For every ∞-Lie algebra cocycle of suitable degree we give a refined ∞-Chern-Weil homomorphism that sends these $\infty$-bundles to classes in ordinary differential cohomology that lift the corresponding curvature characteristic forms.

As a first example we show that applied to the canonical 3-cocycle on a semisimple Lie algebra $\mathfrak{g}$, this construction reproduces the Cech-Deligne cocycle representative for the first differential Pontryagin class that was found by Brylinski-MacLaughlin. This is the Chern-Simons circle 3-bundle with connection. If its class vanishes there is a lift to a String 2-group-2-connection on a smooth $String(G)$- principal 2-bundle. As a second example we describe the higher Chern-Weil-homomorphism applied to this String-bundle which is induced by the canonical degree 7-cocycle on $\mathfrak{g}$. This yields a differential refinement of the fractional second Pontryagin class – the Chern-Simons circle 7-bundle – which is not seen by the ordinary Chern-Weil homomorphism. We end by indicating how this serves to define differential string structures.

References

In

• Urs Schreiber, differential cohomology in a cohesive topos

this is section 3.3.13 (general theory) and sections 4.1 and 4.4 (application to String- and Fivebrane structures).

A survey talk with some related material is at

• Urs Schreiber, $\infty$-Chern-Simons functionals, Talk at Higher Structures 2011, Göttingen (pdf)

For more references are listed at

• differential cohomology in an (∞,1)-topos – references