Schreiber
Cocycles for differential characteristic classes

Content

Abstract

We define for every L-∞-algebra 𝔀\mathfrak{g} a smooth ∞-group GG integrating it, and define GG-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 𝔀\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)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 𝔀\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

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, GΓΆttingen (pdf)

For more references are listed at

Revised on November 29, 2011 12:02:16 by Urs Schreiber (134.76.83.9)