Cech cocycles for differential characteristic classes -- An infinity-Lie theoretic construction

TL;DR

This paper develops a new infinity-Lie theoretic framework to refine secondary characteristic classes from cohomology to differential cohomology, enabling the study of higher geometric structures and their obstructions.

Contribution

It introduces a novel construction of differential characteristic classes as morphisms between infinity-groupoids, extending classical theory to higher connected covers and smooth infinity-groups.

Findings

Realizes differential characteristic classes as morphisms in infinity-groupoids.

Analyzes homotopy fibers to understand differential obstruction problems.

Applies to higher twisted differential structures like spin, string, and fivebrane structures.

Abstract

What are called secondary characteristic classes in Chern-Weil theory are a refinement of ordinary characteristic classes of principal bundles from cohomology to differential cohomology. We consider the problem of refining the construction of secondary characteristic classes from cohomology sets to cocycle spaces; and from Lie groups to higher connected covers of Lie groups by smooth infinity-groups, i.e., by smooth groupal A-infinity-spaces. Namely, we realize differential characteristic classes as morphisms from infinity-groupoids of smooth principal infinity-bundles with connections to infinity-groupoids of higher U(1)-gerbes with connections. This allows us to study the homotopy fibers of the differential characteristic maps thus obtained and to show how these describe differential obstruction problems. This applies in particular to the higher twisted differential spin structures called twisted differential string structures and twisted differential fivebrane structures.