Cech cocycles for differential characteristic classes -- An infinity-Lie theoretic construction
Domenico Fiorenza, Urs Schreiber, Jim Stasheff
TL;DR
The paper develops an infinity-Lie theoretical framework to refine secondary characteristic classes from cohomology to differential cocycles, modeling differential characteristic maps as morphisms between smooth ∞-groupoids of principal ∞-bundles with connections and higher U(1)-gerbes. By integrating L∞-algebras to smooth ∞-groups and constructing differential refinements exp_Δ(g)_{diff} and exp_Δ(g)_{conn}, it defines an ∞-Chern-Weil homomorphism that lifts classical Chern-Weil theory to nonabelian and higher settings. Concrete instantiations include a differential refinement of the first fractional Pontryagin class via String-structures and a differential refinement of the second fractional Pontryagin class via Fivebrane-structures, with homotopy fibers encoding twisted differential structures tied to Green-Schwarz anomaly cancellation. The results provide a unified, higher-categorical treatment of differential obstructions and twisted differential backgrounds (e.g., twisted String and Fivebrane structures) with potential applications in string theory and dualities. Overall, the work delivers a comprehensive, differential-geometric generalization of Chern-Weil theory within the smooth ∞-topos, enabling explicit cocycle-level constructions and their obstructions in high-degree, nonabelian settings.
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.