Cech cocycles for differential characteristic classes -- An infinity-Lie theoretic construction
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.
Cite
@article{arxiv.1011.4735,
title = {Cech cocycles for differential characteristic classes -- An infinity-Lie theoretic construction},
author = {Domenico Fiorenza and Urs Schreiber and Jim Stasheff},
journal= {arXiv preprint arXiv:1011.4735},
year = {2013}
}
Comments
90 pages. Added discussion of computations of homotopy fibers of the infinity-Chern-Weil homomorphism. As an application a section on higher twisted differential spin structures added; in particular twisted string structures and twisted fivebrane structures are briefly discussed. Expanded introduction. References added