Singular Chains on Lie groups, the Cartan relations and Chern-Weil theory

Hybrid.
Contact: Tobias Barthel, Viktoriya Ozornova, Teter Teichner, Aru Ray

The Lie algebra of vector fields on a manifold acts on differential forms by Lie derivatives and contractions, and these operations are related by the Cartan relations. We will explain an interpretation of these relations from the point of view of Lie theory, and describe how this leads to a categorification of the Chern-Weil homomorphism.

For a Lie group G, we consider the space of smooth singular chains C(G), which is a differential graded Hopf algebra. We show that the category of sufficiently local modules over C(G) can be described infinitesimally, as the category of representations of a dg-Lie algebra which is universal for the Cartan relations. If G is compact and simply connected, the equivalence of categories can be promoted to an A-infinity equivalence of dg-categories, which are also A-infinity equivalent to the category of infinity local systems on the classifying space BG. The equivalence can be realized explicitly to provide a categorification of the Chern-Weil homomorphism.
The talk is based on joint works with A. Quintero and S. Pineda, and work in progress with M. Rivera and F. Bischoff.