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

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.

This result allows for a categorification of the Chern-Weil construction of characteristic classes. The categories mentioned above are quasi-equivalent to the category of infinity-local systems on the classifying space of $G$. The Chern-Weil homomorphism can be promoted to a Chern-Weil functor taking values in the dg-category of infinity-local systems. The Chern-Weil homomorphism is then recovered by applying the functor to the endomorphisms of the unit object. If time permits, I will discuss how a monoidal version of the equivalence above is related to string topology of classifying spaces and other results in Lie theory.

The talk is based on joint works with A. Quintero and S. Pineda, and work in progress with M. Rivera.