Chern-Weil theory for $\infty$-local systems

Abstract

Let $G$ be a compact connected Lie group. We show that the category ${\mathbf{Loc}}_{\infty }(BG)$ of $\infty$-local systems on the classifying space of $G$, can be described infinitesimally as the category ${\mathbf{InfLoc}}_{\infty }(\mathfrak{g})$ of basic $\mathfrak{g}$-$L_{\infty }$ spaces. Moreover, we show that, given a principal bundle $\pi :P\to X$ with structure group $G$ and any connection $\theta$ on $P$, there is a DG functor $${\mathcal{C}\mathcal{W}}_{\theta }:{\mathbf{InfLoc}}_{\infty }(\mathfrak{g})⟶{\mathbf{Loc}}_{\infty }(X),$$ which corresponds to the pullback functor by the classifying map of $P$. The DG functors associated to different connections are related by an $A_{\infty }$-natural isomorphism. This construction provides a categorification of the Chern-Weil homomorphism, which is recovered by applying the functor ${\mathcal{C}\mathcal{W}}_{\theta }$ to the endomorphisms of the constant local system.

43 pages. All comments are very welcome