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

Abstract:

Let $G$ be a compact connected Lie group with Lie algebra $\mathfrak {g}$. We show that the category $\operatorname {\mathbf {Loc}} _\infty (BG)$ of $\infty$-local systems on the classifying space of $G$ can be described infinitesimally as the category ${\operatorname {\mathbf {Inf}\mathbf {Loc}}} _{\infty }(\mathfrak {g})$ of basic $\mathfrak {g}$-$L_\infty$ spaces. Moreover, we show that, given a principal bundle $\pi \colon P \to X$ with structure group $G$ and any connection $\theta$ on $P$, there is a differntial graded (DG) functor \begin{equation*} \mathscr {CW}_{\theta } \colon \mathbf {Inf}\mathbf {Loc}_{\infty }(\mathfrak {g}) \longrightarrow \mathbf {Loc}_{\infty }(X), \end{equation*} 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 $\mathscr {CW}_{\theta }$ to the endomorphisms of the constant $\infty$-local system.