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

TL;DR

This paper develops a categorified version of the Chern-Weil theory for $ abla$-local systems, relating $ ext{Lie}(G)$-algebraic structures to geometric data on principal bundles, and introduces DG functors connecting these concepts.

Contribution

It introduces a categorification of the Chern-Weil homomorphism via DG functors between categories of $ abla$-local systems, linking algebraic and geometric perspectives.

Findings

Establishes an equivalence between $ ext{Loc}_ infty(BG)$ and basic $ ext{g}$-$L_ infty$ spaces.

Constructs a DG functor $ ext{CW}_ heta$ relating algebraic and geometric categories.

Shows that different connections yield $A_ infty$-isomorphic functors, ensuring consistency.

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 $\pi \colon P \rightarrow X$ with structure group $G$ and any connection $\theta$ on $P$, there is a DG functor $$\mathcal{CW}_{\theta} \colon \mathbf{InfLoc}_{\infty}(\mathfrak{g}) \longrightarrow \mathbf{Loc}_{\infty}(X), $$ 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 $\mathcal{CW}_{\theta}$ to the endomorphisms of the constant local system.