Chern-Weil theory for ∞-local systems

Let G G be a compact connected Lie group with Lie algebra g \mathfrak {g} . We show that the category L o c ∞ ⁡ ( B G ) \operatorname {\mathbf {Loc}} _\infty (BG) of ∞ \infty -local systems on the classifying space of G G can be described infinitesimally as the category I n f L o c ∞ ( g ) {\operatorname {\mathbf {Inf}\mathbf {Loc}}} _{\infty }(\mathfrak {g}) of basic g \mathfrak {g} - L ∞ L_\infty spaces. Moreover, we show that, given a principal bundle π : P → X \pi \colon P \to X with structure group G G and any connection θ \theta on P P , there is a differntial graded (DG) functor C W θ : I n f L o c ∞ ( g ) ⟶ L o c ∞ ( X ) , \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 P . The DG functors associated to different connections are related by an A ∞ A_\infty -natural isomorphism. This construction provides a categorification of the Chern-Weil homomorphism, which is recovered by applying the functor C W θ \mathscr {CW}_{\theta } to the endomorphisms of the constant ∞ \infty -local system.