Microlocal sheaf categories and the $J$-homomorphism

Abstract

Let $X$ be a smooth manifold and $\mathbf{k}$ be a commutative (or at least $\mathbb{E}_2$) ring spectrum. Given a smooth exact Lagrangian $L\hookrightarrow T^*X$, the microlocal sheaf theory (following Kashiwara--Schapira) naturally assigns a locally constant sheaf of categories on $L$ with fiber equivalent to the category of $\mathbf{k}$-spectra $\mathrm{Mod}(\mathbf{k})$. We show that the classifying map for the local system of categories factors through the stable Gauss map $L\rightarrow U/O$ and the delooping of the $J$-homomorphism $U/O\rightarrow B\mathrm{Pic}(\mathbf{S})$. As an application, combining with previous results of Guillermou [Gui], we recover a result of Abouzaid--Kragh [AbKr] on the triviality of the composition $L\rightarrow U/O\rightarrow B\mathrm{Pic}(\mathbf{S})$, when $L$ is in addition compact.