The center of the categorified ring of differential operators

Abstract

Let $\mathcal{Y}$ be a derived algebraic stack satisfying some mild conditions. The purpose of this paper is three-fold. First, we introduce and study $\mathbb{H}(\mathcal{Y})$, a monoidal DG category that might be regarded as a categorification of the ring of differential operators on $\mathcal{Y}$. When $\mathcal{Y}={\mathrm{LS}}_G$ is the derived stack of $G$-local systems on a smooth projective curve, we expect $\mathbb{H}({\mathrm{LS}}_g)$ to act on both sides of the geometric Langlands correspondence, compatibly with the conjectural Langlands functor. Second, we construct a novel theory of D-modules on derived algebraic stacks. In contrast to usual D-modules, this new theory, to be denoted by ${\mathcal{D}}^{\mathrm{der}}$, is sensitive to the derived structure. Third, we identify the Drinfeld center of $\mathbb{H}(\mathcal{Y})$ with ${\mathcal{D}}^{\mathrm{der}}(L\mathcal{Y})$, the DG category of ${\mathcal{D}}^{\mathrm{der}}$-modules on the loop stack $L\mathcal{Y}$: = $\mathcal{Y}{\times }_{\mathcal{Y}\times \mathcal{Y}}\mathcal{Y}$.