# The center of the categorified ring of differential operators

### Dario Beraldo

Université Paul Sabatier, Toulouse, France

## 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 $L\mathcal Y$: = $\mathcal Y \times_{\mathcal Y \times \mathcal Y}\mathcal Y$.

## Cite this article

Dario Beraldo, The center of the categorified ring of differential operators. J. Eur. Math. Soc. 23 (2021), no. 6, pp. 1999–2049

DOI 10.4171/JEMS/1048