The center of the categorified ring of differential operators

TL;DR

This paper introduces a categorified ring of differential operators on derived stacks, develops a new D-module theory sensitive to derived structures, and links the Drinfeld center of this categorification to derived D-modules on loop stacks, advancing geometric Langlands insights.

Contribution

It defines H( ext{Y}), a monoidal DG category as a categorification of differential operators, and establishes its center as a new derived D-module category on loop stacks, connecting to Langlands program.

Findings

H( ext{Y}) is a monoidal DG category generalizing differential operators.

A new derived D-module theory, D^{der}, sensitive to derived structures, is constructed.

The Drinfeld center of H( ext{Y}) is identified with D^{der} on the loop stack.

Abstract

Let \Y be a derived algebraic stack satisfying some mild conditions. The purpose of this paper is three-fold. First, we introduce and study H(\Y), a monoidal DG category that might be regarded as a categorification of the ring of differential operators on \Y. When \Y = \LS_G is the derived stack of G-local systems on a smooth projective curve, we expect H(\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. Contrarily to usual D-modules, this new theory, to be denoted by D^{der}, is sensitive to the derived structure. Third, we identify the Drinfeld center of H(\Y) with D^{der}(L\Y), the DG category of D^{der}-modules on the loop stack of \Y.