JournalsjemsVol. 23, No. 6pp. 1999–2049

The center of the categorified ring of differential operators

  • Dario Beraldo

    Université Paul Sabatier, Toulouse, France
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Let Y\mathcal 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)\mathbb H(\mathcal Y), a monoidal DG category that might be regarded as a categorification of the ring of differential operators on Y\mathcal Y. When Y=LSG\mathcal Y = \mathrm {LS}_G is the derived stack of GG-local systems on a smooth projective curve, we expect H(LSg)\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 Dder\mathcal D^{\mathrm{der}}, is sensitive to the derived structure. Third, we identify the Drinfeld center of H(Y)\mathbb H(\mathcal Y) with Dder(LY)\mathcal D^{\mathrm{der}}(L\mathcal Y), the DG category of Dder\mathcal D^{\mathrm{der}}-modules on the loop stack LYL\mathcal Y: = Y×Y×YY\mathcal Y \times_{\mathcal Y \times \mathcal Y}\mathcal Y.

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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