Quantum geometric Langlands correspondence in positive characteristic: The GLN case

Abstract

We prove a version of the quantum geometric Langlands conjecture in characteristic $p$. Namely, we construct an equivalence of certain localizations of derived categories of twisted crystalline $\mathcal{D}$-modules on the stack of rank $N$ vector bundles on an algebraic curve $C$ in characteristic $p$. The twisting parameters are related in the way predicted by the conjecture and are assumed to be irrational (i.e., not in ${\mathbb{F}}_p$). We thus extend some previous results Braverman and Bezrukavnikov concerning a similar problem for the usual (nonquantum) geometric Langlands. In the course of the proof, we introduce a generalization of $p$-curvature for line bundles with nonflat connections, define quantum analogues of Hecke functors in characteristic $p$, and construct a Liouville vector field on the space of de Rham local systems on $C$.