Quantum geometric Langlands correspondence in positive characteristic: the GL(N) case

TL;DR

This paper proves a version of the quantum geometric Langlands conjecture in characteristic p for GL(N), establishing an equivalence of derived categories of twisted crystalline D-modules on vector bundle stacks.

Contribution

It extends the quantum geometric Langlands framework to positive characteristic, introducing new tools like generalized p-curvature and quantum Hecke functors.

Findings

Constructed equivalence of derived categories of twisted crystalline D-modules

Introduced generalized p-curvature for line bundles with non-flat connections

Defined quantum Hecke functors in characteristic p

Abstract

We prove a version of 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 the results of arXiv:math/0602255 concerning the similar problem for the usual (non-quantum) geometric Langlands. In the course of the proof, we introduce a generalization of $p$-curvature for line bundles with non-flat connections, define quantum analogs of Hecke functors in characteristic $p$ and construct a Liouville vector field on the space of de Rham local systems on $C$.