Quantum geometric Langlands correspondence in positive characteristic: the \(\mathrm{GL}_N\) case. (English) Zbl 1430.14029

Summary: 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 \(C\).


14D24 Geometric Langlands program (algebro-geometric aspects)
14G17 Positive characteristic ground fields in algebraic geometry
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