15 May 2016 Quantum geometric Langlands correspondence in positive characteristic: The GLN case
Roman Travkin
Duke Math. J. 165(7): 1283-1361 (15 May 2016). DOI: 10.1215/00127094-3449780


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 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 Fp). 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.


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Roman Travkin. "Quantum geometric Langlands correspondence in positive characteristic: The GLN case." Duke Math. J. 165 (7) 1283 - 1361, 15 May 2016. https://doi.org/10.1215/00127094-3449780


Received: 31 December 2013; Revised: 9 June 2015; Published: 15 May 2016
First available in Project Euclid: 19 February 2016

zbMATH: 06591241
MathSciNet: MR3498867
Digital Object Identifier: 10.1215/00127094-3449780

Primary: 14D24
Secondary: 14G17

Keywords: $D$-modules , $p$-curvature , Azumaya algebra , characteristic $p$ , geometric Langlands , quantization , quantum Hecke functors

Rights: Copyright © 2016 Duke University Press


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Vol.165 • No. 7 • 15 May 2016
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