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# Mathematics > Algebraic Geometry

# Title: An analytic version of the Langlands correspondence for complex curves

(Submitted on 26 Aug 2019 (v1), last revised 13 Jul 2021 (this version, v4))

Abstract: The Langlands correspondence for complex curves is traditionally formulated in terms of sheaves rather than functions. Recently, Langlands asked whether it is possible to construct a function-theoretic version. In this paper we use the algebra of commuting global differential operators (quantum Hitchin Hamiltonians and their complex conjugates) on the moduli space of G-bundles of a complex algebraic curve to formulate a function-theoretic correspondence. We conjecture the existence of a canonical self-adjoint extension of the symmetric part of this algebra acting on an appropriate Hilbert space and link its spectrum with the set of opers for the Langlands dual group of G satisfying a certain reality condition, as predicted earlier by Teschner. We prove this conjecture for G=GL(1) and in the simplest non-abelian case.

## Submission history

From: Edward Frenkel [view email]**[v1]**Mon, 26 Aug 2019 13:37:21 GMT (63kb)

**[v2]**Mon, 4 Nov 2019 19:58:45 GMT (64kb)

**[v3]**Wed, 6 May 2020 17:47:15 GMT (65kb)

**[v4]**Tue, 13 Jul 2021 12:27:03 GMT (66kb)

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