An analytic version of the Langlands correspondence for complex curves

TL;DR

This paper develops a function-theoretic approach to the Langlands correspondence for complex curves using differential operators on moduli spaces, linking spectra with opers and proving the conjecture in specific cases.

Contribution

It formulates a new function-theoretic version of the Langlands correspondence using differential operators and proves the conjecture for certain groups.

Findings

Proposes a function-theoretic formulation of Langlands correspondence.

Establishes a link between the spectrum of differential operators and opers.

Proves the conjecture for G=GL(1) and the simplest non-abelian case.

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.