Quantization of the Quantum Hitchin System and the Real Geometric Langlands Correspondence

Abstract

Main resultsWe are going to propose a natural quantisation condition for the Hitchin system, and explain how it can be reformulated in terms of a function Y(a, t). The function Y(a, t) relevant for this task is found to be the generating function for the variety of opers within the space of all local systems as predicted in [6,9]. However, the condition on Y expressing the quantisation condition turns out to be different from the types of conditions considered in [1]. Our derivation is essentially complete for Hitchin systems associated to the Lie algebra sl 2 in genus 0 and 1, which may be called the Gaudin and elliptic Calogero-Moser models assciated to the group SL(2, C). It reduces to a conjecture of E. Frenkel [11] for g > 1, as will be discussed below.Reformulating the quantisation conditions in terms of Y can be done using the Separation of Variables (SOV) method pioneered by Sklyanin [12]. This method may be seen as a more concrete procedure to construct the geometric Langlands correspondence relating opers to Dmodules (eigenvalue equations), as was pointed out in [11]. In our case it will be found that the SOV method relates single-valued solutions of the eigenvalue equations to opers having real holonomy. This problem is closely related to the classification of projective structures on C with real holonomy which has been studied in [13]. Using complex Fenchel-Nielsen coordinates we will reformulate this description in terms of the generating function for the variety of opers.From the point of view of the geometric Langlands correspondence we obtain a correspondence between opers with real holonomy and D-modules admitting single-valued solutions. We expect that a generalisation to more general local systems with real holonomy will exist. We propose to call such correspondences the real geometric Langlands correspondence. Separation of variables for the classical Hitchin integrable system 2.1 Integrability and special geometryA complex symplectic manifold M with holomorphic symplectic form Ω is called an algebraic integrable system if it can be described as a Lagrangian torus fibration π : M → B with fibres being principally polarised abelian varieties. Algebraic integrability is equivalent to the fact that the base B is a special Kähler manifold satisfying certain integrality conditions [14].These connections may be reformulated conveniently in terms of a covering of M with local charts carrying action-angle coordinates consisting of a tuple a = (a 1 , . . . , a d ) of coordinates for the base B, and complex coordinates z = (z 1 , . . . , z d ) for the torus fibres( 2.3)The transformation z D := τ −1 b · z gives an equivalent representation of the torus fibres Θ b . It

Is There an Analytic Theory of Automorphic Functions for Complex Algebraic Curves?

“…In other words, these are the connections with monodromy in the split real form GL 1 (R) of GL 1 (C). This dovetails nicely with the conjecture of Teschner [50] in the case of G = SL 2 . We expect an analogous statement to hold for a general reductive group G, see [16].…”

“…In other words, the objects of the analytic theory of automorphic forms on Bun G can be constructed from the objects of the geometric Langlands theory in roughly the same way as the correlation functions of CFT are constructed from conformal blocks. This was predicted in [24] and [50]. An important difference with the CFT is that whereas in CFT the monodromy of conformal blocks is typically unitary, here we expect the monodromy to be in a split real group.…”

“…As far as I know, the system (4.4) was first considered by Teschner [50], in the case of G = SL 2 (a similar idea was also proposed in [24]). Teschner did not consider (4.4) as a spectral problem in the sense of self-adjoint operators acting on a Hilbert space.…”

“…There is, however, another possibility: rather than looking for the eigenfunctions of Hecke operators, one can look for the eigenfunctions of global differential operators on Bun G . These eigenfunctions and the corresponding eigenvalues have been recently studied for G = SL 2 by Joerg Teschner [50]. In a joint work with Pavel Etingof and David Kazhdan [16], we propose a canonical self-adjoint extension of the algebra of these differential operators and study the corresponding spectral problem.…”

“…If G = SL 2 , then L G = PGL 2 and PGL 2 -opers are the same as projective connections. Teschner [50] proposed that in this case, the eigenvalues correspond to the projective connections with monodromy taking values in the split real form PGL 2 (R) of PGL 2 (C) (up to conjugation by an element of PGL 2 (C)). Such projective connections have been described by W.M.…”