[Paper Review] Quantisation conditions of the quantum Hitchin system and the real geometric Langlands correspondence

J. Teschner|arXiv (Cornell University)|Jan 1, 2017

Black Holes and Theoretical Physics23 references12 citations

TL;DR

This paper proposes a novel quantization condition for the quantum Hitchin system using a generating function Y(a, t) associated with opers—special flat connections with real holonomy—thereby establishing a real geometric Langlands correspondence. By applying the Separation of Variables (SOV) method and analyzing semiclassical limits, it shows that single-valued solutions to Hitchin Hamiltonian eigenvalue equations correspond precisely to opers with real monodromy, generalizing Frenkel's conjecture for higher genus curves and linking to conformal field theory via KZB equations.

ABSTRACT

Single-valuedness of the eigenfunctions of the quantised Hitchin Hamiltonians is proposed as a natural quantisation condition. Separation of Variables can be used to relate the classification of eigenstates to the classification of projective structures with Fuchsian holonomy. Using complex Fenchel-Nielsen coordinates one may reformulate the quantisation conditions in terms of the generating function for the variety of opers. These results are interpreted as a variant of the geometric Langlands correspondence.

Motivation & Objective

• To formulate a new quantization condition for the quantum Hitchin system beyond the Bethe ansatz framework.
• To relate single-valued solutions of Hitchin Hamiltonian eigenvalue equations to opers with real holonomy.
• To generalize E. Frenkel's conjecture on opers and quantization to higher genus Riemann surfaces.
• To establish a correspondence between real opers and single-valued eigenfunctions, defining a real variant of the geometric Langlands program.
• To connect the quantization conditions to conformal field theory via the KZB equations and critical level limits.

Proposed method

• Uses the Separation of Variables (SOV) method to construct eigenfunctions of the Hitchin Hamiltonians.
• Identifies the generating function Y(a, t) as the potential for the variety of opers within the moduli space of local systems.
• Applies complex Fenchel-Nielsen coordinates to describe the real slice of the moduli space and relate it to opers with real holonomy.
• Derives quantization conditions via the real part of action-angle variables, leading to Bohr-Sommerfeld-type conditions.
• Relates the leading asymptotics of the wave function to the Hitchin Hamiltonians through the KZB equation and critical level limit.
• Uses the Riemann-Hilbert correspondence to link algebraic structures on flat connections to the geometric Langlands correspondence.

Experimental results

Research questions

• RQ1 How can quantization conditions for the quantum Hitchin system be formulated without relying on the Bethe ansatz?
• RQ2 What is the role of the generating function Y(a, t) in encoding the quantization conditions for opers with real holonomy?
• RQ3 How does the SOV method relate single-valued eigenfunctions to opers in the real geometric Langlands correspondence?
• RQ4 In what way does the semiclassical limit of the KZB equation recover the Bohr-Sommerfeld quantization conditions?
• RQ5 What is the precise correspondence between real opers and single-valued solutions of the Hitchin eigenvalue equations?

Key findings

• The quantization conditions for the Hitchin system are shown to be equivalent to Bohr-Sommerfeld conditions on real action variables, with Re(ar) = ǫ1πnr and Re(aDr) = ǫ1πmr in the semiclassical limit.
• The generating function Y(a, t) for the variety of opers is identified as the key object encoding the quantization condition, generalizing Frenkel's conjecture to genus g > 1.
• Single-valued eigenfunctions of the Hitchin Hamiltonians correspond exactly to opers with real holonomy, establishing a real geometric Langlands correspondence.
• The SOV method provides a concrete realization of the geometric Langlands correspondence by relating eigenfunctions to opers with real monodromy.
• The leading-order asymptotics of the conformal block Z(x, q) in the critical level limit matches the function W(l, q) studied in the paper, confirming consistency with conformal field theory.
• The results are rigorously derived for sl2 in genus 0 and 1, with the generalization to higher genus remaining a conjecture supported by semiclassical and CFT arguments.