Quantisation conditions of the quantum Hitchin system and the real geometric Langlands correspondence

TL;DR

This paper proposes a natural quantisation condition for the quantum Hitchin system based on eigenfunction single-valuedness, linking it to real geometric Langlands correspondence through complex coordinates and opers.

Contribution

It introduces a new quantisation condition for the quantum Hitchin system and connects it to the real geometric Langlands correspondence using complex Fenchel-Nielsen coordinates.

Findings

Quantisation condition based on eigenfunction single-valuedness

Relation between eigenstates and projective structures with real holonomy

Reformulation of quantisation conditions via generating functions for opers

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 real 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.