Classical limit of the geometric Langlands correspondence for $SL(2, \mathbb{C})$

TL;DR

This paper provides an explicit description of the integrable structure of Hitchin moduli spaces, using separation of variables, and relates it to the classical limit of the geometric Langlands correspondence for SL(2,C).

Contribution

It introduces explicit parameterizations of Hitchin moduli space strata and adapts integrable model techniques to connect with the classical geometric Langlands framework.

Findings

Explicit parameterizations of Hitchin moduli space strata

Application of separation of variables to Hitchin spaces

Connection to the classical limit of Drinfeld's geometric Langlands construction

Abstract

The goal of this paper is to give an explicit description of the integrable structure of the Hitchin moduli spaces. This is done by introducing explicit parameterisations for the different strata of the Hitchin moduli spaces, and by adapting the Separation of Variables method from the theory of integrable models to the Hitchin moduli spaces. The resulting description exhibits a clear analogy with Drinfeld's first construction of the geometric Langlands correspondence. It can be seen as a classical limit of a version of Drinfeld's construction which is adapted to the complex number field.