Higher-Rank Mathieu Opers, Toda Chain, and Analytic Langlands Correspondence

TL;DR

This paper links the solutions of certain oper connections on a punctured sphere to the quantum Toda chain's Yang-Yang function, providing new insights into the analytic Langlands correspondence and solving related quantization problems.

Contribution

It constructs solutions to the Riemann-Hilbert problem for higher-rank opers using a nonlinear integral equation and connects these to the Toda chain's Yang-Yang function, confirming a conjecture.

Findings

Generated solutions match the Yang-Yang function of the quantum Toda chain.

Reformulated Toda chain quantization conditions via connection problems.

Interpreted results as a variant of the Analytic Langlands Correspondence.

Abstract

We study the Riemann-Hilbert problem associated to flat sections of oper connections of arbitrary rank on the twice-punctured Riemann sphere with irregular singularities of the mildest type. We construct the solutions in terms of the solutions to a single non-linear integral equation. It follows from this construction that the generating function of the submanifold of opers coincides with the Yang-Yang function of the quantum Toda chain, proving a conjecture by Nekrasov, Rosly and Shatashvili. In this way we may furthermore reformulate the quantization conditions of the Toda chain in terms of the connection problem, for which we also provide a solution. We finally interpret our results as a variant of the Analytic Langlands Correspondence for the real version of the Hitchin system corresponding to the Toda chain.