A journey from the Hitchin section to the oper moduli

TL;DR

This paper explores quantum curves linked to Hitchin spectral curves, proving Gaiotto's conjecture for rank 2 Higgs bundles and connecting quantum curves with opers via topological recursion.

Contribution

It develops a theory of quantum curves associated with Hitchin spectral curves and provides a detailed proof of Gaiotto's conjecture for rank 2 cases.

Findings

Proof of Gaiotto's conjecture for rank 2 Higgs bundles

Identification of quantum curves with opers via topological recursion

Development of quantum curve theory for Hitchin spectral curves

Abstract

This paper provides an introduction to the mathematical notion of \emph{quantum curves}. We start with a concrete example arising from a graph enumeration problem. We then develop a theory of quantum curves associated with Hitchin spectral curves. A conjecture of Gaiotto, which predicts a new construction of opers from a Hitchin spectral curve, is explained. We give a step-by-step detailed description of the proof of the conjecture for the case of rank $2$ Higgs bundles. Finally, we identify the two concepts of \textit{quantum curve} arising from the topological recursion formalism with the limit oper of Gaiotto's conjecture.