A journey from the Hitchin section to the oper moduli

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.