Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles

Quantum curves were introduced in the physics literature. We develop a mathematical framework for case associated with Hitchin spectral curves. In this context, quantum curve is Rees $\mathcal{D}$-module on smooth projective algebraic curve, whose semi-classical limit produces of Higgs bundle. give method quantization by concretely constructing one-parameter deformation families opers. We propose variant topological recursion Eynard--Orantin and Mirzakhani context singular show that PDE version provides all-order WKB analysis $\mathcal{D}$-modules, defined as meromorphic $SL(2,\mathbb{C})$-Higgs bundles. Topological can be considered process prove these two quantizations, one via construction opers, other type, agree holomorphic Classical differential equations such Airy equation typical example. Through classical examples, we see relate bundles, conjecture Gaiotto, invariants, Gromov--Witten invariants