The Algebra & Geometry of Modern Physics

This talk aims at explaining our recent solution of Gaiotto's conjecture. The conjecture is about the precise construction of opers from a flat family of connections associated with the Hitchin component of the moduli space of Higgs bundles. It has been solved very recently in collaboration with Olivia Dumitrescu, Laura Fredrickson, Georgios Kydonakis, Rafe Mazzeo and Andrew Neitzke. This theorem leads us to developing a mathematical theory of quantum curves for Hitchin spectral curves. In this talk, the result of our joint work on the conjecture is outlined. Then we give a holomorphic formula for non-Abelian Hodge correspondence for the Hitchin components.