Let $C$ be a compact Riemann surface. A holomorphic quadratic differential on $C$ determines a spectral curve $\Sigma$ over $C$. Given a holomorphic quadratic differential, we can associate two types of differential operators. One is an oper on $C$, also known as a holomorphic Schrödinger operator. The other is a family of first-order differential equations related to Teichmüller theory, which appear in the study of Hitchin's integrable system. Recently, Gaiotto conjectured a precise relationship between these two. I will describe the proof of this conjecture, in joint work with Olivia Dumitrescu, Georgios Kydonakis, Rafe Mazzeo, Motohico Mulase and Andrew Neitzke.

A spectral curve $\Sigma$ is also the input data for topological recursion. It has been conjectured that the invariants produced by topological recursion provide a quantization of the spectral curve. Dumitrescu–Mulase package the invariants together into a formal power series which they call a "quantum curve." It is expected that the WKB analysis of our oper should be given by the topological recursion formulated by Dumitrescu–Mulase.

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