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From the Hitchin component to opers

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Laura Fredrickson
Stanford University
Thu, 2019-08-01 10:00 - 11:00
MPIM Lecture Hall

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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