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Quantization of Hitchin spectral curves as opers

Banff International Research Station for Mathematical Innovation and Discovery

Topological recursion of Eynard and Orantin is known to produce solutions of Pain\-leve equations through the process of quantization of spectral curves. Recently, a similar quantization procedure is discovered for arbitrary Hitchin spectral curves. This time the topological recursion that is required for quantization is not the Eynard-Orantin type. It is a recursive system of PDEs, and the result of quantization turns out to be an "oper." The correspondence between the Hitchin spectral curve and the oper is exactly the same as the scaling limit construction conjectured by D. Gaiotto. This conjecture is recently solved in a joint paper of Olivia Dumitrescu, Laura Fredrickson, Georgios Kydonakis, Rafe Mazzeo, Andrew Neitzke, and myself. I will explain how the quantization fits into the WKB analysis of the quantum curve through the PDE recursion. The talk is based on a joint work with Olivia Dumitrescu.

Mathematics; Ordinary differential equations; Dynamical systems and ergodic theory; Dynamical systems

video/mp4

Author affiliation: University of California, Davis

2017-04-05

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/61109

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/