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Topological recursion and quantum curves Bouchard, Vincent


Given a spectral curve, the Eynard-Orantin topological recursion constructs an infinite sequence of meromorphic differentials. Those can be assembled into a wavefunction, which is believed to be the WKB asymptotic solution of a differential equation that is a quantization of the original spectral curve. This connection may have implications for many areas of enumerative geometry. In a recent paper we proved that this statement is true for a large class of genus zero spectral curves, that includes most (if not all) cases previously studied in the literature. I would also like to report on recent progress for genus one spectral curves. We consider the family of elliptic spectral curves in Weierstrass normal form, and show explicitly that the perturbative wavefunction constructed from the topological recursion is not the right object to consider; one must include non-perturbative corrections. However, we can prove, to order five in hbar, that the non-perturbative wavefunction is the WKB solution of a quantization of the spectral curve, albeit a non-trivial one. We obtain along the way interesting identities for cycle integrals of elliptic functions. This is joint work with N. Chidambaram, B. Eynard and T. Dauphinee.

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