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Topological recursion and quantum curves Bouchard, Vincent
Description
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.
Item Metadata
Title |
Topological recursion and quantum curves
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2016-09-27T14:31
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Description |
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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Extent |
52 minutes
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: University of Alberta
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Series | |
Date Available |
2017-03-29
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0343380
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Faculty
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Rights URI | |
Aggregated Source Repository |
DSpace
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Item Media
Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International