# Quantizing Weierstrass

@article{Bouchard2016QuantizingW,
title={Quantizing Weierstrass},
author={Vincent Bouchard and Nitin Kumar Chidambaram and Tyler Dauphinee},
journal={arXiv: Mathematical Physics},
year={2016}
}
• Published 2 October 2016
• Mathematics
• arXiv: Mathematical Physics
We study the connection between the Eynard-Orantin topological recursion and quantum curves for the family of genus one spectral curves given by the Weierstrass equation. We construct quantizations of the spectral curve that annihilate the perturbative and non-perturbative wave-functions. In particular, for the non-perturbative wave-function, we prove, up to order hbar^5, that the quantum curve satisfies the properties expected from matrix models. As a side result, we obtain an infinite…
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## References

SHOWING 1-10 OF 35 REFERENCES
The spectral curve and the Schrödinger equation of double Hurwitz numbers and higher spin structures
• Mathematics
• 2013
We derive the spectral curves for q-part double Hurwitz numbers, r-spin simple Hurwitz numbers, and arbitrary combinations of these cases, from the analysis of the unstable (0, 1)-geometry. We
Geometry of Spectral Curves and All Order Dispersive Integrable System
• Mathematics
• 2012
We propose a definition for a Tau function and a spinor kernel (closely related to Baker{Akhiezer functions), where times parametrize slow (of order 1=N) deformations of an algebraic plane curve.
Orbifold Hurwitz numbers and Eynard-Orantin invariants
• Mathematics
• 2012
We prove that a generalisation of simple Hurwitz numbers due to Johnson, Pandharipande and Tseng satisfy the topological recursion of Eynard and Orantin. This generalises the Bouchard-Marino
Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial
We study three-dimensional Chern-Simons theory with complex gauge group SL(2,ℂ), which has many interesting connections with three-dimensional quantum gravity and geometry of hyperbolic 3-manifolds.
Hurwitz numbers, matrix models and enumerative geometry
• Mathematics
• 2007
We propose a new, conjectural recursion solution for Hurwitz numbers at all genera. This conjecture is based on recent progress in solving type B topological string theory on the mirrors of toric
Reconstructing WKB from topological recursion
• Mathematics
• 2017
We prove that the topological recursion reconstructs the WKB expansion of a quantum curve for all spectral curves whose Newton polygons have no interior point (and that are smooth as affine curves).
Mirror symmetry for orbifold Hurwitz numbers
• Mathematics
• 2013
We study mirror symmetry for orbifold Hurwitz numbers. We show that the Laplace transform of orbifold Hurwitz numbers satisfy a dierential recursion, which is then proved to be equivalent to the
All-order asymptotics of hyperbolic knot invariants from non-perturbative topological recursion of A-polynomials
• Mathematics
• 2012
We propose a conjecture to compute the all-order asymptotic expansion of the colored Jones polynomial of the complement of a hyperbolic knot, J_N(q = exp(2u/N)) when N goes to infinity. Our
Open string amplitudes and large order behavior in topological string theory
We propose a formalism inspired by matrix models to compute open and closed topological string amplitudes in the B-model on toric Calabi-Yau manifolds. We find closed expressions for various open