# Quantizing Weierstrass

@article{Bouchard2016QuantizingW, title={Quantizing Weierstrass}, author={Vincent Bouchard and Nitin Kumar Chidambaram and Tyler Dauphinee}, journal={arXiv: Mathematical Physics}, year={2016} }

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…

## 9 Citations

Topological Recursion and Genus One Quantum Curves: An Accessible Exploration

- Mathematics
- 2017

We explore the connection between Eynard-Orantin Topological Recursion (EOTR) and the asymptotic solutions to differential equations constructed with the WKB method (named for its creators Wentzel,…

Holomorphic Anomaly and Quantum Mechanics

- Physics
- 2016

We show that the all-orders WKB periods of one-dimensional quantum mechanical oscillators are governed by the refined holomorphic anomaly equations of topological string theory. We analyze in detail…

Voros coefficients for the hypergeometric differential equations and Eynard–Orantin’s topological recursion: Part II: For confluent family of hypergeometric equations

- MathematicsJournal of Integrable Systems
- 2019

We show that each member of the confluent family of the Gauss hypergeometric equations is realized as quantum curves for appropriate spectral curves. As an application, relations between the Voros…

From topological recursion to wave functions and PDEs quantizing hyperelliptic curves

- Mathematics
- 2019

Starting from loop equations, we prove that the wave functions constructed from topological recursion on families of spectral curves with a global involution satisfy a system of partial differential…

From CFT to Ramond super-quantum curves

- Mathematics
- 2017

A bstractAs we have shown in the previous work, using the formalism of matrix and eigenvalue models, to a given classical algebraic curve one can associate an infinite family of quantum curves, which…

2-Parameter $$\tau $$-Function for the First Painlevé Equation: Topological Recursion and Direct Monodromy Problem via Exact WKB Analysis

- Mathematics
- 2020

We show that a 2-parameter family of $$\tau $$
-functions for the first Painleve equation can be constructed by the discrete Fourier transform of the topological recursion partition function for a…

Witten–Reshetikhin–Turaev Function for a Knot in Seifert Manifolds

- Mathematics
- 2020

In this paper, for a Seifert loop (i.e., a knot in a Seifert three-manifold), first we give a family of an explicit function $\Phi(q; N)$ whose special values at roots of unity are identified with…

Summary of results in topological recursion

- Mathematics
- 2018

1 General properties The topological recursion (TR) is an axiomatic construction of a family of correlation functions ωg,n indexed by two integers g, n ≥ 0, from the initial data of a spectral curve…

Cycles of curves, cover counts, and central invariants

- Mathematics
- 2019

The first topic of this dissertation is the moduli space of curves. I define half-spin relations, specialising Pandharipande-Pixton-Zvonkine’s spin relations, to reprove Buryak-Shadrin-Zvonkine’s…

## References

SHOWING 1-10 OF 34 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…

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…

A Holomorphic and background independent partition function for matrix models and topological strings

- Mathematics, Physics
- 2011

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

- Mathematics
- 2008

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…

Identification of the Givental Formula with the Spectral Curve Topological Recursion Procedure

- Mathematics
- 2014

We identify the Givental formula for the ancestor formal Gromov–Witten potential with a version of the topological recursion procedure for a collection of isolated local germs of the spectral curve.…