Voros Coefficients for the Hypergeometric Differential Equations and Eynard-Orantin's Topological Recursion - Part I : For the Weber Equation

TL;DR

This paper explores the quantization of spectral curves related to the Weber hypergeometric equation using topological recursion, establishing a link between Voros coefficients and free energy, with explicit formulas involving Bernoulli numbers.

Contribution

It introduces a novel quantization scheme for spectral curves and relates Voros coefficients to free energy for the Weber equation, extending to the confluent hypergeometric family.

Findings

Established a relation between Voros coefficients and free energy for the Weber equation.

Derived explicit formulas for free energy using Bernoulli numbers.

Generalized results to other confluent hypergeometric equations in the second part.

Abstract

We develop the theory of quantization of spectral curves via the topological recursion. We formulate a quantization scheme of spectral curves which is not necessarily admissible in the sense of Bouchard and Eynard. The main result of this paper and the second part [IKoT] establishes a relation between the Voros coefficients for the quantum curves and the free energy for spectral curves associated with the confluent family of Gauss hypergeometric differential equations. We focus on the Weber equation in this article, and generalize the result for the other members of the confluent family in the second part. We also find explicit formulas of free energy for those spectral curves in terms of the Bernoulli numbers.