[Paper Review] Voros Coefficients and the Topological Recursion for a Class of the Hypergeometric Differential Equations associated with the Degeneration of the 2-dimensional Garnier System

Yumiko Takei|arXiv (Cornell University)|May 17, 2020

Algebraic structures and combinatorial models16 references4 citations

TL;DR

This paper establishes a direct correspondence between Voros coefficients in exact WKB analysis and the free energy computed via Eynard-Orantin topological recursion for hypergeometric differential equations associated with 2-dimensional degenerate Garnier systems. It proves that the Voros coefficients for the (1,4) and (2,3) types are expressible as difference values of the free energy, with explicit closed-form expressions derived using Bernoulli polynomials and shift operators, generalizing earlier results on confluent Gauss hypergeometric equations.

ABSTRACT

In my joint papers with Iwaki and Koike ([IKoT1, IKoT2]) we found an intriguing relation between the Voros coefficients in the exact WKB analysis and the free energy in the topological recursion introduced by Eynard and Orantin in the case of the confluent family of the Gauss hypergeometric differential equations. In this paper we discuss its generalization to the case of the hypergeometric differential equations associated with $2$-dimensional degenerate Garnier systems.

Motivation & Objective

• To generalize the known relation between Voros coefficients and topological recursion free energy from confluent Gauss hypergeometric equations to hypergeometric equations linked to 2-dimensional degenerate Garnier systems.
• To establish a precise mathematical correspondence between the Voros coefficients—defined via contour integrals of WKB solution logarithmic derivatives—and the free energy from Eynard-Orantin topological recursion.
• To derive explicit analytical expressions for the Voros coefficients of the (1,4) and (2,3) hypergeometric differential equations using this correspondence.
• To validate the method through detailed computation for the (1,4) case and extend the result to the (2,3) case via similar techniques.

Proposed method

• Utilizes exact WKB analysis to define Voros coefficients as contour integrals of the logarithmic derivative of WKB solutions, capturing parametric Stokes phenomena.
• Applies Eynard-Orantin topological recursion to spectral curves derived from the classical limits of the differential equations, computing correlation functions and free energies $F_g$.
• Establishes a key identity linking the regularized Voros coefficient $V_{\text{reg}}$ to the free energy $F$ via shift operators and difference equations involving $\hbar\partial_{\lambda_{\infty}}$.
• Employs generating functions of Bernoulli polynomials $B_m(X)$ to express the free energy and Voros coefficients in closed form.
• Uses the three-term difference equation $F(\lambda_{\infty}+\hbar,t;\hbar) - 2F(\lambda_{\infty},t;\hbar) + F(\lambda_{\infty}-\hbar,t;\hbar) = 0$ to constrain solutions and prove uniqueness.
• Applies operator inversion techniques using the identity $e^{-Xw}(e^w - 1)\left(\frac{1}{w} + \sum_{m=0}^\infty \frac{B_{m+1}(X)}{m+1}\frac{w^m}{m!}\right) = 1$ to solve for the Voros coefficient in terms of $F_0$ and its derivatives.

Experimental results

Research questions

• RQ1 Can the relation between Voros coefficients and topological recursion free energy, previously observed in confluent Gauss hypergeometric equations, be extended to hypergeometric equations associated with 2-dimensional degenerate Garnier systems?
• RQ2 What is the explicit analytical form of the Voros coefficients for the (1,4) and (2,3) hypergeometric differential equations?
• RQ3 How do the free energy and Voros coefficient relate through shift operators and difference equations in this generalized setting?
• RQ4 Is the free energy $F_g$ uniquely determined by the classical spectral curve and the admissibility condition in the context of these equations?
• RQ5 Can the topological recursion framework be used to compute parametric Stokes phenomena via Voros coefficients in higher-dimensional Garnier systems?

Key findings

• The Voros coefficient for the (1,4) hypergeometric equation is given explicitly by $V(\lambda_{\infty},t,\nu_{\infty},\hbar) = \sum_{m=1}^{\infty} \frac{B_{m+1}(\nu_{\infty})}{m(m+1)} \left(\frac{\hbar}{\lambda_{\infty}}\right)^m$, involving Bernoulli polynomials.
• For the (2,3) hypergeometric equation, the Voros coefficient vanishes identically: $V(\lambda_{\infty},t,\nu_{\infty},\hbar) = 0$, indicating absence of nontrivial parametric Stokes phenomena in this case.
• The free energy $F_0$ for the (1,4) curve is $F_0 = \frac{1}{2}\log(-3\lambda_{\infty}^2)$, which is used to construct the full free energy series $F_g$ via recursion.
• The difference between the full solution $F$ and its regularized version $\hat{F}$ vanishes at all genera, proving that $F_g = \hat{F}_g$ for all $g \geq 2$, ensuring consistency of the solution.
• The relation $V_{\text{reg}} = e^{-\nu_{\infty}\hbar\partial_{\lambda_{\infty}}} (e^{\hbar\partial_{\lambda_{\infty}}} - 1) F$ holds, linking the Voros coefficient to the free energy through shift operators.
• The method successfully generalizes the WKB-topological recursion correspondence to higher-dimensional Garnier systems, providing a framework for computing parametric Stokes phenomena in complex differential systems.