[Paper Review] Voros Coefficients for the Hypergeometric Differential Equations and Eynard-Orantin's Topological Recursion - Part II : For the Confluent Family of Hypergeometric Equations

Kohei Iwaki, Tatsuya Koike+1 more|arXiv (Cornell University)|Oct 6, 2018

Nonlinear Waves and Solitons26 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 the confluent family of Gauss hypergeometric equations. It proves that the Voros coefficient equals the difference of free energies for quantum curves derived from spectral curves, and provides explicit expressions in terms of Bernoulli numbers and polynomials, generalizing earlier results on the Weber curve to a broader class of quantum curves including Kummer, Bessel, and Legendre types.

ABSTRACT

We show that the 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 coefficients of those equations and the free energy of their classical limit computed by the topological recursion are established. We will also find explicit expressions of the free energy and the Voros coefficients in terms of the Bernoulli numbers and Bernoulli polynomials.

Motivation & Objective

• To establish a correspondence between Voros coefficients from exact WKB analysis and free energies computed via Eynard-Orantin topological recursion for quantum curves.
• To generalize the known relation between Voros coefficients and free energy—previously established for the Weber curve—to the entire confluent family of Gauss hypergeometric equations.
• To derive explicit expressions for both the free energy and Voros coefficients in terms of Bernoulli numbers and polynomials.
• To demonstrate that each member of the confluent family of hypergeometric equations arises as a quantum curve for an appropriate spectral curve.

Proposed method

• Utilizes Eynard-Orantin topological recursion to compute free energies $ F_g $ from spectral curves associated with the confluent family of hypergeometric equations.
• Applies exact WKB analysis to define Voros coefficients as integrals of the WKB expansion's logarithmic derivative along specific contours.
• Establishes a key identity showing the Voros coefficient equals the difference of free energies, generalizing the Weber curve result from Part I.
• Employs contiguity relations and difference equations involving Bernoulli numbers and polynomials to derive closed-form expressions for both free energy and Voros coefficients.
• Uses formal power series solutions to difference equations involving $ rac{1}{ar{h}}rac{d}{dar{\lambda}} $ to solve for logarithmic generating functions.
• Applies properties of Bernoulli polynomials $ B_n(x) $ and generating functions to express solutions in terms of $ B_n $ and $ B_n(x) $.

Experimental results

Research questions

• RQ1 How are the Voros coefficients of the confluent family of hypergeometric equations related to the free energy computed by topological recursion?
• RQ2 Can the explicit formula for the free energy of the Weber curve be generalized to other quantum curves in the confluent family?
• RQ3 What role do Bernoulli numbers and polynomials play in expressing the free energy and Voros coefficients for these quantum curves?
• RQ4 How does the topological recursion structure relate to the exact WKB analysis in the context of confluent hypergeometric equations?

Key findings

• The Voros coefficient for each quantum curve in the confluent family is equal to the difference of the free energy computed via topological recursion.
• Explicit expressions for both the free energy and Voros coefficients are derived in terms of Bernoulli numbers $ B_n $ and Bernoulli polynomials $ B_n(x) $.
• The free energy for the quantum Gauss curve is expressed as a series involving $ (ar{h}rac{d}{dar{\lambda}})^{-k} $ and Bernoulli numbers, with a closed-form solution via difference equations.
• The method recovers the known formula for the Weber curve's free energy in terms of Bernoulli numbers, confirming consistency with prior results.
• Solutions to difference equations involving $ rac{1}{ar{h}}rac{d}{dar{\lambda}} $ are constructed using generating functions and Bernoulli polynomials, yielding explicit series for $ F(ar{\lambda}) $.
• The formal power series solutions to homogeneous difference equations $ F(ar{\lambda}+ar{h}) - F(ar{\lambda}) = 0 $ and $ F(ar{\lambda}+ar{h}) - 2F(ar{\lambda}) + F(ar{\lambda}-ar{h}) = 0 $ are shown to be constant or linear in $ ar{\lambda} $, respectively, due to vanishing derivatives.