# WKB solutions of difference equations and reconstruction by the topological recursion

## TL;DR

This paper establishes a connection between topological recursion and WKB solutions of $$-difference equations, applying it to Gromov-Witten invariants of  and confirming a conjecture by Dubrovin and Yang.

## Contribution

It extends determinantal formulas and topological type properties to $$-difference systems and applies these to prove a conjecture relating Gromov-Witten invariants and topological recursion.

## Key findings

- Correlation functions reconstructed from topological recursion match the $$-difference system's solutions.
- Proved the conjecture linking Gromov-Witten invariants of  to topological recursion.
- Extended the framework of topological recursion to $$-difference equations.

## Abstract

The purpose of this article is to analyze the connection between Eynard-Orantin topological recursion and formal WKB solutions of a $\hbar$-difference equation: $\Psi(x+\hbar)=\left(e^{\hbar\frac{d}{dx}}\right) \Psi(x)=L(x;\hbar)\Psi(x)$ with $L(x;\hbar)\in GL_2( (\mathbb{C}(x))[\hbar])$. In particular, we extend the notion of determinantal formulas and topological type property proposed for formal WKB solutions of $\hbar$-differential systems to this setting. We apply our results to a specific $\hbar$-difference system associated to the quantum curve of the Gromov-Witten invariants of $\mathbb{P}^1$ for which we are able to prove that the correlation functions are reconstructed from the Eynard-Orantin differentials computed from the topological recursion applied to the spectral curve $y=\cosh^{-1}\frac{x}{2}$. Finally, identifying the large $x$ expansion of the correlation functions, proves a recent conjecture made by B. Dubrovin and D. Yang regarding a new generating series for Gromov-Witten invariants of $\mathbb{P}^1$.