WKB solutions of difference equations and reconstruction by the topological recursion

Abstract

The purpose of this article is to analyze the connection between Eynard-Orantin topological recursion and formal WKB solutions of a $\hslash$-difference equation: $\Psi (x+\hslash )=\left(e^{\hslash \frac{d}{dx}}\right)\Psi (x)=L(x;\hslash )\Psi (x)$ with $L(x;\hslash )\in G{L}_2((\mathbb{C}(x))[\hslash ])$. In particular, we extend the notion of determinantal formulas and topological type property proposed for formal WKB solutions of $\hslash$-differential systems to this setting. We apply our results to a specific $\hslash$-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 ${\mathbb{P}}^1$.

41 pages, 2 figures, published version in Nonlinearity

References in corpus (3)

• Reconstructing WKB from topological recursion
• Generating series for GUE correlators
• Integrable differential systems of topological type and reconstruction by the topological recursion

Cited by in corpus (3)

• On tau-functions for the Toda lattice hierarchy
• Matrix models for stationary Gromov-Witten invariants of the Riemann sphere
• Quantum Curve for strip geometries, Topological Recursion and open GW/DT invariants