WKB solutions of difference equations and reconstruction by the topological recursion

International audienceThe purpose of this article is to analyze the connection between Eynard-Orantin topological recursion and formal WKB solutions of a difference equation: Ψ(x +\hbar) = exp(\frac{d}{dx}) Ψ(x) = L(x;\hbar)Ψ(x) with L(x;\hbar) ∈ 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-differential systems to this setting. We apply our results to a specific 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