WKB solutions of difference equations and reconstruction by the topological recursion

Abstract

The purpose of this article is to analyze the connection between EynardOrantin topological recursion and formal WKB solutions of a -difference equation:. 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 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

On Gromov–Witten invariants of $\mathbb{P}^1$

“…Note added: Recently, O. Marchal [26] gave a proof of the Main Conjecture of the present paper by using the Chekhov-Eynard-Orantin topological recursion [27,16,17].…”

On tau-functions for the KdV hierarchy

“…Applying ∂ to both sides of this equation yields (59). This shows compatibility between the 1st and the 3rd equations in (11).…”

“…], one could use alternatively the Sato Grassmannian approach [13,26,63,72] to prove the identity (19). It would be interesting to investigate if the identity (19) could also be proved, say for V = C((x)), by using the approach of Bergère and Eynard with the employment of topological recursion (loop equation) [1,8,10,24,29,45,59,69], or by using matrix models together with appropriate Riemann-Hilbert problems (isomonodromic deformations) [12,16,17,27,29,37,39,49,57,60,66], or by using OPE from appropriate vertex algebras [5,26,47,72]; we expect that at least for the initial data f (x) ∈ V having the bispectral property defined by Duistermaat and Grünbaum [43] (see also the M -bispectrality given in Section 6.2) this might be possible. We also note that the matrix resolvent method and some of the above-mentioned methods extend to new situations (see e.g.…”

“…We also note that the matrix resolvent method and some of the above-mentioned methods extend to new situations (see e.g. [14,15,37,38,39,40], [3], [8,9,11,59] and [16,17,18,51]).…”

Gromov–Witten invariants of the Riemann sphere

“…We will prove it in Section 2.4. (A different proof was given recently by O. Marchal [27].) Theorem 3.…”