Painlev'e 2 equation with arbitrary monodromy parameter, topological recursion and determinantal formulas

TL;DR

This paper proves that determinantal formulas for Painlevé 2 match correlation functions from topological recursion on their spectral curves, for any non-zero monodromy parameter, extending previous results and introducing new proof techniques.

Contribution

It establishes the equivalence between determinantal formulas and topological recursion correlation functions for Painlevé 2 with arbitrary monodromy, using novel methods without insertion operators.

Findings

Determinantal formulas match topological recursion correlation functions for Painlevé 2.

The method applies to two different Lax pairs with non-connected spectral curves.

Explicit computations up to genus 3 illustrate the results.

Abstract

The goal of this article is to prove that the determinantal formulas of the Painlev'e 2 system identify with the correlation functions computed from the topological recursion on their spectral curve for an arbitrary non-zero monodromy parameter. The result is established for two different Lax pairs associated to the Painlev'e 2 system, namely the Jimbo-Miwa Lax pair and the Harnad-Tracy-Widom Lax pair, whose spectral curves are not connected by any symplectic transformation. We provide a new method to prove the topological type property without using the insertion operators. In the process, taking the time parameter t to infinity gives that the symplectic invariants F(g) computed from the Hermite-Weber curve and the Bessel curve are equal to respectively. This result generalizes similar results obtained from random matrix theory in the special case where {\theta} = 0. We believe that this approach should apply for all 6 Painlev'e equations with arbitrary monodromy parameters. Explicit computations up to g = 3 are provided along the paper as an illustration of the results.