Integrable differential systems of topological type and reconstruction by the topological recursion

TL;DR

This paper proves that certain rational Lax pair systems with genus zero spectral curves satisfy the topological type property, leading to their correlators obeying topological recursion, with applications to minimal models and Painlevé systems.

Contribution

It establishes the topological type property for a class of integrable systems and links their correlators to topological recursion, extending the scope of these methods.

Findings

Systems satisfy the topological type property under specified conditions.

Determinantal correlators follow the topological recursion.

Applicable to minimal models and Painlevé systems.

Abstract

Starting from a $d\times d$ rational Lax pair system of the form $\hbar \partial_x \Psi= L\Psi$ and $\hbar \partial_t \Psi=R\Psi$ we prove that, under certain assumptions (genus $0$ spectral curve and additional conditions on $R$ and $L$), the system satisfies the "topological type property". A consequence is that the formal $\hbar$-WKB expansion of its determinantal correlators, satisfy the topological recursion. This applies in particular to all $(p,q)$ minimal models reductions of the KP hierarchy, or to the six Painlev\'e systems.