Higher genera Catalan numbers and Hirota equations for extended nonlinear Schroedinger hierarchy

TL;DR

This paper proves a conjecture linking the partition function of a specific Dubrovin--Frobenius manifold to the extended nonlinear Schrödinger hierarchy, using Hirota equations and Lax representations.

Contribution

It establishes that the partition function is a tau-function of the extended nonlinear Schrödinger hierarchy by constructing Hirota quadratic equations and deriving the Lax representation.

Findings

Partition function is a tau-function of the extended NLS hierarchy.

Constructed Hirota quadratic equations for the partition function.

Derived the Lax representation from Hirota equations.

Abstract

We consider the Dubrovin--Frobenius manifold of rank $2$ whose genus expansion at a special point controls the enumeration of a higher genera generalization of the Catalan numbers, or, equivalently, the enumeration of maps on surfaces, ribbon graphs, Grothendieck's dessins d'enfants, strictly monotone Hurwitz numbers, or lattice points in the moduli spaces of curves. Liu, Zhang, and Zhou conjectured that the full partition function of this Dubrovin--Frobenius manifold is a tau-function of the extended nonlinear Schr\"odinger hierarchy, an extension of a particular rational reduction of the Kadomtsev--Petviashvili hierarchy. We prove a version of their conjecture specializing the Givental--Milanov method that allows to construct the Hirota quadratic equations for the partition function, and then deriving from them the Lax representation.