Hurwitz numbers and integrable hierarchy of Volterra type

TL;DR

This paper reveals that the generating function for Hurwitz numbers on the Riemann sphere is a tau function of the lattice KP hierarchy, connecting it to integrable systems and difference-differential operators.

Contribution

It demonstrates that Hurwitz number generating functions are tau functions of the lattice KP hierarchy and identifies the associated Lax operator as an exponential of a difference-differential operator.

Findings

The Lax operator is expressed as L = e^{ak L} with ak L = ext{partial}_s - v e^{- ext{partial}_s}

The operator ak L satisfies Lax equations forming a continuum Bogoyavlensky-Itoh hierarchy

Logarithmic string equations are derived and confirmed via operator factorization

Abstract

A generating function of the single Hurwitz numbers of the Riemann sphere $\mathbb{CP}^1$ is a tau function of the lattice KP hierarchy. The associated Lax operator $L$ turns out to be expressed as $L = e^{\mathfrak{L}}$, where $\mathfrak{L}$ is a difference-differential operator of the form $\mathfrak{L} = \partial_s - ve^{-\partial_s}$. $\mathfrak{L}$ satisfies a set of Lax equations that form a continuum version of the Bogoyavlensky-Itoh (aka hungry Lotka-Volterra) hierarchies. Emergence of this underlying integrable structure is further explained in the language of generalized string equations for the Lax and Orlov-Schulman operators of the 2D Toda hierarchy. This leads to logarithmic string equations, which are confirmed with the help of a factorization problem of operators.