Generalized string equations for double Hurwitz numbers

TL;DR

This paper derives generalized string equations for double Hurwitz numbers using fermionic operators, connecting them to integrable hierarchies and spectral curves, thus offering new insights into their mathematical structure.

Contribution

It introduces a new set of generalized string equations for double Hurwitz numbers derived from fermionic bilinear relations, linking to integrable systems and spectral curves.

Findings

Generalized string equations resemble those of c=1 string theory.

A classical limit yields the Lambert curve as a spectral curve.

Connections to the Toda hierarchy and random matrix models.

Abstract

The generating function of double Hurwitz numbers is known to become a tau function of the Toda hierarchy. The associated Lax and Orlov-Schulman operators turn out to satisfy a set of generalized string equations. These generalized string equations resemble those of $c = 1$ string theory except that the Orlov-Schulman operators are contained therein in an exponentiated form. These equations are derived from a set of intertwining relations for fermiom bilinears in a two-dimensional free fermion system. The intertwiner is constructed from a fermionic counterpart of the cut-and-join operator. A classical limit of these generalized string equations is also obtained. The so called Lambert curve emerges in a specialization of its solution. This seems to be another way to derive the spectral curve of the random matrix approach to Hurwitz numbers.