Gelfand--Dickey hierarchy, generalized BGW tau-function, and $W$-constraints

TL;DR

This paper demonstrates that the generalized BGW tau-function for the Gelfand--Dickey hierarchy satisfies specific $W$-constraints, establishing a correspondence between parameters and linear equations in the hierarchy.

Contribution

It introduces $W$-constraints of the second kind for the generalized BGW tau-function and establishes a parameter correspondence, advancing understanding of the hierarchy's structure.

Findings

Tau-function satisfies $W$-constraints of the second kind.

One-to-one correspondence between parameters and operators.

Provides new linear equations for the hierarchy.

Abstract

Let $r\geq 2$ be an integer. The generalized BGW tau-function for the Gelfand--Dickey hierarchy of $(r-1)$ dependent variables (aka the $r$-reduced KP hierarchy) is defined as a particular tau-function that depends on $(r-1)$ constant parameters $d_1,\dots,d_{r-1}$. In this paper we show that this tau-function satisfies a family of linear equations, called the $W$-constraints of the second kind. The operators giving rise to the linear equations also depend on $(r-1)$ constant parameters. We show that there is a one-to-one correspondence between the two sets of parameters.