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

Abstract

Let r ⩾ 2 be an integer. The generalized Brezin–Gross–Witten (BGW) tau-function for the Gelfand–Dickey hierarchy of ( r − 1 ) dependent variables (aka the r-reduced Kadomtsev–Petviashvili hierarchy) is defined as a particular tau-function that depends on ( r − 1 ) constant parameters d 1 , … , 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.