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

Abstract

AbstractLetr⩾2be an integer. The generalized Brezin–Gross–Witten (BGW) tau-function for the Gelfand–Dickey hierarchy of(r−1)dependent variables (aka ther-reduced Kadomtsev–Petviashvili hierarchy) is defined as a particular tau-function that depends on(r−1)constant parametersd1,…,dr−1. In this paper we show that this tau-function satisfies a family of linear equations, called theW-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.