KdV type hierarchies, the string equation and $W_{1+\infty}$ constraints

TL;DR

This paper explores the relationship between partition-based reductions of the KP hierarchy, matrix KdV equations, and $W_{1+ abla}$ constraints, establishing that certain tau-functions satisfy vacuum constraints under specific conditions.

Contribution

It demonstrates that tau-functions of reduced KP hierarchies satisfying a string equation also fulfill $W_{1+ abla}$ vacuum constraints, linking hierarchies, algebraic constraints, and vertex operator realizations.

Findings

Tau-functions of reduced KP hierarchies satisfy $W_{1+ abla}$ constraints.

Partition-based reductions lead to matrix KdV type equations.

String equations imply vacuum constraints in the $W_{1+ abla}$ algebra.

Abstract

To every partition $n=n_1+n_2+\cdots+n_s$ one can associate a vertex operator realization of the Lie algebras $a_{\infty}$ and $\hat{gl}_n$. Using this construction we make reductions of the $s$--component KP hierarchy, reductions which are related to these partitions. In this way we obtain matrix KdV type equations. Now assuming that (1) $\tau$ is a $\tau$--function of the $[n_1,n_2,\ldots,n_s]$--th reduced KP hierarchy and (2) $\tau$ satisfies a `natural' string equation, we prove that $\tau$ also satisfies the vacuum constraints of the $W_{1+\infty}$ algebra.