The $[n_1,n_2,\ldots,n_s]$--th reduced KP hierarchy and $W_{1+\infty}$ constraints

TL;DR

This paper explores the connection between reduced KP hierarchies associated with partitions and $W_{1+ abla}$ constraints, revealing their equivalence through vertex operator realizations and matrix KdV equations.

Contribution

It introduces a novel link between partition-based reductions of the KP hierarchy and $W_{1+ abla}$ constraints via vertex operator constructions.

Findings

Reduced KP hierarchies correspond to specific $W_{1+ abla}$ constraints.

Vertex operator realizations relate partitions to Lie algebra representations.

Matrix KdV equations emerge from these reductions.

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 obtain reductions of the $s$--component KP hierarchy, reductions which are related to these partitions. In this way we obtain matrix KdV type equations. We show that the following two constraints on a KP $\tau$--function are equivalent (1) $\tau$ is a $\tau$--function of the $[n_1,n_2,\ldots,n_s]$--th reduced KP hierarchy which satisfies string equation, $L_{-1}\tau=0$, (2) $\tau$ satisfies the vacuum constraints of the $W_{1+\infty}$ algebra. Talk given at the V International Conference on Mathematical Physics, String Theory and Quantum Gravity at Alushta, June 10-20 1994