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

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 ${\widehat{gl}}_n$. 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 $[n_1,n_2,\dots ,n_s]$--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

13 pages of amstex

References in corpus (1)

• Quasifinite highest weight modules over the Lie algebra of differential operators on the circle