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

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

1994-11-10

arxiv.org/abs/hep-th/9411071v1

Theor.Math.Phys.104:783-792,1995; Teor.Mat.Fiz.104:32-42,1995

Physics

High Energy Physics

High Energy Physics - Theory

13 pages of amstex

Scientific paper

10.1007/BF02066653

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

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