The $n$--TH Reduced BKP Hierarchy, the String Equation and $BW_{1+\infty}$--Constraints

Abstract

We study the BKP hierarchy and its $n$--reduction, for the case that $n$ is odd. This is related to the principal realization of the basic module of the twisted affine Lie algebra $\hat{sl}_n^{(2)}$. We show that the following two statements for a BKP $\tau$ function are equivalent: (1) $\tau$ is is $n$--reduced and satisfies the string equation, i.e. $L_{-1}\tau=0$, where $L_{-1}$ is an element of some `natural' Virasoro algebra. (2) $\tau$ satisfies the vacuum constraints of the $BW_{1+\infty}$ algebra. Here $BW_{1+\infty}$ is the natural analog of the $W_{1+\infty}$ algebra, which plays a role in the KP case.