Representation theory of the vertex algebra $W_{1 + \infty}$

In our paper~\cite{KR} we began a systematic study of representations of the universal central extension $\widehat{\Cal D}\/$ of the Lie algebra of differential operators on the circle. This study was continued in the paper~\cite{FKRW} in the framework of vertex algebra theory. It was shown that the associated to $\widehat {\Cal D}\/$ simple vertex algebra $W_{1+ \infty, N}\/$ with positive integral central charge $N\/$ is isomorphic to the classical vertex algebra $W (gl_N)$, which led to a classification of modules over $W_{1 + \infty, N}$. In the present paper we study the remaining non-trivial case, that of a negative central charge $-N$. The basic tool is the decomposition of $N\/$ pairs of free charged bosons with respect to $gl_N\/$ and the commuting with $gl_N\/$ Lie algebra of infinite matrices $\widehat{gl}$.