  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
$\widehat{gl}$.

High Energy Physics - Theory