Representations of the vertex algebra $W_{1+\mathrm{\infty }}$ with a negative integer central charge

Let $\text{D}$ be the Lie algebra of regular differentialoperators on ${\C} \setminus {0}$,and${\hD}= {\D} + {\C} C$bethecentralextensionof${\D}$\mathrm{.}Let$W_{1+\infty,-N}$ be the vertex algebra associated to the irreducible vacuum $\text{hD}$-module with the central charge $c=-N$. We show that $W_{1+\mathrm{\infty },-N}$ is a subalgebra of the Heisenberg vertex algebra M(1) with $2 N$generators,andconstruct2N-dimensionalfamilyofirreducible$W_{1+\infty,-N}$-modules\mathrm{.}Consideringthesemodulesas$\hD$-modules, we identify the corresponding highest weights.