On a family of vertex operator superalgebras

This paper is to study vertex operator superalgebras which are strongly generated by their weight-$2$ and weight-$\frac{3}{2}$ homogeneous subspaces. Among the main results, it is proved that if such a vertex operator superalgebra $V$ is simple, then $V_{(2)}$ has a canonical commutative associative algebra structure equipped with a non-degenerate symmetric associative bilinear form and $V_{(\frac{3}{2})}$ is naturally a $V_{(2)}$-module equipped with a $V_{(2)}$-valued symmetric bilinear form and a non-degenerate ($\mathbb{C}$-valued) symmetric bilinear form, satisfying a set of conditions. On the other hand, assume that $A$ is any commutative associative algebra equipped with a non-degenerate symmetric associative bilinear form and assume that $U$ is an $A$-module equipped with a symmetric $A$-valued bilinear form and a non-degenerate ($\mathbb{C}$-valued) symmetric bilinear form, satisfying the corresponding conditions. Then we construct a Lie superalgebra $\mathcal{L}(A,U)$ and a simple vertex operator superalgebra $L_{\mathcal{L}(A,U)}(\ell,0)$ for every nonzero number $\ell$ such that $L_{\mathcal{L}(A,U)}(\ell,0)_{(2)}=A$ and $L_{\mathcal{L}(A,U)}(\ell,0)_{(\frac{3}{2})}=U$.