Some Properties For Vertex Operator Superalgebras

As a natural generalization of Vertex operator algebras, the theory of Vertex op-erator superalgebras is closely connected to that of super string, which plays a crucial role in physics. In this thesis we study the structure of vertex operator superalgebras which have strong CFT type and satisfy C2-confiniteness and some certain rationali-ties. It is proved that the weight one subspace which carries the natural structure of Lie algebra, is indeed reductive. Naturally given a fixed vertex operator superalgebra V it will be not expected that the Cartan subalgebras of V1.is arbitrary, we need to find some invariant for this vertex operator superalgebra to control the dimension d of the Cartan subalgebras of V1.Surprisingly, it is proved that d is less than the ef-fective central charge c, and meanwhile it is also proved that the dimension of V1/2 is less than 2~c+1. Currently, we can not completely interpret the C2-cofinite condition, however, such condition allow us to view a vertex operator superalgebra as a integrable representation for some of its certain subalgebras. Finally, we prove that the map Zn:V(?)n(?)C((z1)…(zn))→V((z1))…((zn)) v(1)(?)…v(n)(?)fY(v(1),z1…Y(v(n),zn)1 is injective for all positive integers.