Higher Genus Characters for Vertex Operator Superalgebras on Sewn Riemann Surfaces

We review our recent results on computation of the higher genus characters for vertex operator superalgebras modules. The vertex operator formal parameters are associated to local parameters on Riemann surfaces formed in one of two schemes of (self- or tori- ) sewing of lower genus Riemann surfaces. For the free fermion vertex operator superalgebra we present a closed formula for the genus two continuous orbifold partition functions (in either sewings) in terms of an infinite dimensional determinant with entries arising from the original torus Szeg? kernel. This partition function is holomorphic in the sewing parameters on a given suitable domain and possesses natural modular properties. Several higher genus generalizations of classical (including Fay’s and Jacobi triple product) identities show up in a natural way in the vertex operator algebra approach.

Regularity of vertex operator superalgebras

We prove that the regularity of a vertex operator superalgebra can be reduced to the semisimplicity of the category of its weak modules.Moreover,the rationality can be replaced by requiring that each 1/2Z+-graded module is a direct sum of irreducible weak modules.

Bimodules associated to vertex operator superalgebras

Let V be a vertex operator superalgebra and m, n ∈ 1/2 ?+. We construct an A n (V)-A m (V)-bimodule A n,m (V) which characterizes the action of V from the level m subspace to level n subspace of an admissible V-module. We also construct the Verma type admissible V-module from an A m (V)-module by using bimodules

Representations of Code Vertex Operator Superalgebras

We study the representations of code vertex operator superalgebras resulting from a binary linear code which contains codewords of odd weight. We also show that there exists only one set of seven mutually orthogonal conformal vectors with central charge 1/2 in the Hamming code vertex operator superalgebra MH7. Phrthermore, we classify all the irreducible weak MH7-modules.

C2-Cofiniteness of Cyclic-Orbifold Vertex Operator Superalgebras

In this paper, it is proved that for any C2-cofinite, simple, CFT-type vertex operator superalgebra V and a finite solvable group G consisting of automorphisms of V, the fixed point subalgebra Vc is a C2-cofinite vertex operator superalgebra.

Constructing tensor products of modules for C2-cofinite vertex operator superalgebras

For any C2-cofinite vertex product and the P（z）-tensor product finite length are proved to exist, which operator superalgebra V, the tensor of any two admissible V-modules of are shown to be isomorphic, and their constructions are given explicitly in this paper.