The Representations Of Vertex Operator Superalgebras

Vertex operator superalgebras can be considered as natural generalizations of vertex operator algebras. The supersymmetry, which plays an important role in two-dimensional conformal field theories, is one of the main reasons to study vertex operator superalgebras. There has been a rapid development in the theory of vertex operator superalgebras over the past decades. Several important types of vertex operator superalgebras have been studied since 1990 by H. Tsukada. The Zhu's A(V)-theory on vertex operator algebras was generalized to A(V)-theoty on vertex operator superalgebras by V. G. Kac and W. Wang in 1996. They also studied in detail three classes of vertex operator superalgebras, i.e., the vertex operator su-peralgebras associated to the affine Kac-Moody superalgebras, the Neveu-Schwarz algebras, and the free fermions. The representations of these vertex operator su-peralgebras were also studied. By the " local system of vertex operators " for a (super) vector space, H. Li proved that any local system of vertex operators on a (super) vector space M has a natural vertex (super)algebra structure with M as a module and studied the representations of vertex operator superalgebras in 1996. For more results on the theory of vertex operator superalgebras and their repre-sentations on can refer to Xu'book pressed in 1998. In this paper, we study the representations of vertex operator superalgebras and related associative algebras, bimodules associated to vertex operator superalgebras and the representations of code vertex operator superalgebras obtained by combining the minimal vertex op-erator superalgebra L(1/2,0)(?)L(1/2,1/2) with a binary linear code which contains codewords of odd weight. The present paper includes three main parts.In the first part, let V be a vertex operator superalgebra. We construct a sequence of associative algebras An(V) for n∈1/2Z+. It is also exposed that there is a pair of functors between the category of An(V)-modules which are not An-1/2(V)- modules and the category of admissible V-modules. The functors exhibit a bijection between the simple modules in each category. We also construct a generalized Verma admissible V-module Mn(U) from an An(V)-module U which is not an An-1/2(V)-module. Furthermore, we study the theory of representations of vertex operator superalgebras by associative algebras An(V), n∈1/2Z+.In the second part, let V be a vertex operator superalgebra and m,n∈1/2Z+. We construct an An(V)-Am(V)-bimodule An,m(V) which characterizes the action of V from the level m subspace to level n subspace of an admissible V-module. We study the properties of the An(V)-Am(V)-bimodule An,m(V) and discuss relations between An(V)-modules and admissible V-modules. We also construct a Verma type admissible V-module M(U)=(?)n∈Z An,m(V)(?)Am(v) U, which is proved to be isomorphic to the M(U) defined in the first part of this paper.In the third part, we study the structure of the tensor product of vertex op-erator superalagebras. Furthermore, we study the representations of code vertex operator superalgebras resulting from a binary linear code which contains code-words of odd weight. We prove that the code vertex operator superalgebra MD is rational, and Miyamoto's construction of induced modules is generalized to code vertex operator superalgebra. We show that there exists only one set of mutually orthogonal seven conformal vectors with central charge 1/2 in the Hamming code vertex operator superalgebra MH7, and we classify all irreducible MH7-modules. As in [29] and [31], our main tool is the theory of induced modules developed by Dong and Lin in [33] and fusion rules of the rational vertex operator algebra L(1/2,0) with central charge 1/2 (see [26]).