Local Systems of Vertex Operators, Vertex Superalgebras and Modules

We give an analogue for vertex operator algebras and superalgebras of the notion of endomorphism ring of a vector space by means of a notion of “local system of vertex operators” for a (super) vector space. We first prove 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. Then we prove that for a vertex (operator) superalgebra V , giving a V -module M is equivalent to giving a vertex (operator) superalgebra homomorphism from V to some local system of vertex operators on M . As applications, we prove that certain lowest weight modules for some wellknown infinite-dimensional Lie algebras or Lie superalgebras have natural vertex operator superalgebra structures. We prove the rationality of vertex operator superalgebras associated to standard modules for an affine algebra. We also give an analogue of the notion of the space of linear homomorphisms from one module to another for a Lie algebra by introducing a notion of “generalized intertwining operators.” We prove that G(M1,M2), the space of generalized intertwining operators from one module M1 to another module M2 for a vertex operator superalgebra V , is a generalized V -module. Furthermore, we prove that for a fixed vertex operator superalgebra V and three V -modules M i (i = 1, 2, 3), giving an intertwining operator of type ( M3 M1,M2 ) is equivalent to giving a V -homomorphism from M1 to G(M2,M3).