A unitary tensor product theory for unitary vertex operator algebra modules

Abstract

Let V be a unitary vertex operator algebra (VOA) satisfying the following conditions: (1) V is of CFT type. (2) Every N-gradable weak V -module is completely reducible. (3) V is C2-cofinite. Let Rep(V)be the category of unitary V -modules, and let C be a subcategory of Rep(V) whose objects are closed under taking tensor product. Then C is a ribbon fusion category. For any objects Wi; Wj of C, we define a sesquilinear form on the tensor product Wi bWj. We show that if these sesquilinear forms are positive definite (i.e., when they are inner products), then the ribbon category C is unitary. We show that if the unitary V -modules and a generating set of intertwining operators in C satisfy certain energy bounds, then these sesquilinear forms are positive definite. Our result can be applied to the modular tensor categories associated to unitary Virasoro VOAs, and unitary affine VOAs of type An; Dn; G2, and more.