Virasoro vertex operator algebras, the (nonmeromorphic) operator product expansion and the tensor product theory

Abstract A theory of tensor products of modules for a vertex operator algebra is being developed by Lepowsky and the author. To use this theory, one first has to verify that the vertex operator algebra satisfies certain conditions. We show in the present paper that for any vertex operator algebra containing a vertex operator subalgebra isomorphic to a tensor product algebra of minimal Virasoro vertex operator algebras (vertex operator algebras associated to minimal models), the tensor product theory can be applied. In particular, intertwining operators for such a vertex operator algebra satisfy the (nonmeromorphic) commutativity (locality) and the (nonmeromorphic) associativity (operator product expansion). Combined with a result announced by Lepowsky and the author in 1994, the results of the present paper also show that the category of modules for such a vertex operator algebra has a natural structure of a braided tensor category. In particular, for any pair p, q of relatively prime positive integers larger than 1, the category of minimal modules of central charge 1−6[( p−q ) 2 / pq ] for the Virasoro algebra has a natural structure of a braided tensor category.

摘要 莱波夫斯基（Lepowsky）和作者正在发展一种顶点算子代数的模的张量积理论。要运用这一理论，首先必须验证该顶点算子代数满足某些条件。在本文中我们表明，对于任何包含一个同构于极小维拉索罗（Virasoro）顶点算子代数（与极小模型相关联的顶点算子代数）的张量积代数的顶点算子子代数的顶点算子代数，张量积理论都可以应用。特别地，这样一个顶点算子代数的交织算子满足（非亚纯的）交换性（局域性）和（非亚纯的）结合性（算符乘积展开）。结合莱波夫斯基和作者在1994年宣布的一个结果，本文的结果还表明，这样一个顶点算子代数的模范畴具有一个辫张量范畴的自然结构。特别地，对于任何一对大于1且互质的正整数p，q，维拉索罗代数的中心荷为\(1 - 6[(p - q)^2 / pq]\)的极小模范畴具有一个辫张量范畴的自然结构。