Rigidity and Modularity of Vertex Tensor Categories

Abstract

Let V be a simple vertex operator algebra satisfying the following conditions: (i) V (n) = 0 for n < 0, V (0) = C1 and V ′ is isomorphic to V as a V -module. (ii) Every N-gradable weak V -module is completely reducible. (iii) V is C 2 -cofinite. (In the presence of Condition (i), Conditions (ii) and (iii) are equivalent to a single condition, namely, that every weak V -module is completely reducible.) Using the results obtained by the author in the formulation and proof of the general version of the Verlinde conjecture and in the proof of the Verlinde formula, we prove that the braided tensor category structure on the category of V -modules is rigid, balanced and nondegenerate. In particular, the category of V -modules has a natural structure of modular tensor category. We also prove that the tensor-categorical dimension of an irreducible V -module is the reciprocal of a suitable matrix element of the fusing isomorphism under a suitable basis. IntroductionIn the present paper, we prove the rigidity and modularity of the braided tensor category of modules for a vertex operator algebra satisfying certain natural conditions (see below). Finding proofs of these properties has been an open problem for many years. Our proofs in this paper are based on the results obatined by the author [H7] in the formulation and proof of the general version of the Verlinde conjecture and in the proof of the Verlinde formula.In MS2] also demonstrated that the theory of these polynomial equations is actually a conformal-field-theoretic analogue of the theory of the tensor categories. This work of Moore and Seiberg greatly advanced our understanding of the structure of conformal field theories and the name "modular tensor category" was suggested by I. Frenkel for the theory of these Moore-Seiberg equations. Later, the precise notion of modular tensor category was introduced to summarize the properties of these polynomial equations and has played a central role in the development of conformal field theories and three-dimensional topological field theories. See for example [T] and [BK] for the theory of modular tensor categories, their applications and references to many important works of mathematicians and physicists.Mathematically To prove that a semisimple braided tensor category actually carries a modular tensor category structure, we need to prove that it is rigid, balanced and nondegenerate. The balancing isomorphisms or twists in these braided tensor categories are actually trivial to construct and the balancing axioms are easy to prove. On the other hand, the rigidity has been an open problem for many years after the braided tensor category structure on the category of modules for a vertex operator algebra satisfying the conditions in [HL1]-[HL4] [H1] was constructed. The main difficulty is that from the construction, 2 it is not clear why the numbers determining the module maps given by the sequences in the axioms for the rigidity are not 0. The nondegeneracy of the semisimple braided tensor category of modules for a...

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