Vertex operator algebras and the Verlinde conjecture.pdf

a r X i v : m a t h / 0 4 0 6 2 9 1 v 3 [ m a t h .Q A ] 2 3 N o v 2 0 0 7 Vertex operator algebras and the Verlinde conjecture Yi-Zhi Huang Abstract We prove the Verlinde conjecture in the following general form: Let V be a simple vertex operator algebra satisfying the following condi- tions: (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 C2-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.) Then the ma- trices formed by the fusion rules among the irreducible V -modules are diagonalized by the matrix given by the action of the modular transformation τ 7→ ?1/τ on the space of characters of irreducible V -modules. Using this result, we obtain the Verlinde formula for the fusion rules. We also prove that the matrix associated to the modular transformation τ 7→ ?1/τ is symmetric. 0 Introduction In the present paper, we formulate and prove a general version of the Ver- linde conjecture and prove the Verlinde formula for fusion rules using the representation theory of vertex operator algebras. The Verlinde conjecture [V] in conformal field theory states that the ac- tion of the modular transformation τ 7→ ?1/τ on the space of characters of a rational conformal field theory diagonalizes the fusion rules. Except for some particular examples (see below), the general Verlinde conjecture has been an open problem for twenty years. In [MS1], Moore and Seiberg showed on a physical level of rigor that this conjecture follows from the axioms for rational conformal field theories (see [K], [S1], [S2] and [S3] for axioms for conformal 1 field theories and [MS2] for axioms and assumptions for rational conformal field theories on a physical level of rigor). Axioms for rational conformal field theories are in fact much stronger than statements such as the Verlin