Vertex operator algebras and the Verlinde conjecture

We prove the Verlinde conjecture in the following general form: 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 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 matrices 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.