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 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.) 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 ↦ -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 ↦ -1/τ is symmetric.