Verlinde conjecture and modular tensor categories

Let V be a simple vertex operator algebra satisfying the following conditions: (i) V(n) = 0 for n < 0, V(0) = C1 and the contragredient module V ′ is isomorphic to V as a V -module. (ii) Every N-gradable weak V -module is completely reducible. (iii) V is C2-cofinite. We announce a proof of the Verlinde conjecture for V , that is, of the statement that the matrices formed by the fusion rules among 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. We discuss some consequences of the Verlinde conjecture, including the Verlinde formula for the fusion rules, a formula for the matrix given by the action of τ 7→ −1/τ and the symmetry of this matrix. We also announce a proof of the rigidity and nondegeneracy property of the braided tensor category structure on the category of V -modules when V satisfies in addition the condition that irreducible V -modules not equivalent to V has no nonzero elements of weight 0. In particular, the category of V -modules has a natural structure of modular tensor category.