Vertex operator algebras, the Verlinde conjecture, and modular tensor categories

Abstract

Let V be a simple vertex operator algebra satisfying the following conditions: (i) V(n) = 0 for n < 0, V0 = 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 C 2-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 → -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 → -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 have no nonzero elements of weight 0. In particular, the category of V-modules has a natural structure of modular tensor category. © 2005 by The National Academy of Sciences of the USA.