Vertex operator algebras, the Verlinde conjecture, and modular tensor categories
- Department of Mathematics, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019
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Communicated by G. D. Mostow, Yale University, New Haven, CT, December 31, 2004 (received for review August 17, 2004)
Abstract
Let V be a simple vertex operator algebra satisfying the following conditions: (i) V
(
n
) = 0 for n < 0, 
, and the contragredient module V' is isomorphic to V as a V-module; (ii) every 
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.
