Vertex operator algebras, the Verlinde conjecture and modular tensor categories

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

2004-12-13

arxiv.org/abs/math/0412261v1

Proc.Nat.Acad.Sci. 102 (2005) 5352-5356

Mathematics

Quantum Algebra

18 pages. To appear in the Proc. Natl. Acad. Sci. USA

Scientific paper

10.1073/pnas.0409901102

Let V be a simple vertex operator algebra satisfying the following conditions: (i) The homogeneous subspaces of V of weights less than 0 are 0, the homogeneous subspace of V of weight 0 is spanned by the vacuum and V' is isomorphic to V as a V-module. (ii) Every weak V-module gradable by nonnegative integers 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 \tau\mapsto -1/\tau 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 \tau\mapsto -1/\tau 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.

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