Vertex operator algebras and the Verlinde conjecture

Details Vertex operator algebras and the Verlinde conjecture Vertex operator algebras and the Verlinde conjecture

2004-06-15

arxiv.org/abs/math/0406291v3

Mathematics

Quantum Algebra

62 pages. To appear in Communications in Contemporary Mathematics. Referee's comments have been taken into account

Scientific paper

We prove the Verlinde conjecture in the following general form: 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. (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 \tau\mapsto -1/\tau 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 \tau\mapsto -1/\tau is symmetric.

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