Vertex operator algebras and the Verlinde conjecture

TL;DR

This paper proves the Verlinde conjecture for a class of vertex operator algebras, showing that fusion rule matrices are diagonalized by modular transformations, leading to the Verlinde formula.

Contribution

It establishes the Verlinde conjecture for certain vertex operator algebras under specific conditions, connecting fusion rules with modular transformations.

Findings

Fusion rule matrices are diagonalized by modular transformation matrices.

The Verlinde formula for fusion rules is derived.

The modular transformation matrix is shown to be symmetric.

Abstract

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.