Fusion rules and modular transformations in 2D conformal field theory

We study conformal field theories with a finite number of primary fields with respect to some chiral algebra. It is shown that the fusion rules are completely determined by the behavior of the characters under the modular group. We illustrate with some examples that conversely the modular properties of the character can be derived from the fusion rules. We propose how these results can be used to find restrictions on the values of the central charge and conformal dimensions. (orig.)