Conformal Blocks, Fusion Rules And The Verlinde Formula

. A Rational Conformal Field Theory (RCFT) is a functor which associates to any Riemann surface with marked points a finite-dimensional vector space, so that certain axioms are satisfied; the Verlinde formula computes the dimension of these vector spaces. For some particular RCFTs associated to a compact Lie group G (the WZW models), these spaces have a beautiful algebro-geometric interpretation as spaces of generalized theta functions, that is, sections of a determinant bundle (or its powers) over the moduli space of G-bundles on a Riemann surface. In this paper we explain the formalism of the Verlinde formula: the dimension of the spaces are encoded in a finite-dimensional Z-algebra, the fusion ring of the theory; everything can be expressed in terms of the characters of this ring. We show how to compute these characters in the case of the WZW model and thus obtain an explicit formula for the dimension of the space of generalized theta functions. Dedicated to F. Hirzebruch Introducti..