Fusion rings and geometry

The algebraic structure of fusion rings in rational conformal field theories is analyzed in detail in this paper. A formalism which closely prallels classical tools in the study of the cohomology of homogeneous spaces is developed for fusion rings, in general, and for current algebra theories, in particular. It is shown that fusion rings lead to a natural orthogonal polynomial structure. The rings are expressed through generators and relations. The relations are then derived from some potentials leading to an identification of the fusion rings with deformations of affine varieties. In general, the fusion algebras are mapped to affine varieties which are the locus of the relations. The connection with modular transformations is investigated in this picture. It is explained how chiral algebras, arising in N=2 superconformal field theory, can be derived from fusion rings. In particular, it is argued that theories of the type SU(N)k/SU(N-1) are the N=2 counterparts of Grassmann manifolds and that there is a natural identification of the chiral fields with Schubert varieties, which is a graded algebra isomorphism. (orig.)