FLAT CONNECTIONS FOR CHARACTERS IN IRRATIONAL CONFORMAL FIELD-THEORY

Following the paradigm on the sphere, we begin the study of irrational conformal field theory (ICFT) on the torus. In particular, we find th at the affine-Virasoro characters of ICFT satisfy heatlike differentia l equations with fiat connections. As a first example, we solve the sy stem for the general g/h coset construction, obtaining an integral rep resentation for the general coset characters. In a second application, we solve for the high-level characters of the general ICFT on simple g, noting a simplification for the subspace of theories which possess a nontrivial symmetry group. Finally, we give a geometric formulation of the system in which the flat connections are generalized Laplacians on the centrally extended loop group.