  Following the paradigm on the sphere, we begin the study of irrational
conformal field theory (ICFT) on the torus. In particular, we find that the
affine-Virasoro characters of ICFT satisfy heat-like differential equations
with flat connections. As a first example, we solve the system for the general
$g/h$ coset construction, obtaining an integral representation 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 non-trivial 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.
