Conformal field theory and genus-zero function fields

One-loop partition functions of rational conformal field theories are finite linear combinations of modular invariants associated with projective modular functions of a modular subgroup. We show that, for normal subgroups with a genus-zero fundamental region, the functions which lead to physically acceptable partition functions are extremely limited in number, and can be found explicitly. We also show that the conformal charge and weights of theories which factorize on these subgroups can only take on certain discrete values. (orig.)