Congruence subgroups and rational conformal field theory

Abstract

We address here the question of whether the characters of an RCFT are modular functions for some level N, i.e. whether the representation of the modular group SL_2(Z) coming from any RCFT is trivial on some congruence subgroup. We prove that if the matrix T, associated to $(\matrix{1&1\cr 0&1})\in{\rm SL}_2(\Z)$, has ODD order, then this must be so. When the order of T is even, we present a simple test which if satisfied -- and we conjecture it always will be -- implies that the characters for that RCFT will also be level N. We use this to explain three curious observations in RCFT made by various authors.