Congruence property in orbifold theory

Abstract

Let V V be a rational, selfdual, C 2 C_2 -cofinite vertex operator algebra of CFT type, and G G a finite automorphism group of V . V. It is proved that the kernel of the representation of the modular group on twisted conformal blocks associated to V V and G G is a congruence subgroup. In particular, the q q -character of each irreducible twisted module is a modular function on the same congruence subgroup. In the case V V is the Frenkel-Lepowsky-Meurman’s moonshine vertex operator algebra and G G is the monster simple group, the generalized McKay-Thompson series associated to any commuting pair in the monster group is a modular function.