Congruence Property in Orbifold Theory

TL;DR

This paper proves that for certain vertex operator algebras and finite automorphism groups, the associated modular group representations are congruence subgroups, ensuring modularity of twisted module characters, with applications to moonshine theory.

Contribution

It establishes the congruence property for the modular group action on twisted conformal blocks in orbifold theory, extending modularity results to a broad class of vertex operator algebras.

Findings

Kernel of modular group representation is a congruence subgroup.

Twisted module characters are modular functions on the same subgroup.

Generalized McKay-Thompson series are modular functions.

Abstract

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