Congruence subgroups and rational conformal field theory

TL;DR

This paper investigates the modular properties of characters in rational conformal field theory, establishing conditions under which these characters are modular functions for some congruence subgroup, and providing tests for such properties.

Contribution

The authors prove that if the T-matrix has odd order, the RCFT characters are modular functions for some level N, and propose a test for even order cases, advancing understanding of RCFT modularity.

Findings

If T has odd order, characters are modular functions for some level N.

A simple test for even order T matrices suggests characters may also be level N.

The results explain previously observed phenomena in RCFT character modularity.

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.