Congruence Property In Conformal Field Theory

TL;DR

This paper proves that the modular representations in rational conformal field theories have a congruence subgroup structure, establishing a deep connection between algebraic and number-theoretic properties of these theories.

Contribution

It demonstrates the congruence subgroup property for modular representations of rational vertex operator algebras and explores the implications for Galois symmetry and anomalies.

Findings

Kernel of modular group representation is a congruence subgroup

q-characters are modular functions on the same subgroup

Galois symmetry of modular representations established

Abstract

The congruence subgroup property is established for the modular representations associated to any modular tensor category. This result is used to prove that the kernel of the representation of the modular group on the conformal blocks of any rational, C_2-cofinite vertex operator algebra is a congruence subgroup. In particular, the q-character of each irreducible module is a modular function on the same congruence subgroup. The Galois symmetry of the modular representations is obtained and the order of the anomaly for those modular categories satisfying some integrality conditions is determined.