Implications of an arithmetical symmetry of the commutant for modular invariants

TL;DR

This paper reveals an arithmetical symmetry in the commutant of modular matrices that simplifies the classification of modular invariants across all affine simple Lie algebras, especially for SU(3)_k.

Contribution

It introduces a universal arithmetical symmetry in the commutant of S and T matrices, enabling classification of modular invariants without detailed commutant analysis.

Findings

Symmetry holds for all affine simple Lie algebras at all levels.

Classified SU(3)_k modular invariants for specific coprimality conditions.

No detailed commutant knowledge needed for classification.

Abstract

We point out the existence of an arithmetical symmetry for the commutant of the modular matrices S and T. This symmetry holds for all affine simple Lie algebras at all levels and implies the equality of certain coefficients in any modular invariant. Particularizing to SU(3)_k, we classify the modular invariant partition functions when k+3 is an integer coprime with 6 and when it is a power of either 2 or 3. Our results imply that no detailed knowledge of the commutant is needed to undertake a classification of all modular invariants.