On the Classification of Modular Fusion Algebras

TL;DR

This paper classifies small-dimensional strongly-modular and nondegenerate fusion algebras, providing explicit realizations and fusion graphs, advancing understanding of their structure in conformal field theory.

Contribution

It offers the first complete classification of strongly-modular fusion algebras up to dimension four and nondegenerate cases below dimension 24, including explicit realizations.

Findings

Classified all strongly-modular fusion algebras of dimensions 2, 3, and 4.

Classified all nondegenerate strongly-modular fusion algebras of dimension less than 24.

Provided polynomial realizations and fusion graphs for these algebras.

Abstract

We introduce the notion of (nondegenerate) strong-modular fusion algebras. Here strongly-modular means that the fusion algebra is induced via Verlinde's formula by a representation of the modular group SL(2,Z) whose kernel contains a congruence subgroup. Furthermore, nondegenerate means that the conformal dimensions of possibly underlying rational conformal field theories do not differ by integers. Our main result is the classification of all strongly-modular fusion algebras of dimension two, three and four and the classification of all nondegenerate strongly-modular fusion algebras of dimension less than 24. We use the classification of the irreducible representations of the finite groups SL(2,Z_{p^l}) where p is a prime and l a positive integer. Finally, we give polynomial realizations and fusion graphs for all simple nondegenerate strongly-modular fusion algebras of dimension less than 24.