On the classification of fusion rings

TL;DR

This paper classifies fusion rings of rational conformal field theories with up to six primary fields, showing all such fusion rules can be realized by current algebras, supporting their fundamental role.

Contribution

It provides a complete classification of fusion rings for small rational conformal field theories and demonstrates their realization via current algebras.

Findings

All classified fusion rules can be realized by current algebras.

The classification supports the conjecture linking rational conformal field theories to current algebras.

A comprehensive catalogue of possible fusion rings for theories with up to six primary fields.

Abstract

The fusion rules and modular matrix of a rational conformal field theory obey a list of properties. We use these properties to classify rational conformal field theories with not more than six primary fields and small values of the fusion coefficients. We give a catalogue of fusion rings which can arise for these field theories. It is shown that all such fusion rules can be realized by current algebras. Our results support the conjecture that all rational conformal field theories are related to current algebras.