Towards a Classification of Fusion Rule Algebras in Rational Conformal Field Theories

TL;DR

This paper reviews the properties and classifications of Fusion Rule Algebras in Rational Conformal Field Theories, exploring known results, connections to graph theory, and potential for new insights.

Contribution

It summarizes existing classifications of FRA's, examines FRA's generated by a fundamental field, and discusses novel graph-theoretic approaches for further research.

Findings

Complete classification for up to 4 fields

Classification of FRA's generated by a fundamental field

Potential links between FRA's and graph theory

Abstract

We review the main topics concerning Fusion Rule Algebras (FRA) of Rational Conformal Field Theories. After an exposition of their general properties, we examine known results on the complete classification for low number of fields ($\leq 4$). We then turn our attention to FRA's generated polynomially by one (real) fundamental field, for which a classification is known. Attempting to generalize this result, we describe some connections between FRA's and Graph Theory. The possibility to get new results on the subject following this ``graph'' approach is briefly discussed.