Fusion Algebras Induced by Representations of the Modular Group

TL;DR

This paper explores how representations of the modular group induce fusion algebras, analyzing their properties and connections to rational conformal field theories, including conformal dimensions and central charges.

Contribution

It identifies specific representations that produce 'good' fusion algebras and links these to known rational conformal field theories, providing new insights into their structure.

Findings

Certain representations lead to well-structured fusion algebras

Calculated conformal dimensions and central charges for these theories

Most induced fusion algebras correspond to known rational models

Abstract

Using the representation theory of the subgroups SL_2(Z_p) of the modular group we investigate the induced fusion algebras in some simple examples. Only some of these representations lead to 'good' fusion algebras. Furthermore, the conformal dimensions and the central charge of the corresponding rational conformal field theories are calculated. Two series of representations which can be realized by unitary theories are presented. We show that most of the fusion algebras induced by admissible representations are realized in well known rational models.