Nonabelian parafermions and their dimensions

TL;DR

This paper generalizes abelian parafermions to nonabelian groups, deriving their fusion rules and dimensions, revealing integer dimensions for simple groups and similarities to $Z_n$ parafermions for cover groups, with detailed examples.

Contribution

It introduces a nonabelian generalization of parafermions and determines their allowed dimensions using Vafa equations, expanding the theoretical framework of parafermionic systems.

Findings

Dimensions are integers for simple groups.

Dimensions match $Z_n$ parafermions for cover groups.

Examples of integral parafermionic systems are provided.

Abstract

We propose a generalization of the Zamolodchikov-Fateev parafermions which are abelian, to nonabelian groups. The fusion rules are given by the tensor product of representations of the group. Using Vafa equations we get the allowed dimensions of the parafermions. We find for simple groups that the dimensions are integers. For cover groups of simple groups, we find, for $n.G.m$, that the dimensions are the same as $Z_n$ parafermions. Examples of integral parafermionic systems are studied in detail.