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arXiv:1003.2342 · Physics Letters B

Nonabelian parafermions and their dimensions

Roman Dovgard, Doron Gepner

Published May 13, 2010• 0 citations
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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 Zn parafermions. Examples of integral parafermionic systems are studied in detail.

DOI
10.1016/j.physletb.2010.05.011

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Abstract We explicitly work out the details of some simple examples anyon theories. We start with a fusion ring and using consistency conditions we build F-matrices, then the possible braidings (R-matrices). We consider Z2 fusion rules, Fibonacci fusion rules and Z3 fusion rules as simple cases. We consider Ising fusion rules, obtaining eight possible (modular) unitary braided theories. We then describe several more abelian theories, including the toric code, and general ZN anyons. These can be extended to describe all possible abelian braided theories in terms of a limited number of so-called “prime” theories.
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