Automorphisms of the affine SU(3) fusion rules

TL;DR

This paper classifies all automorphisms of the affine SU(3) fusion rules at any level k, revealing the structure of automorphism groups depending on divisibility conditions and employing cyclotomic arithmetic techniques.

Contribution

It provides a complete classification of automorphisms for affine SU(3) fusion rules using cyclotomic extension arithmetic, extending methods to other algebras.

Findings

Automorphism group is Z_2 generated by charge conjugation when k divisible by 3.

Automorphism group is Z_2 x Z_2 generated by charge conjugation and another automorphism when k not divisible by 3.

Techniques are applicable to other algebraic structures beyond SU(3).

Abstract

We classify the automorphisms of the (chiral) level-k affine SU(3) fusion rules, for any value of k, by looking for all permutations that commute with the modular matrices S and T. This can be done by using the arithmetic of the cyclotomic extensions where the problem is naturally posed. When k is divisible by 3, the automorphism group (Z_2) is generated by the charge conjugation C. If k is not divisible by 3, the automorphism group (Z_2 x Z_2) is generated by C and the Altsch\"uler--Lacki--Zaugg automorphism. Although the combinatorial analysis can become more involved, the techniques used here for SU(3) can be applied to other algebras.