Non-cyclotomic fusion categories

TL;DR

This paper demonstrates that not all fusion categories can be defined over cyclotomic fields, providing specific examples from subfactors and analyzing their minimal fields of definition and Galois properties.

Contribution

It shows counterexamples to the conjecture that all fusion categories are cyclotomic, and determines the minimal fields and Galois groups for several key examples.

Findings

Certain fusion categories from subfactors are not cyclotomic.

The double of the even part of the Haagerup subfactor is cyclotomic.

Minimal fields of definition and Galois groups are identified for these categories.

Abstract

Etingof, Nikshych and Ostrik ask in arXiv:math.QA/0203060 if every fusion category can be completely defined over a cyclotomic field. We show that this is not the case: in particular one of the fusion categories coming from the Haagerup subfactor arXiv:math.OA/9803044 and one coming from the newly constructed extended Haagerup subfactor arXiv:0909.4099 can not be completely defined over a cyclotomic field. On the other hand, we show that the double of the even part of the Haagerup subfactor is completely defined over a cyclotomic field. We identify the minimal field of definition for each of these fusion categories, compute the Galois groups, and identify their Galois conjugates.