Noah Snyder and I have just uploaded to the arxiv our paper Non-cyclotomic fusion categories.
In their paper On fusion categories, Etingof, Nikshych and Ostrik asked if every fusion category can be defined over a cyclotomic field. This is supported by the following facts:
- The representation category of any finite group has a complete rational form over a cyclotomic field.
- The semisimplified representation category of any quantum group at a root of unity has a complete rational form over a cyclotomic field.
- The Frobenius-Perron dimension of any object in a fusion category is a cyclotomic integer.
A fusion category is a rigid monoidal category which is semisimple and has finitely many (isomorphism classes of) simple objects. It is tempting to think of them as the common generalization of representation categories of finite groups and of representation categories of quantum groups at roots of unity (these provide many of the most interesting examples), but our result shows that this is a little dangerous. In particular, Etingof, Nikshych and Ostrik’s question can be answered in the negative:
Theorem (Morrison-Snyder): one of the fusion categories coming from the Haagerup subfactor cannot be defined over any cyclotomic field.
This shows that there are fusion categories, namely those coming from the smallest “exotic” (i.e. apparently unrelated to finite groups or quantum groups) subfactors, which exhibit more complicated behaviour than those coming from finite groups and quantum groups.
Subfactors provide examples of fusion categories via two “even parts”, called the principal even part and the dual even part. We show that the principal even parts of each of the Haagerup subfactor and the recently constructed extended Haagerup subfactor are fusion categories which cannot be defined over any cyclotomic field.
Our approach makes detailed use of the construction of the Haagerup subfactor via its planar algebra in Emily Peters’ thesis, and the corresponding construction of the extended Haagerup subfactor. This approach proceeds by establishing that any subfactor planar algebra with the desired principal graph must contain an element which has moments (traces of powers) and “twisted moments” which evaluate to particular scalars, and then finding such an element inside a larger (but non-subfactor) planar algebra. The essential insight of the current paper is that the principal even part of such a subfactor must contain the twisted moments in its field of definition.
Given an object in a fusion category, and an endomorphism of , we can choose maps and , define and to be the dual maps, and abuse notation by thinking of as a map by composing copies of itself. At this point, we can define the third twisted moment of , by the following diagram:
This diagram evaluates to a scalar, but this scalar depends on the choices of , , , and so it is not an invariant. We choose where and are the projections onto the two simple objects and . For any complete rational form you can find inside it. Now notice that and are well-defined up to a choice of scalar. (The former just uses that is self-dual. The latter uses something very particular about the even part of Haagerup.)
We want to remove the dependence of the third twisted moment of on the choices of T’s and B’s. So define the normalised third twisted moment to be the third twisted moment divided by the theta symbols and the -th power of the circle; this expression is homogeneous of degree zero with respect to the ambiguity coming from our choices, and so must lie in the ground field for any rational form.
Next, we observe that the third twisted moment is relatively easy to compute, in the original planar algebra. We choose and as in the diagram below:
Thus third twisted moment is given by the following diagram in the planar algebra:
and a short calculation using the structure constants of the planar algebra (i.e. using the relations in the generators mod relations presentations of the planar algebra) shows that the normalised third twisted moment is not cyclotomic.