# Cyclotomic integers, fusion categories, and subfactors (March)

Frank Calegari, Scott Morrison, and I recently uploaded to the arxiv our paper Cyclotomic integers, fusion categories, and subfactors. In this paper we give two applications of cyclotomic number theory to quantum algebra.

1. A complete list of possible Frobenius-Perron dimensions in the interval (2, 76/33) for an object in a fusion category.
2. Given a family of graphs G_n obtained from a graph G by attaching a chain of n edges to a chosen vertex, an effective bound on the greatest n so that G_n can be the principal graph of a subfactor.

Neither of these results look like they involve number theory. The connection comes from a result of Etingof, Nikshych, and Ostrik which says that the dimension of every object in a fusion category is a cyclotomic integer.

A possible subtitle to this paper is

What’s so special about $(\sqrt{3} + \sqrt{7})/2$?

It turns out that $(\sqrt{3} + \sqrt{7})/2$ is the smallest number satisfying the following properties:

1. $x \in \mathbb{Z}[\zeta_N]$ for some N
2. x is real
3. x is maximal among its Galois conjugates
4. x is not of the form $2 \cos(2 \pi/N)$

It then follows from Etingof-Nikshych-Ostrik that $(\sqrt{3} + \sqrt{7})/2$ is the smallest number larger than 2 which can possibly be the FP dimension of an object in a fusion category. In fact, $(\sqrt{3} + \sqrt{7})/2$ is the FP dimension of the object in a fusion category which is constructed by Victor Ostrik in the appendix (based on an earlier subfactor construction of Feng Xu’s).

In ENO’s theorem “dimension” can mean any way of assigning a complex number to each object which is multiplicative under tensor product and additive under direct sum. The Frobenius-Perron dimension is the unique such way of assigning positive real numbers. Since the index of a finite depth subfactor is the FP dimension of a Frobenius algebra object in a fusion category, this result also has applications to finite depth subfactors. The first such application was given by Asaeda & Yasuda who proved a special case of the second result where G is the principal graph of the Haagerup subfactor.

This project was a Math Overflow baby. Scott and I had a lot of numerical evidence for the second theorem from our computer search for new subfactors, but no idea of how to attack it other than finding the right number theorist. Enter Math Overflow where we posted this question and were lucky enough that the right number theorist read it.