Modular categories, integrality and Egyptian fractions

TL;DR

This paper explores classification strategies for integral modular categories by imposing specific conditions, leading to partial classifications and improved bounds, especially for categories of odd dimension and low rank.

Contribution

It introduces new conditions that simplify the classification of integral modular categories and provides classifications for certain cases up to rank 11.

Findings

Classified modular categories with twist order 2, 3, 4, or 6.

Achieved classification of odd-dimensional modular categories up to rank 11.

Provided improved bounds on classification complexity using number-theoretic techniques.

Abstract

It is a well-known result of Etingof, Nikshych and Ostrik that there are finitely many inequivalent integral modular categories of any fixed rank $n$. This follows from a double-exponential bound on the maximal denominator in an Egyptian fraction representation of 1. A na\"ive computer search approach to the classification of rank $n$ integral modular categories using this bound quickly overwhelms the computer's memory (for $n\geq 7$). We use a modified strategy: find general conditions on modular categories that imply integrality and study the classification problem in these limited settings. The first such condition is that the order of the twist matrix is 2,3,4 or 6 and we obtain a fairly complete description of these classes of modular categories. The second condition is that the unit object is the only simple non-self-dual object, which is equivalent to odd-dimensionality. In this case we obtain a (linear) improvement on the bounds and employ number-theoretic techniques to obtain a classification for rank at most 11 for odd-dimensional modular categories.