On the Classification of Weakly Integral Modular Categories

TL;DR

This paper classifies all weakly integral modular categories up to rank 7, revealing that integral ones are pointed and providing a detailed analysis of their structure through Galois and invertible object actions.

Contribution

It completes the classification of weakly integral modular categories up to rank 7 and analyzes the interplay of Galois and invertible object actions in these categories.

Findings

All integral modular categories of rank ≤ 7 are pointed.

Weakly integral modular categories of ranks 6 and 7 have dimension 4m, with m odd and square-free.

The Galois group and invertible object actions provide key insights into the structure of these categories.

Abstract

We classify all modular categories of dimension $4m$, where $m$ is an odd square-free integer, and all ranks $6$ and $7$ weakly integral modular categories. This completes the classification of weakly integral modular categories through rank $7$. Our results imply that all integral modular categories of rank at most $7$ are pointed (that is, every simple object has dimension $1$). All strictly weakly integral (weakly integral but non-integral) modular categories of ranks $6$ and $7$ have dimension $4m$, with $m$ an odd square free integer, so their classification is an application of our main result. The classification of rank $7$ integral modular categories is facilitated by an analysis of two actions on modular categories: the Galois group of the field generated by the entries of the $S$-matrix and the group of isomorphism classes of invertible simple objects. The interplay of these two actions is of independent interest, and we derive some valuable arithmetic consequences from their actions.