Classification of Rank 6 Modular Categories with Galois Group $\langle (012)(345)\rangle$

TL;DR

This paper classifies rank 6 modular tensor categories with a specific Galois group, revealing their structure and showing that certain symmetries lead to nonunitarizable, nonphysical categories, thus advancing the understanding of low-rank MTCs.

Contribution

It applies classification methods to a specific rank 6 case with a particular Galois group, identifying all such categories and their modular data structures.

Findings

All rank 6 MTCs with the given Galois group are conjugate to known categories.

Certain symmetries imply nonunitarizable (nonphysical) MTCs.

Identified the modular data as conjugate to specific known subcategories.

Abstract

Modular Tensor Categories (MTC's) arise in the study of certain condensed matter systems. There is an ongoing program to classify MTC's of low rank, up to modular data. We present an overview of the methods to classify modular tensor categories of low rank, applied to the specific case of a rank 6 category with Galois group $\langle(012)(345)\rangle$, and show that certain symmetries in this case imply nonunitarizable (hence, nonphysical) MTC's. We show that all the rank 6 MTC's with this Galois group have modular data conjugate to either the product of the semion category with $(A_1, 5)_{\frac{1}{2}}$ or a certain modular subcategory of $\mathcal{C}(\mathfrak{so}_5, 9, e^{j\pi i/9})$ with gcd$(18, j) = 1$.