[Paper Review] On the classification of almost square-free modular categories

Jingcheng Dong, Sonia Natale|arXiv (Cornell University)|Dec 29, 2016

Algebraic structures and combinatorial models19 references3 citations

TL;DR

This paper classifies almost square-free modular categories—those with Frobenius-Perron dimension $dq^n$, where $q>2$ is prime and $d$ is square-free—showing they are integral, nilpotent, and hence group-theoretical. For $q=2$, it provides a complete structural description via $G$-equivariantizations of braided $G$-crossed fusion categories, including cases involving Ising categories and pointed modular categories.

ABSTRACT

Let $C$ be a modular category of Frobenius-Perron dimension $dq^n$, where $q$ is a prime number and $d$ is a square-free integer. We show that if $q>2$ then $C$ is integral and nilpotent. In particular, $C$ is group-theoretical. In the general case, we describe the structure of $C$ in terms of equivariantizations of group-crossed braided fusion categories.

Motivation & Objective

• To classify modular categories of Frobenius-Perron dimension $dq^n$, where $q>2$ is prime, $d$ is square-free, and $\gcd(q,d)=1$.
• To determine the structure of strictly weakly integral almost square-free modular categories when $q=2$.
• To extend the classification of modular categories beyond the known cases of small dimensions or specific primes.
• To provide a structural framework for such categories using group-theoretical and equivariantization techniques.
• To clarify the role of Tannakian categories and braided $G$-crossed fusion categories in the classification.

Proposed method

• Use of Frobenius-Perron dimension analysis to deduce integrality and nilpotency in the $q>2$ case.
• Application of results on group-theoretical fusion categories and their Drinfeld centers to show equivalence to $\mathrm{Rep}(D^\omega G)$ for some $\omega$.
• Employment of $G$-equivariantization and de-equivariantization techniques to relate modular categories to braided $G$-crossed fusion categories.
• Utilization of the Müger centralizer and core decomposition to analyze the structure of the neutral component after de-equivariantization.
• Application of results on weakly anisotropic braided fusion categories and the classification of their pointed and adjoint components.
• Use of Deligne tensor products and known classification of Ising categories to describe components in the $q=2$ case.

Experimental results

Research questions

• RQ1 Are all almost square-free modular categories with $q>2$ integral and nilpotent?
• RQ2 Can such categories be fully classified in group-theoretical terms when $q>2$?
• RQ3 What is the structure of strictly weakly integral almost square-free modular categories when $q=2$?
• RQ4 How do Tannakian subcategories and equivariantization affect the classification of these modular categories?
• RQ5 Can all such categories be realized as equivariantizations of braided $G$-crossed fusion categories with specific component structures?

Key findings

• For $q>2$, every almost square-free modular category of dimension $dq^n$ is integral and nilpotent, hence group-theoretical.
• When $q=2$, such categories are either equivalent to a Deligne tensor product $\mathcal{I} \boxtimes \mathcal{B}$ with $\mathcal{I}$ an Ising category and $\mathcal{B}$ pointed, or arise as $G$-equivariantizations of braided $G$-crossed fusion categories.
• If the adjoint subcategory has dimension 2, the category is equivalent to $\mathcal{I} \boxtimes \mathcal{B}$ for some pointed modular category $\mathcal{B}$.
• When the core category $\mathcal{E}$ has non-trivial pointed-adjoint intersection isomorphic to $\mathrm{sVect}$, then $\mathcal{E} \cong \mathcal{I} \boxtimes \mathcal{B}$, placing the category in the third class of the classification.
• Examples exist (e.g., $\mathcal{I}_1 \boxtimes \mathcal{I}_2 \boxtimes \mathcal{B}$) that fall into the equivariantization classes but not into the pure tensor product class.
• The classification is complete for $q=2$ and extends to $dq^4$ dimensions for odd $q$, where such categories are necessarily pointed.