On modular categories with Frobenius-Perron dimension congruent to 2 modulo 4

Abstract

We contribute to the classification of modular categories $\mathcal{C}$ with $\operatorname{FPdim}(\mathcal{C})\equiv 2 \pmod 4$. We prove that such categories have group of invertibles of even order, and that they factorize as $\mathcal C\cong \widetilde{\mathcal C} \boxtimes \operatorname{sem}$, where $\widetilde{\mathcal C}$ is an odd-dimensional modular category and $\operatorname{sem}$ is the rank 2 pointed modular category. This reduces the classification of these categories to the classification of odd-dimensional modular categories. It follows that modular categories $\mathcal C$ with $\operatorname{FPdim}(\mathcal{C})\equiv 2 \pmod 4$ of rank up to 46 are pointed. More generally, we prove that if $\mathcal C$ is a weakly integral MTC and $p$ is an odd prime dividing the order of the group of invertibles that has multiplicity one in $\operatorname{FPdim}(\mathcal C)$, then we have a factorization $\mathcal C \cong \widetilde{\mathcal C} \boxtimes \operatorname{Vec}_{\mathbb Z_p}^{\chi},$ for $\widetilde{\mathcal C}$ an MTC with dimension not divisible by $p$.