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

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Abstract
We contribute to the classification of modular categories C \mathcal {C} with F P d i m ( C ) ≡ 2 ( mod 4 ) FPdim(\mathcal {C})\equiv 2 \pmod 4 . We prove that such categories have group of invertibles of even order and that they factorize as C ≅ C ~ ⊠ P \mathcal C\cong \widetilde {\mathcal C} \boxtimes \mathcal P , where C ~ \widetilde {\mathcal C} is an odd-dimensional modular category and P \mathcal P is a pointed modular category of dimension 2. This reduces the classification of these categories to the classification of odd-dimensional modular categories. It follows that modular categories C \mathcal {C} with F P d i m ( C ) ≡ 2 ( mod 4 ) FPdim(\mathcal {C})\equiv 2 \pmod 4 of rank up to 46 (except for 34) are pointed. More generally, we prove that if C \mathcal {C} is a weakly integral modular category and p p is an odd prime dividing the order of the group of invertibles that has multiplicity one in F P d i m ( C ) FPdim(\mathcal {C}) , then we have a factorization C ≅ C ~ ⊠ P \mathcal {C} \cong \tilde {\mathcal {C}} \boxtimes \mathcal {P} , for C ~ \tilde {\mathcal {C}} a modular category with dimension not divisible by p p and P \mathcal {P} a pointed modular category of dimension p p .
Funding Information
  • National Science Foundation (2146392)
  • National Science Foundation (889000)
  • Simons Foundation (2146392)
  • Simons Foundation (889000)

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