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Modular categories with transitive Galois actions

In this paper, we study modular categories whose Galois group actions on their simple objects are transitive. We show that such modular categories admit unique factorization into prime transitive factors. The representations of $SL_2(\mathbb{Z})$ associated with transitive modular categories are proven to be minimal and irreducible. Together with the Verlinde formula, we characterize prime transitive modular categories as the Galois conjugates of the adjoint subcategory of the quantum group modular category $\mathcal{C}(\mathfrak{sl}_2,p-2)$ for some prime $p > 3$. As a consequence, we completely classify transitive modular categories. Transitivity of super-modular categories can be similarly defined. A unique factorization of any transitive super-modular category into s-simple transitive factors is obtained, and the split transitive super-modular categories are completely classified.