Modular Categories with Transitive Galois Actions

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