Modular categories with transitive Galois actions

TL;DR

This paper classifies transitive modular categories by their Galois actions, showing they decompose into prime factors, and characterizes their representations, extending to super-modular categories.

Contribution

It provides a complete classification of transitive modular and super-modular categories, introducing their prime factorization and Galois conjugate characterization.

Findings

Prime transitive modular categories are Galois conjugates of quantum group categories.

Representations of SL_2(Z) for these categories are minimal and irreducible.

Complete classification of transitive super-modular categories and their factors.

Abstract

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.