Schopieray's Galois-modular extension conjecture

TL;DR

This paper proves Schopieray's conjecture that every pseudounitary premodular fusion category can be embedded into a modular category as a Galois-closed subcategory, advancing understanding of fusion category extensions.

Contribution

The paper confirms Schopieray's Galois-modular extension conjecture specifically for pseudounitary categories, providing new insights into the structure of fusion categories.

Findings

Proves the Galois-modular extension conjecture for pseudounitary categories

Provides comments on the minimal nondegenerate extension problem

Establishes conditions for Galois submodules in modular categories

Abstract

Plavnik, Schopieray, Yu, and Zhang have drawn attention to those (automatically premodular) fusion subcategories of modular fusion categories which are submodules for the Galois action on the ambient category. In particular, they showed that a subcategory is a Galois submodule if and only if its centralizer is integral. In the other direction, Schopieray has conjectured that every premodular fusion category can be embedded as a Galois-closed subcategory of a modular category; Schopieray calls such an embedding a "Galois-modular extension." We prove Schopieray's conjecture for pseudounitary categories. Along the way we record some general comments about the minimal nondegenerate extension problem for braided fusion categories.