Schopieray’s Galois-modular extension conjecture

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; in the nonpseudounitary case, we prove that every braided fusion category admits a “Galois nondegenerate extension” but we do not know it to be modular. Along the way we record some general comments about the minimal nondegenerate extension problem for braided fusion categories.