Nondegenerate extensions of near-group braided fusion categories

TL;DR

This paper classifies near-group braided fusion categories that satisfy the minimal modular extension conjecture and explores the conditions under which certain categories have nondegenerate braided extensions, revealing new examples and counterexamples.

Contribution

It provides a classification of near-group braided fusion categories satisfying the minimal modular extension conjecture and identifies families of categories that violate it.

Findings

Classified near-group braided fusion categories satisfying the conjecture

Identified families of Tambara-Yamagami categories violating the conjecture

Extended results to categories related to extraspecial p-groups

Abstract

This is a study of weakly integral braided fusion categories with elementary fusion rules to determine which possess nondegenerately braided extensions of theoretically minimal dimension, or equivalently in this case, which satisfy the minimal modular extension conjecture. We classify near-group braided fusion categories satisfying the minimal modular extension conjecture; the remaining Tambara-Yamagami braided fusion categories provide arbitrarily large families of braided fusion categories with identical fusion rules violating the minimal modular extension conjecture. These examples generalize to braided fusion categories with the fusion rules of the representation categories of extraspecial $p$-groups for any prime $p$, which possess a minimal modular extension only if they arise as the adjoint subcategory of a twisted double of an extraspecial $p$-group.