Categorical Fermionic Actions and Minimal Modular Extensions
César Galindo, César F. Venegas-Ramírez
TL;DR
This work develops a comprehensive obstruction-theoretic framework for when braided fusion categories admit minimal non-degenerate extensions, distinguishing modularizable (Tannakian) and non-modularizable (super-Tannakian) centers. By de-equivariantizing with respect to the maximal central Tannakian subcategory and introducing fermionic actions of finite super-groups on fermionic fusion categories, the authors reduce the extension problem to cohomological data, notably an $H^4$-anomaly $O_4$ and fermionic liftings governed by $O_3$. The main result states that a modularizable Braided fusion category $\\mathcal{B}$ has a minimal non-degenerate extension iff $O_4(\\mathcal{B})=0$, while a non-modularizable case requires a minimal extension of the de-equivariantized piece, a compatible fermionic super-group action, and vanishing $O_4(\\mathcal{S}^G)$. The work also provides explicit computations and examples, including Ising and rank-four pointed spin-braided categories, and situates these results alongside recent advances (e.g., JFR) in the broader program of classifying modular extensions of braided fusion categories.
Abstract
We define fermionic actions of finite super-groups on fermionic fusion categories and establish necessary and sufficient conditions for their existence. Our main result characterizes when a braided fusion category admits a minimal non-degenerate extension in terms of cohomological obstructions. This characterization for braided fusion categories with non-Tannakian Müger center involves the fermionic structures and fermionic actions introduced in this work.