Categorical Fermionic actions and minimal modular extensions

Abstract: The purpose of this paper is to study minimal modular extensions of braided fusion categories, with emphasis on minimal modular extensions of super-Tannakian fusion categories. We define actions of finite super-groups on fermionic fusion categories and spin-braided fusion categories. Similar to the case of groups, our motivation stems from the study of fusion categories containing the representation category of a super-group. We develop many results analogous to the Tannakian case, including cohomological obstructions, relationships with braided $G$-crossed categories, and minimal modular extensions. We apply our general framework to the construction and classification of minimal modular extensions of super-groups and braided fusion categories.