Categorical fermionic actions and minimal modular extensions

This talk is based on a joint work done with César Galindo at Universidad de los Andes from Colombia. The purpose of this talk is to present some results on  minimal modular extensions of braided fusion categories doing emphases in 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 came from the study of fusion categories containing the representation category of a super-group. We develop many analog results to the Tannakian case, including cohomological obstructions, relation with braided $G$-crossed categories and minimal modular extensions.  We apply the general results to the construction and classification of minimal modular extensions of super-groups and braided fusion categories. In particular, we exhibit some examples of braided fusion categories without minimal modular extensions.