On the Classification of Bosonic and Fermionic One-Form Symmetries in $$2+1$$d and ’t Hooft Anomaly Matching

Motivated by the fundamental role that bosonic and fermionic symmetries play in physics, we study finite (non-invertible) one-form symmetries in $$2+1$$ 2 + 1 d consisting of topological lines with bosonic and fermionic self-statistics. We refer to these lines as Bose–Fermi–Braided (BFB) symmetries and argue that they can be classified. Unlike the case of generic anyonic lines, BFB symmetries are closely related to groups. In particular, when BFB lines are non-invertible, they are non-intrinsically non-invertible. Moreover, BFB symmetries are, in a categorical sense, weakly group theoretical. Using this understanding, we study invariants of renormalization group flows involving non-topological QFTs with BFB symmetry.