Invertibility of Condensation Defects and Symmetries of 2 + 1d QFTs

Abstract

We characterize discrete (anti-)unitary symmetries and their non-invertible generalizations in $2+1$d topological quantum field theories (TQFTs) through their actions on line operators and fusion spaces. We explain all possible sources of non-invertibility that can arise in this context. Our approach gives a simple $2+1$d proof that non-invertible generalizations of unitary symmetries exist if and only if a bosonic TQFT contains condensable bosonic line operators (i.e., these non-invertible symmetries are necessarily "non-intrinsic"). Moving beyond unitary symmetries and their non-invertible cousins, we define a non-invertible generalization of time-reversal symmetries and derive various properties of TQFTs with such symmetries. Finally, using recent results on 2-categories, we extend our results to corresponding statements in $2+1$d quantum field theories that are not necessarily topological.