Time-reversal and spatial-reflection symmetry localization anomalies in (2+1)-dimensional topological phases of matter

We study a class of anomalies associated with time-reversal and spatial-reflection symmetry in (2+1)-dimensional bosonic topological phases of matter. In these systems, the topological quantum numbers of the quasiparticles, such as the fusion rules and braiding statistics, possess a Z₂ symmetry which can be associated with either time reversal (denoted Z₂T) or spatial reflections. Under this symmetry, correlation functions of all Wilson loop operators in the low-energy topological quantum field theory (TQFT) are invariant. However, the theories that we study possess a severe anomaly associated with the failure to consistently localize the symmetry action to the quasiparticles, precluding even defining a consistent notion of symmetry fractionalization in such systems. We present simple sufficient conditions which determine when Z₂T symmetry localization anomalies exist in general. We present an infinite series of TQFTs with such anomalies, some examples of which include USp(4)₂ Chern-Simons (CS) theory and SO(4)₄ CS theory. The theories that we find with these Z₂T anomalies can all be obtained by gauging the unitary Z₂ subgroup of a different TQFT with a Z₄T symmetry. We further show that the anomaly can be resolved in several distinct ways: (1) the true symmetry of the theory is Z₄T, or (2) the theory can be considered to be a theory of fermions, with T²=(-1)^Nf corresponding to fermion parity. Finally, we demonstrate that theories with the Z₂T localization anomaly can be compatible with Z₂T if they are ``pseudorealized'' at the surface of a (3+1)D symmetry-enriched topological phase. The ``pseudorealization'' refers to the fact that the bulk (3+1)D system is described by a dynamical Z₂ gauge theory and thus only a subset of the quasiparticles are truly confined to the surface.