Abstract
We study a class of anomalies associated with time-reversal and spatial reflection symmetry in (2+1)D 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 Z₂^ T) 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 notion of symmetry fractionalization. We present simple sufficient conditions which determine when Z₂^ T symmetry localization anomalies exist. We present an infinite series of TQFTs with such anomalies, some examples of which include USp(4)₂ and SO(4)₄ Chern-Simons (CS) theory. The theories that we find with these Z₂^ T anomalies can be obtained by gauging the unitary Z₂ subgroup of a different TQFT with a Z₄^ T symmetry. We show that the anomaly can be resolved in several ways: (1) the true symmetry of the theory is Z₄^ T, or (2) the theory can be considered to be a theory of fermions, with T² = (-1)^N_f corresponding to fermion parity. Finally, we demonstrate that theories with the Z₂^ T localization anomaly can be compatible with Z₂^ T if they are "pseudo-realized" at the surface of a (3+1)D symmetry-enriched topological phase. The "pseudo-realization" 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 confined to the surface.