Title: Time-reversal and spatial reflection symmetry localization anomalies in (2+1)D topological phases of matter,
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 $\mathbb{Z}_2$ symmetry which can be
associated with either time-reversal (denoted $\mathbb{Z}_2^{\bf T})$ 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
$\mathbb{Z}_2^{\bf T}$ 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 $\mathbb{Z}_2^{\bf T}$ anomalies can be obtained by gauging the
unitary $\mathbb{Z}_2$ subgroup of a different TQFT with a $\mathbb{Z}_4^{\bf
T}$ symmetry. We show that the anomaly can be resolved in several ways: (1) the
true symmetry of the theory is $\mathbb{Z}_4^{\bf T}$, or (2) the theory can be
considered to be a theory of fermions, with ${\bf T}^2 = (-1)^{N_f}$
corresponding to fermion parity. Finally, we demonstrate that theories with the
$\mathbb{Z}_2^{\bf T}$ localization anomaly can be compatible with
$\mathbb{Z}_2^{\bf T}$ 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 $\mathbb{Z}_2$
gauge theory and thus only a subset of the quasiparticles are confined to the
surface.