SciPost Phys. 8, 028 (2020) ·
published 18 February 2020

· pdf
Certain patterns of symmetry fractionalization in topologically ordered
phases of matter are anomalous, in the sense that they can only occur at the
surface of a higher dimensional symmetryprotected topological (SPT) state. An
important question is to determine how to compute this anomaly, which means
determining which SPT hosts a given symmetryenriched topological order at its
surface. While special cases are known, a general method to compute the anomaly
has so far been lacking. In this paper we propose a general method to compute
relative anomalies between different symmetry fractionalization classes of a
given (2+1)D topological order. This method applies to all types of symmetry
actions, including anyonpermuting symmetries and general spacetime reflection
symmetries. We demonstrate compatibility of the relative anomaly formula with
previous results for diagnosing anomalies for $\mathbb{Z}_2^{\bf T}$ spacetime
reflection symmetry (e.g. where timereversal squares to the identity) and
mixed anomalies for $U(1) \times \mathbb{Z}_2^{\bf T}$ and $U(1) \rtimes
\mathbb{Z}_2^{\bf T}$ symmetries. We also study a number of additional
examples, including cases where spacetime reflection symmetries are
intertwined in nontrivial ways with unitary symmetries, such as
$\mathbb{Z}_4^{\bf T}$ and mixed anomalies for $\mathbb{Z}_2 \times
\mathbb{Z}_2^{\bf T}$ symmetry, and unitary $\mathbb{Z}_2 \times \mathbb{Z}_2$
symmetry with nontrivial anyon permutations.