Anomalies and symmetry fractionalization in reflection-symmetric   topological order
  One of the central ideas regarding anomalies in topological phases of matter is that they imply the existence of higher-dimensional physics, with an anomaly in a D-dimensional theory typically being cancelled by a bulk (D+1)-dimensional symmetry-protected topological phase (SPT). We demonstrate that for some topological phases with reflection symmetry, anomalies may actually be cancelled by a D-dimensional SPT, provided that it comes embedded in an otherwise trivial (D+1)-dimensional bulk. We illustrate this for the example of $\mathbb{Z}_N$ topological order enriched with reflection symmetry in (2+1)D, and along the way establish a classification of anomalous reflection symmetry fractionalization patterns. In particular, we show that anomalies occur if and only if both electric and magnetic quasiparticle excitations possess nontrivial fractional reflection quantum numbers. 
Ethan Lake
cond-mat.str-el