Decorated Z2 symmetry defects and their time-reversal anomalies

We discuss an isomorphism between the possible anomalies of (d+1)-dimensional quantum field theories with Z2 unitary global symmetry, and those of d-dimensional quantum field theories with time-reversal symmetry T. This correspondence is an instance of symmetry defect decoration. The worldvolume of a Z2 symmetry defect is naturally invariant under T, and bulk Z2 anomalies descend to T anomalies on these defects. We illustrate this correspondence in detail for (1+1)d bosonic systems where the bulk Z2 anomaly leads to a Kramers degeneracy in the symmetry defect Hilbert space and exhibits examples. We also discuss (1+1)d fermion systems protected by Z2 global symmetry where interactions lead to a Z8 classification of anomalies. Under the correspondence, this is directly related to the Z8 classification of (0+1)d fermions protected by T. Finally, we consider (3+1)d bosonic systems with Z2 symmetry where the possible anomalies are classified by Z2×Z2. We construct topological field theories realizing these anomalies and show that their associated symmetry defects support anyons that can be either fermions or Kramers doublets.