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.$