Decorated $Z_2$ symmetry defects and their time-reversal anomalies

We discuss an isomorphism between the possible anomalies of $(d+1)$-dimensional quantum field theories with $Z_2$ 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 $Z_2$ symmetry defect is naturally invariant under $T$, and bulk $Z_2$ anomalies descend to $T$ anomalies on these defects. We illustrate this correspondence in detail for $(1+1)d$ bosonic systems where the bulk $Z_2$ 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 $Z_2$ global symmetry where interactions lead to a $Z_8$ classification of anomalies. Under the correspondence, this is directly related to the $Z_8$ classification of $(0+1)d$ fermions protected by T. Finally, we consider $(3+1)d$ bosonic systems with $Z_2$ symmetry where the possible anomalies are classified by $Z_2×Z_2$. 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.$