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.
Córdova et al. (Thu,) studied this question.