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 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) x 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.