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Decorated $\mathbb{Z}_{2}$ Symmetry Defects and Their Time-Reversal Anomalies

We discuss an isomorphism between the possible anomalies of $(d+1)$-dimensional quantum field theories with $\mathbb{Z}_{2}$ unitary global symmetry, and those of $d$-dimensional quantum field theories with time-reversal symmetry $\mathsf{T}$. This correspondence is an instance of symmetry defect decoration. The worldvolume of a $\mathbb{Z}_{2}$ symmetry defect is naturally invariant under $\mathsf{T},$ and bulk $\mathbb{Z}_{2}$ anomalies descend to $\mathsf{T}$ anomalies on these defects. We illustrate this correspondence in detail for $(1+1)d$ bosonic systems where the bulk $\mathbb{Z}_{2}$ anomaly leads to a Kramers degeneracy in the symmetry defect Hilbert space, and exhibit examples. We also discuss $(1+1)d$ fermion systems protected by $\mathbb{Z}_{2}$ global symmetry where interactions lead to a $\mathbb{Z}_{8}$ classification of anomalies. Under the correspondence, this is directly related to the $\mathbb{Z}_{8}$ classification of $(0+1)d$ fermions protected by $\mathsf{T}$. Finally, we consider $(3+1)d$ bosonic systems with $\mathbb{Z}_{2}$ symmetry where the possible anomalies are classified by $\mathbb{Z}_{2}\times \mathbb{Z}_{2}$. 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.