Decorated ${\mathbb{Z}}_2$ Symmetry Defects and Their Time-Reversal Anomalies

Abstract

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.

29 pages, 7 figures