Decorated <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi mathvariant="double-struck">Z</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math> symmetry defects and their time-reversal anomalies
Clay Córdova, Kantaro Ohmori, Shu-Heng Shao, F. Yan
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 exhibits 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 ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$. 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.