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

TL;DR

This work reveals a Smith isomorphism between $\mathbb{Z}_2$ anomalies in $d$-dimensional QFTs and time-reversal anomalies in $(d-1)$-dimensional theories, realized through symmetry-defect decoration. By examining the defect worldvolume of a $\mathbb{Z}_2$ symmetry line, the authors show that bulk anomalies inflow to anomalous $\mathsf{T}$ symmetry on the defect, with concrete demonstrations in $(1+1)d$ bosonic and fermionic systems and a $(3+1)d$ bosonic example. Key results include the identification of Kramers degeneracy on defect Hilbert spaces when the bulk is anomalous, the $\mathbb{Z}_8$ classification in $(1+1)d$ fermions mapping to $(0+1)d$ $\mathsf{T}$ anomalies, and the extension to $(3+1)d$ where $\mathbb{Z}_2\times\mathbb{Z}_2$ anomalies correspond to $(2+1)d$ TRS anomalies realized in discrete gauge theories with defect lines carrying fermions or Kramers doublets. The findings illuminate how symmetry defects encode bulk topological data and demonstrate a unified, cobordism-based perspective on anomalies across dimensions with explicit constructions via twisted compactification and inflow.

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.

Figures (7)

• Figure 1: The action of a $\mathbb{Z}_2$ symmetry defect. The defect is topological and small changes in its position do not modify correlation functions. However, when the defect crosses a local operator $\phi$, the correlation changes by $\pm1$ depending on whether $\phi$ is even or odd under the action of the symmetry.
• Figure 2: The $\mathsf{T}$ symmetry on the symmetry defect in a spontaneously broken phase. The defect separates two regions with degenerate ground states labelled by $\pm$. The action of the $\mathbb{Z}_{2}$ symmetry exchanges the vacua, and combining with $\mathsf{CPT}$ leads to a symmetry of the original configuration that changes the worldvolume orientation of the defect.
• Figure 3: The crossing relation of a $\mathbb{Z}_2$ line $\cal L$ (shown in black) in a local patch of the $(1+1)d$ geometry. The lines can be recombined at the cost of a phase $\alpha=\pm1$. The $\mathbb{Z}_{2}$ symmetry is anomalous (i.e. there is an obstruction to orbifolding), if and only if $\alpha=-1$.
• Figure 4: The defect Hilbert space ${\cal H}_{\cal L}$ of a $\mathbb{Z}_2$ line quantized on a circle $S^1$. A state in the defect Hilbert space is mapped to an operator living at the end of the $\mathbb{Z}_2$ line via the operator-state correspondence.
• Figure 5: When the $\mathbb{Z}_2$ is anomalous (i.e. $\alpha=-1$), the two resolutions (sides) of an intersection of lines (center) lead to different configurations.