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.
Highlights
Global symmetries and anomalies are crucial tools in the analysis of quantum field theory
Since anomalies of d dimensional theories are determined by inflow from (d þ 1)-dimensional symmetry-protected topological phases (SPTs), our discussion can be interpreted as a connection between SPTs
The precise mathematical relationship behind (1.1) is sometimes referred to as a Smith isomorphism [9,10], which we review in Appendix
Summary
Global symmetries and anomalies are crucial tools in the analysis of quantum field theory. A rich class of phenomena are associated with systems with discrete global symmetry In this case the associated anomalies are finite order and can be carried by gapped or gapless systems. A precise understanding of the physics of anomalies is a central tool in many recent developments including the theory of topological insulators. Since anomalies of d dimensional theories are determined by inflow from (d þ 1)-dimensional symmetry-protected topological phases (SPTs), our discussion can be interpreted as a connection between SPTs. there is an isomorphism: Z2 anomalies in d dimensions. We illustrate this correspondence in a simple class of models described by ð1 þ 1Þd bosonic quantum field theories (QFTs) with Z2 global symmetry. We discuss the extension to ð1 þ 1Þd fermionic systems and give an example application in ð3 þ 1Þd bosonic systems
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have