Time-reversal and spatial reflection symmetry localization anomalies in (2+1)D topological phases of matter

Abstract

We study a class of anomalies associated with time-reversal and spatial reflection symmetry in (2+1)D topological phases of matter. In these systems, the topological quantum numbers of the quasiparticles, such as the fusion rules and braiding statistics, possess a ${\mathbb{Z}}_2$ symmetry which can be associated with either time-reversal (denoted ${\mathbb{Z}}_2^{\mathbf{T}})$ or spatial reflections. Under this symmetry, correlation functions of all Wilson loop operators in the low energy topological quantum field theory (TQFT) are invariant. However, the theories that we study possess a severe anomaly associated with the failure to consistently localize the symmetry action to the quasiparticles, precluding even defining a notion of symmetry fractionalization. We present simple sufficient conditions which determine when ${\mathbb{Z}}_2^{\mathbf{T}}$ symmetry localization anomalies exist. We present an infinite series of TQFTs with such anomalies, some examples of which include USp$(4{)}_2$ and SO$(4{)}_4$ Chern-Simons (CS) theory. The theories that we find with these ${\mathbb{Z}}_2^{\mathbf{T}}$ anomalies can be obtained by gauging the unitary ${\mathbb{Z}}_2$ subgroup of a different TQFT with a ${\mathbb{Z}}_4^{\mathbf{T}}$ symmetry. We show that the anomaly can be resolved in several ways: (1) the true symmetry of the theory is ${\mathbb{Z}}_4^{\mathbf{T}}$, or (2) the theory can be considered to be a theory of fermions, with ${\mathbf{T}}^2=(-1{)}^{N_f}$ corresponding to fermion parity. Finally, we demonstrate that theories with the ${\mathbb{Z}}_2^{\mathbf{T}}$ localization anomaly can be compatible with ${\mathbb{Z}}_2^{\mathbf{T}}$ if they are "pseudo-realized" at the surface of a (3+1)D symmetry-enriched topological phase. The "pseudo-realization" refers to the fact that the bulk (3+1)D system is described by a dynamical ${\mathbb{Z}}_2$ gauge theory and thus only a subset of the quasiparticles are confined to the surface.

v3: typos corrected

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• 2-Group Symmetries in Class S
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• Three-dimensional dualities with bosons and fermions
• Codimension-2 defects and higher symmetries in (3+1)D topological phases
• Anomalies of Non-Invertible Symmetries in (3+1)d
• Relative Anomalies in (2+1)D Symmetry Enriched Topological States
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• Higher-group symmetry in finite gauge theory and stabilizer codes
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• Absolute anomalies in (2+1)D symmetry-enriched topological states and exact (3+1)D constructions
• Exceptional Chern-Simons-Matter Dualities
• A study of time reversal symmetry of abelian anyons
• Folding approach to topological orders enriched by mirror symmetry
• Higher-group symmetry of (3+1)D fermionic ${\mathbb{Z}}_2$ gauge theory: logical CCZ, CS, and T gates from higher symmetry
• Fractionalization and Anomalies in Symmetry-Enriched U(1) Gauge Theories
• Galois Orbits of TQFTs: Symmetries and Unitarity
• Soft symmetries of topological orders
• On dimensions of (2+1)D abelian bosonic topological systems on non-orientable manifolds
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• Anomalous gapped boundaries between surface topological orders in higher-order topological insulators and superconductors with inversion symmetry
• Generalization of anomaly formula for time reversal symmetry in (2+1)D abelian bosonic TQFTs