More on time-reversal anomaly of 2+1d topological phases

Abstract

We prove an explicit formula conjectured recently by Wang and Levin for the anomaly of time-reversal symmetry in 2+1 dimensional fermionic topological quantum field theories. The crucial step is to determine the crosscap state in terms of the modular S matrix and ${\mathsf{T}}^2$ eigenvalues, generalizing the recent analysis by Barkeshli et al. in the bosonic case.

arXiv comment9 pages; v2:heuristic explanation added, title in the journal version changed to "Derivation of the Time-Reversal Anomaly for (2+1)-Dimensional Topological Phases"

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• Anomaly and global inconsistency matching: $\theta$-angles, $SU(3)/U(1{)}^2$ nonlinear sigma model, $SU(3)$ chains and its generalizations
• Cobordism Conjecture, Anomalies, and the String Lamppost Principle
• Modified instanton sum in QCD and higher-groups
• Anomaly matching for phase diagram of massless ${\mathbb{Z}}_N$-QCD
• Phases Of Adjoint QCD${}_3$ And Dualities
• Circle compactification and 't Hooft anomaly
• Anomaly constraint on massless QCD and the role of Skyrmions in chiral symmetry breaking
• Global Anomalies on the Hilbert Space
• Time-Reversal Symmetry, Anomalies, and Dualities in (2+1)$d$
• Anomalies, a mod 2 index, and dynamics of 2d adjoint QCD
• C-P-T anomaly matching in bosonic quantum field theory and spin chains
• Anomalies of Non-Invertible Symmetries in (3+1)d
• 8d gauge anomalies and the topological Green-Schwarz mechanism
• Topological field theories and symmetry protected topological phases with fusion category symmetries
• Time-reversal and spatial reflection symmetry localization anomalies in (2+1)D topological phases of matter
• Symmetry enhancement and closing of knots in 3d/3d correspondence
• `Unhinging' the surfaces of higher-order topological insulators and superconductors
• Relative Anomalies in (2+1)D Symmetry Enriched Topological States
• Quantum Distillation of Hilbert Spaces, Semi-classics and Anomaly Matching
• Anomaly matching in QCD thermal phase transition
• Absolute anomalies in (2+1)D symmetry-enriched topological states and exact (3+1)D constructions
• Mixed global anomalies and boundary conformal field theories
• Microscopic Theory of Surface Topological Order for Topological Crystalline Superconductors
• Pin TQFT and Grassmann integral
• A study of time reversal symmetry of abelian anyons
• Folding approach to topological orders enriched by mirror symmetry
• Categorical Symmetry of the Standard Model from Gravitational Anomaly
• Lattice gauge theory for Haldane conjecture and central-branch Wilson fermion
• High-temperature domain walls of QCD with imaginary chemical potentials
• Anomaly indicators for topological orders with $U(1)$ and time-reversal symmetry
• Anomaly indicators and bulk-boundary correspondences for 3D interacting topological crystalline phases with mirror and continuous symmetries
• Geometric Approach to 3D Interfaces at Strong Coupling
• $3d$ One-form Mixed Anomaly and Entanglement Entropy
• Mirror anomaly in fermionic topological orders
• On dimensions of (2+1)D abelian bosonic topological systems on non-orientable manifolds
• From $\mathcal{N}=4$ Super-Yang-Mills on ${\mathbb{RP}}^4$ to bosonic Yang-Mills on ${\mathbb{RP}}^2$
• Time-Reversal Anomalies of QCD${}_3$ and QED${}_3$
• Anomalies in mirror symmetry enriched topological orders
• Entanglement Spectrum Resolved by Loop Symmetries
• Generalization of anomaly formula for time reversal symmetry in (2+1)D abelian bosonic TQFTs