Global Anomalies on the Hilbert Space

Abstract

We show that certain global anomalies can be detected in an elementary fashion by analyzing the way the symmetry algebra is realized on the torus Hilbert space of the anomalous theory. Distinct anomalous behaviours imprinted in the Hilbert space are identified with the distinct cohomology "layers" that appear in the classification of anomalies in terms of cobordism groups. We illustrate the manifestation of the layers in the Hilbert for a variety of anomalous symmetries and spacetime dimensions, including time-reversal symmetry, and both in systems of fermions and in anomalous topological quantum field theories (TQFTs) in 2+1d. We argue that anomalies can imply an exact bose-fermi degeneracy in the Hilbert space, thus revealing a supersymmetric spectrum of states; we provide a sharp characterization of when this phenomenon occurs and give nontrivial examples in various dimensions, including in strongly coupled QFTs. Unraveling the anomalies of TQFTs leads us to develop the construction of the Hilbert spaces, the action of operators and the modular data in spin TQFTs, material that can be read on its own.

74 pages

References in corpus (71)

• Generalized Global Symmetries
• Symmetry protected topological orders and the group cohomology of their symmetry group
• Topological Superconductivity and Superfluidity
• Symmetry Protected Topological phases of Quantum Matter
• The effects of interactions on the topological classification of free fermion systems
• Fermion Path Integrals And Topological Phases
• Fermionic Symmetry Protected Topological Phases and Cobordisms
• Symmetry-protected topological orders for interacting fermions -- Fermionic topological nonlinear $\sigma$ models and a special group supercohomology theory
• Interacting fermionic topological insulators/superconductors in three dimensions
• Symmetry protected topological phases from decorated domain walls
• Classifying gauge anomalies through SPT orders and classifying gravitational anomalies through topological orders
• Interacting topological phases and modular invariance
• Reflection positivity and invertible topological phases
• Sachdev-Ye-Kitaev Model and Thermalization on the Boundary of Many-Body Localized Fermionic Symmetry Protected Topological States
• The "Parity" Anomaly On An Unorientable Manifold
• Fermionic SPT phases in higher dimensions and bosonization
• Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology
• Dai-Freed anomalies in particle physics
• Spin TQFTs and fermionic phases of matter
• Interaction effect on topological classification of superconductors in two dimensions
• Chern-Simons-matter dualities with $SO$ and $USp$ gauge groups
• The effect of interactions on 2D fermionic symmetry-protected topological phases with Z2 symmetry
• State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter
• A Web of 2d Dualities: ${\mathbf{Z}}_2$ Gauge Fields and Arf Invariants
• A new class of $(2+1)$-d topological superconductor with ${\mathbb{Z}}_8$ topological classification
• Symmetry Protected Topological States of Interacting Fermions and Bosons
• Symmetry Protected Topological phases and Generalized Cohomology
• Global Symmetries, Counterterms, and Duality in Chern-Simons Matter Theories with Orthogonal Gauge Groups
• Anomalies and Bounds on Charged Operators
• Fermionic minimal models
• Phases Of Adjoint QCD${}_3$ And Dualities
• A New SU(2) Anomaly
• Higher Anomalies, Higher Symmetries, and Cobordisms I: Classification of Higher-Symmetry-Protected Topological States and Their Boundary Fermionic/Bosonic Anomalies via a Generalized Cobordism Theory
• Reflection and time reversal symmetry enriched topological phases of matter: path integrals, non-orientable manifolds, and anomalies
• On the cobordism classification of symmetry protected topological phases
• Fermionic Finite-Group Gauge Theories and Interacting Symmetric/Crystalline Orders via Cobordisms
• Classification of 2+1D topological orders and SPT orders for bosonic and fermionic systems with on-site symmetries
• Bosonization in three spatial dimensions and a 2-form gauge theory
• Construction and classification of symmetry protected topological phases in interacting fermion systems
• Beyond Standard Models and Grand Unifications: Anomalies, Topological Terms, and Dynamical Constraints via Cobordisms
• On time-reversal anomaly of 2+1d topological phases
• Symmetry-protected Topological Phases, Generalized Laughlin Argument and Orientifolds
• Anomaly indicators for time-reversal symmetric topological orders
• Interacting fermionic symmetry-protected topological phases in two dimensions
• Anomalies, a mod 2 index, and dynamics of 2d adjoint QCD
• Fermion condensation and super pivotal categories
• Interaction effects on 3D topological superconductors: surface topological order from vortex condensation, the 16 fold way and fermionic Kramers doublets
• Minimalist approach to the classification of symmetry protected topological phases
• Anomaly Matching in the Symmetry Broken Phase: Domain Walls, CPT, and the Smith Isomorphism
• More on time-reversal anomaly of 2+1d topological phases
• Time-reversal and spatial reflection symmetry localization anomalies in (2+1)D topological phases of matter
• Decorated ${\mathbb{Z}}_2$ Symmetry Defects and Their Time-Reversal Anomalies
• Dimensional Reduction and Topological Invariants of Symmetry-Protected Topological Phases
• Topological Superconductors on Superstring Worldsheets
• Absolute anomalies in (2+1)D symmetry-enriched topological states and exact (3+1)D constructions
• Two More Fermionic Minimal Models
• The algebra of anomaly interplay
• Supersymmetry in the nonsupersymmetric Sachdev-Ye-Kitaev model
• Fermionic Matrix Product States and One-Dimensional Short-Range Entangled Phases with Anti-Unitary Symmetries
• A (Dummy's) Guide to Working with Gapped Boundaries via (Fermion) Condensation
• Pin TQFT and Grassmann integral
• Topological Phases on Non-orientable Surfaces: Twisting by Parity Symmetry
• Unwinding Fermionic SPT Phases: Supersymmetry Extension
• Unoriented 3d TFTs
• Boundary supersymmetry of 1+1 d fermionic SPT phases
• Diagrammatic State Sums for 2D Pin-Minus TQFTs
• Supersymmetric boundaries of one-dimensional phases of fermions beyond symmetry-protected topological states
• Domain Walls in 4d N=1 Supersymmetric Yang-Mills
• Non-local Order Parameters and Quantum Entanglement for Fermionic Topological Field Theories
• Anomaly constraint on chiral central charge of (2+1)d topological order
• Classification of Metaplectic Fusion Categories

Cited by in corpus (21)

• Generalized Symmetries in Condensed Matter
• Symmetric Mass Generation
• Anomalies and Symmetry Fractionalization
• Codimension-2 defects and higher symmetries in (3+1)D topological phases
• Fermionic symmetry fractionalization in (2+1)D
• Topological Defect Lines in Two Dimensional Fermionic CFTs
• Anomalies in (2+1)D fermionic topological phases and (3+1)D path integral state sums for fermionic SPTs
• Fermionization of fusion category symmetries in 1+1 dimensions
• Anomaly cascade in (2+1)D fermionic topological phases
• Casimir Energy and Modularity in Higher-dimensional Conformal Field Theories
• Topological Defects in Floquet Circuits
• Symmetry-preserving boundary of (2+1)D fractional quantum Hall states
• Generalized non-unitary Haagerup-Izumi modular data from 3D S-fold SCFTs
• Classification of Fermionic Topological Orders from Congruence Representations
• (3+1)D path integral state sums on curved U(1) bundles and U(1) anomalies of (2+1)D topological phases
• Symmetries of 2d TQFTs and Equivariant Verlinde Formulae for General Groups
• C-P-T Fractionalization
• The 3d $A$-model and generalised symmetries, Part I: bosonic Chern-Simons theories
• Graded Hilbert spaces, quantum distillation and connecting SQCD to QCD
• Genus-One data and Anomaly Detection
• The mixed $0$-form/$1$-form anomaly in Hilbert space: pouring new wine into old bottles