Non-invertible Symmetries and Higher Representation Theory II

Abstract

In this paper we continue our investigation of the global categorical symmetries that arise when gauging finite higher groups and their higher subgroups with discrete torsion. The motivation is to provide a common perspective on the construction of non-invertible global symmetries in higher dimensions and a precise description of the associated symmetry categories. We propose that the symmetry categories obtained by gauging higher subgroups may be defined as higher group-theoretical fusion categories, which are built from the projective higher representations of higher groups. As concrete applications we provide a unified description of the symmetry categories of gauge theories in three and four dimensions based on the Lie algebra $\mathfrak{so}(N)$, and a fully categorical description of non-invertible symmetries obtained by gauging a 1-form symmetry with a mixed 't Hooft anomaly. We also discuss the effect of discrete torsion on symmetry categories, based a series of obstructions determined by spectral sequence arguments.

arXiv comment56 pages + appendix, v2: clarifications and citations added

References in corpus (69)

• Symmetry Fractionalization, Defects, and Gauging of Topological Phases
• Topological Defect Lines and Renormalization Group Flows in Two Dimensions
• Duality and defects in rational conformal field theory
• Generalised twisted partition functions
• On finite symmetries and their gauging in two dimensions
• Kramers-Wannier duality from conformal defects
• Kramers-Wannier-like duality defects in (3+1)d gauge theories
• Non-Invertible Duality Defects in 3+1 Dimensions
• Symmetries and Strings of Adjoint QCD${}_2$
• Higher Gauging and Non-invertible Condensation Defects
• Orbifold groupoids
• The Quantum Double Model with Boundary: Condensations and Symmetries
• Comments on One-Form Global Symmetries and Their Gauging in 3d and 4d
• Non-Invertible Higher-Categorical Symmetries
• Non-Invertible Global Symmetries and Completeness of the Spectrum
• Non-invertible Condensation, Duality, and Triality Defects in 3+1 Dimensions
• Defect lines, dualities, and generalised orbifolds
• Universal Non-Invertible Symmetries
• Anomalies of discrete symmetries in three dimensions and group cohomology
• Asymptotic density of states in 2d CFTs with non-invertible symmetries
• Non-Invertible Symmetries from Holography and Branes
• Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases
• Non-Invertible Symmetries of $\mathcal{N}=4$ SYM and Twisted Compactification
• The Branes Behind Generalized Symmetry Operators
• Non-invertible Time-reversal Symmetry
• Global Symmetries, Counterterms, and Duality in Chern-Simons Matter Theories with Orthogonal Gauge Groups
• Topological Transition on the Conformal Manifold
• Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions
• Orbifold completion of defect bicategories
• Branes and Non-Invertible Symmetries
• Fusion 2-categories and a state-sum invariant for 4-manifolds
• Topological Operators and Completeness of Spectrum in Discrete Gauge Theories
• Exploring Non-Invertible Symmetries in Free Theories
• On Continuous 2-Category Symmetries and Yang-Mills Theory
• Non-invertible Symmetries of Class $\mathcal{S}$ Theories
• Non-invertible 1-form symmetry and Casimir scaling in 2d Yang-Mills theory
• Anomalies of discrete symmetries in various dimensions and group cohomology
• Topological Holography: Towards a Unification of Landau and Beyond-Landau Physics
• Non-invertible self-duality defects of Cardy-Rabinovici model and mixed gravitational anomaly
• Non-invertible Symmetries and Higher Representation Theory I
• Non-Invertible Defects in 5d, Boundaries and Holography
• Discrete Theta Angles, Symmetries and Anomalies
• Decomposition, condensation defects, and fusion
• Generalized Symmetry Breaking Scales and Weak Gravity Conjectures
• Fusion Category Symmetry II: Categoriosities at c = 1 and Beyond
• Continuous Generalized Symmetries in Three Dimensions
• On anomalies and gauging of U(1) non-invertible symmetries in 4d QED
• Non-Invertible Symmetries from Discrete Gauging and Completeness of the Spectrum
• Mixed Anomalies, Two-groups, Non-Invertible Symmetries, and 3d Superconformal Indices
• Numerical evidence for a Haagerup conformal field theory
• Fusion rules of equivariantizations of fusion categories
• Topological symmetry in quantum field theory
• Duality Defects in $E_8$
• Relative Anomalies in (2+1)D Symmetry Enriched Topological States
• Solitonic symmetry beyond homotopy: Invertibility from bordism and noninvertibility from topological quantum field theory
• Gauging Noninvertible Defects: A 2-Categorical Perspective
• Topological Defect Lines in Two Dimensional Fermionic CFTs
• Lorentzian Dynamics and Factorization Beyond Rationality
• Absolute anomalies in (2+1)D symmetry-enriched topological states and exact (3+1)D constructions
• Twisted Quantum Double Model of Topological Orders with Boundaries
• Orbifolds and topological defects
• On Triality Defects in 2d CFT
• Gauge Enhanced Quantum Criticality Between Grand Unifications: Categorical Higher Symmetry Retraction
• Gapped boundaries and string-like excitations in (3+1)d gauge models of topological phases
• Topological operators, noninvertible symmetries and decomposition
• Gapped Boundary Theory of the Twisted Gauge Theory Model of Three-Dimensional Topological Orders
• Neutrino Masses from Generalized Symmetry Breaking
• Explicit Formulas for 2-Characters
• 't Hooft anomalies of discrete gauge theories and non-abelian group cohomology

Cited by in corpus (19)

• Gapped Phases with Non-Invertible Symmetries: (1+1)d
• Anomalies of Non-Invertible Symmetries in (3+1)d
• Non-invertible and higher-form symmetries in 2+1d lattice gauge theories
• The Club Sandwich: Gapless Phases and Phase Transitions with Non-Invertible Symmetries
• Gauging modulated symmetries: Kramers-Wannier dualities and non-invertible reflections
• Generalized Charges, Part II: Non-Invertible Symmetries and the Symmetry TFT
• Emergent generalized symmetries in ordered phases and applications to quantum disordering
• Non-invertible symmetry-protected topological order in a group-based cluster state
• Hasse Diagrams for Gapless SPT and SSB Phases with Non-Invertible Symmetries
• Categorical Symmetry of the Standard Model from Gravitational Anomaly
• Fermionic Non-Invertible Symmetries in (1+1)d: Gapped and Gapless Phases, Transitions, and Symmetry TFTs
• Boundary SymTFT
• Generalized symmetries in singularity-free nonlinear $\sigma$ models and their disordered phases
• Symmetry fractionalization, mixed-anomalies and dualities in quantum spin models with generalized symmetries
• Fiber 2-Functors and Tambara-Yamagami Fusion 2-Categories
• Symmetry fractionalization and duality defects in Maxwell theory
• The 3d $A$-model and generalised symmetries, Part I: bosonic Chern-Simons theories
• (2+1)d Lattice Models and Tensor Networks for Gapped Phases with Categorical Symmetry
• B-brane transport in anomalous (2,2) models and localization