Noninvertible Gauge Symmetry in (2+1)d Topological Orders: A String-Net Model Realization

Abstract

We develop a systematic framework for understanding symmetries in topological phases in 2+1 dimensions using the string-net model, encompassing both gauge symmetries that preserve anyon species and global symmetries permuting anyon species, including both invertible symmetries describable by groups and noninvertible symmetries described by categories. As an archetypal example, we reveal the first noninvertible categorical gauge symmetry of topological orders in 2+1 dimensions: the Fibonacci gauge symmetry of the doubled Fibonacci topological order, described by the Fibonacci fusion 2-category. Our approach involves two steps: first, establishing duality between different string-net models with Morita equivalent input fusion categories that describe the same topological order; and second, constructing symmetry transformations within the same string-net model when the dual models have isomorphic input fusion categories, achieved by composing duality maps with isomorphisms of degrees of freedom between the dual models.

41 pages, 5 Figures. Version 2 has improved clarity, with typos and minor errors corrected. Version 3 has revised the article format and incorporated additional figures and examples into the main text

References in corpus (30)

• Generalized Global Symmetries
• High temperature fractional quantum Hall states
• Exotic non-Abelian anyons from conventional fractional quantum Hall states
• Classifying fractionalization: symmetry classification of gapped Z2 spin liquids in two dimensions
• Non-Invertible Higher-Categorical Symmetries
• Universal Non-Invertible Symmetries
• Non-Invertible Symmetries from Holography and Branes
• Non-Invertible Symmetries of $\mathcal{N}=4$ SYM and Twisted Compactification
• Non-invertible Time-reversal Symmetry
• Non-invertible Symmetries and Higher Representation Theory I
• Symmetry TFTs for 3d QFTs from M-theory
• Hamiltonian and Algebraic Theories of Gapped Boundaries in Topological Phases of Matter
• Electric-magnetic duality in the quantum double models of topological orders with gapped boundaries
• Entanglement Entropy, Quantum Fluctuations, and Thermal Entropy in Topological Phases
• Fusion Surface Models: 2+1d Lattice Models from Fusion 2-Categories
• SymTFTs and Duality Defects from 6d SCFTs on 4-manifolds
• Quantization of Axion-Gauge Couplings and Non-Invertible Higher Symmetries
• Twisted Quantum Double Model of Topological Orders with Boundaries
• Non-invertible and higher-form symmetries in 2+1d lattice gauge theories
• Mapping between Morita equivalent string-net states with a constant depth quantum circuit
• Kennedy-Tasaki transformation and non-invertible symmetry in lattice models beyond one dimension
• Fusion 2-categories with no line operators are grouplike
• Space group symmetry fractionalization in a family of exactly solvable models with Z2 topological order
• Anyon Condensation: Coherent states, Symmetry Enriched Topological Phases, Goldstone Theorem, and Dynamical Rearrangement of Symmetry
• A 2-categorical state sum model
• Electric-Magnetic duality in twisted quantum double model of topological orders
• Extend The Levin-Wen Model To Two-dimensional Topological Orders With Gapped Boundary Junctions
• Symmetry Fractionalized (Irrationalized) Fusion Rules and Two Domain-Wall Verlinde Formulae
• Characteristic Properties of a Composite System of Topological Phases Separated by Gapped Domain Walls via an Exactly Solvable Hamiltonian Model
• Nonabelian Anyon Condensation in 2+1d topological orders: A String-Net Model Realization

Cited by in corpus (1)

• Nonlinear Symmetry-Fragmentation of Nonabelian Anyons In Symmetry-Enriched Topological Phases: A String-Net Model Realization