Diagrammatics, Pentagon Equations, and Hexagon Equations of Topological Orders with Loop- and Membrane-like Excitations
arXiv:2405.19077 · doi:10.1007/JHEP06(2025)238
Abstract
In spacetime dimensions of 4 (i.e., 3+1) and higher, topological orders exhibit spatially extended excitations like loops and membranes, which support diverse topological data characterizing braiding, fusion, and shrinking processes, despite the absence of anyons. Our understanding of these topological data remains less mature compared to 3D, where anyons have been extensively studied and can be fully described through diagrammatic representations. Inspired by recent advancements in field theory descriptions of higher-dimensional topological orders, this paper systematically constructs diagrammatic representations for 4D and 5D topological orders, generalizable to higher dimensions. We introduce elementary diagrams for fusion and shrinking processes, treating them as vectors in fusion and shrinking spaces, respectively, and build complex diagrams by combining these elementary diagrams. Within these vector spaces, we design unitary operations represented by \(F\)-, \(Δ\)-, and \(Δ^2\)-symbols to transform between different bases. We discover \textit{pentagon equations} and \textit{(hierarchical) shrinking-fusion hexagon equations} that impose constraints on the legitimate forms of these unitary operations. We conjecture that all anomaly-free higher-dimensional topological orders must satisfy these conditions and any violations indicate a quantum anomaly. This work opens promising avenues for future research, including the exploration of diagrammatic representations involving braiding and the study of non-invertible symmetries and symmetry topological field theories in higher spacetime dimensions.
arXiv commenttitle adjusted. This paper is part of a series of work with 1703.01926, 2012.13761, 2104.07067, 2208.09228, 2306.14611, and 2307.09983
References in corpus (60)
- Topological insulators and superconductors
- Non-Abelian Anyons and Topological Quantum Computation
- Anyons in an exactly solved model and beyond
- String-net condensation: A physical mechanism for topological phases
- Zoo of quantum-topological phases of matter
- Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order
- Symmetry Fractionalization, Defects, and Gauging of Topological Phases
- Coupling a QFT to a TQFT and Duality
- Theory and classification of interacting 'integer' topological phases in two dimensions: A Chern-Simons approach
- Fermion Path Integrals And Topological Phases
- Superconductors are topologically ordered
- Kramers-Wannier-like duality defects in (3+1)d gauge theories
- Non-Invertible Duality Defects in 3+1 Dimensions
- Higher Gauging and Non-invertible Condensation Defects
- Non-invertible Condensation, Duality, and Triality Defects in 3+1 Dimensions
- Field theory representation of gauge-gravity symmetry-protected topological invariants, group cohomology and beyond
- Non-Abelian Topological Order and Anyons on a Trapped-Ion Processor
- A theory of 2+1D bosonic topological orders
- Twisted Gauge Theory Model of Topological Phases in Three Dimensions
- Diagrammatics for Bose condensation in anyon theories
- Interferometry of non-Abelian Anyons
- Non-Abelian String and Particle Braiding in Topological Order: Modular SL(3,Z) Representation and 3+1D Twisted Gauge Theory
- Quantum Field Theory of Nonabelian Strings and Vortices
- Non-Invertible Symmetries of SYM and Twisted Compactification
- Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions
- A classification of 3+1D bosonic topological orders (I): the case when point-like excitations are all bosons
- Generalized modular transformations in 3+1D topologically ordered phases and triple linking invariant of loop braiding
- Non-Invertible Defects in 5d, Boundaries and Holography
- Layer construction of 3D topological states and string braiding statistics
- Bulk-boundary correspondence in (3+1)-dimensional topological phases
- A classification of 3+1D bosonic topological orders (II): the case when some point-like excitations are fermions
- Bulk-boundary correspondence for three-dimensional symmetry-protected topological phases
- A K matrix Construction of Symmetry Enriched Phases of Matter
- Universal Topological Data for Gapped Quantum Liquids in Three Dimensions and Fusion Algebra for Non-Abelian String Excitations
- Vortex-line condensation in three dimensions: A physical mechanism for bosonic topological insulators
- Constructing symmetric topological phases of bosons in three dimensions via fermionic projective construction and dyon condensation
- Topological quantum field theory of three-dimensional bosonic Abelian-symmetry-protected topological phases
- Projective construction of two-dimensional symmetry-protected topological phases with U(1), SO(3), or SU(2) symmetries
- Multi-kink topological terms and charge-binding domain-wall condensation induced symmetry-protected topological states: Beyond Chern-Simons/BF theory
- Symmetry-protected topological phases with charge and spin symmetries: response theory and dynamical gauge theory in 2D, 3D and the surface of 3D
- Anomalies and gauging of U(1) symmetries
- Braiding with Borromean Rings in (3+1)-Dimensional Spacetime
- Wilson operator algebras and ground states for coupled BF theories
- Composite Particle Theory of Three-dimensional Gapped Fermionic Phases: Fractional Topological Insulators and Charge-Loop Excitation Symmetry
- Generalized Wen-Zee Terms
- On the Symmetry TFT of Yang-Mills-Chern-Simons theory
- Entanglement entropy of (3+1)D topological orders with excitations
- Symmetry Enrichment in Three-Dimensional Topological Phases
- Topological Quantum Field Theory for Abelian Topological Phases and Loop Braiding Statistics in -Dimensions
- Fractional -duality, Classification of Fractional Topological Insulators and Surface Topological Order
- Topological Electromagnetic Responses of Bosonic Quantum Hall, Topological Insulator, and Chiral Semi-Metal phases in All Dimensions
- Clebsch-Gordan and 6j-coefficients for rank two quantum groups
- Three-dimensional anomalous twisted gauge theories with global symmetry: Implications for quantum spin liquids
- Theory of oblique topological insulators
- Compatible braidings with Hopf links, multi-loop, and Borromean rings in -dimensional spacetime
- Topological Orders, Braiding Statistics, and Mixture of Two Types of Twisted Theories in Five Dimensions
- Non-Abelian Fusion, Shrinking and Quantum Dimensions of Abelian Gauge Fluxes
- Fractionalizing Global Symmetry on Looplike Topological Excitations
- Continuum field theory of 3D topological orders with emergent fermions and braiding statistics
- Fusion rules and shrinking rules of topological orders in five dimensions