Fusion rules and shrinking rules of topological orders in five dimensions

Abstract

As a series of work about 5D (spacetime) topological orders, here we employ the path-integral formalism of 5D topological quantum field theory (TQFT) established in Zhang and Ye, JHEP 04 (2022) 138 to explore non-Abelian fusion rules, hierarchical shrinking rules and quantum dimensions of particle-like, loop-like and membrane-like topological excitations in 5D topological orders. To illustrate, we focus on a prototypical example of twisted $BF$ theories that comprise the twisted topological terms of the $BBA$ type. First, we classify topological excitations by establishing equivalence classes among all gauge-invariant Wilson operators. Then, we compute fusion rules from the path-integral and find that fusion rules may be non-Abelian; that is, the fusion outcome can be a direct sum of distinct excitations. We further compute shrinking rules. Especially, we discover exotic hierarchical structures hidden in shrinking processes of 5D or higher: a membrane is shrunk into particles and loops, and the loops are subsequently shrunk into a direct sum of particles. We obtain the algebraic structure of shrinking coefficients and fusion coefficients. We compute the quantum dimensions of all excitations and find that sphere-like membranes and torus-like membranes differ not only by their shapes but also by their quantum dimensions. We further study the algebraic structure that determines anomaly-free conditions on fusion coefficients and shrinking coefficients. Besides $BBA$, we explore general properties of all twisted terms in $5$D. Together with braiding statistics reported before, the theoretical progress here paves the way toward characterizing and classifying topological orders in higher dimensions where topological excitations consist of both particles and spatially extended objects.

arXiv commentAccepted by JHEP (Nov 20, 2023). This paper is part of a series of work with 2104.07067 and 2208.09228

References in corpus (45)

• Fault-tolerant quantum computation by anyons
• Non-Abelian Anyons and Topological Quantum Computation
• Anyons in an exactly solved model and beyond
• Topological entanglement entropy
• Detecting topological order in a ground state wave function
• Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States
• 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
• Coupling a QFT to a TQFT and Duality
• Generalized Symmetries in Condensed Matter
• Superconductors are topologically ordered
• Classifying gauge anomalies through SPT orders and classifying gravitational anomalies through topological orders
• Field theory representation of gauge-gravity symmetry-protected topological invariants, group cohomology and beyond
• Algebraic higher symmetry and categorical symmetry -- a holographic and entanglement view of symmetry
• Quantum Field Theory of Nonabelian Strings and Vortices
• Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions
• Projective Construction of Non-Abelian Quantum Hall Liquids
• Quantized topological terms in weak-coupling gauge theories with symmetry and their connection to symmetry enriched topological phases
• Topological Response Theory of Abelian Symmetry-Protected Topological Phases in Two Dimensions
• A K matrix Construction of Symmetry Enriched Phases of Matter
• Topological quantum field theory of three-dimensional bosonic Abelian-symmetry-protected topological phases
• 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
• Projective construction of two-dimensional symmetry-protected topological phases with U(1), SO(3), or SU(2) symmetries
• Effective Field Theory and Projective Construction for the Z_k Parafermion Fractional Quantum Hall States
• Symmetry-protected topological phases with charge and spin symmetries: response theory and dynamical gauge theory in 2D, 3D and the surface of 3D
• Multi-kink topological terms and charge-binding domain-wall condensation induced symmetry-protected topological states: Beyond Chern-Simons/BF theory
• Braiding with Borromean Rings in (3+1)-Dimensional Spacetime
• Composite Particle Theory of Three-dimensional Gapped Fermionic Phases: Fractional Topological Insulators and Charge-Loop Excitation Symmetry
• Generalized Wen-Zee Terms
• $U(1)\times U(1)$ Symmetry Protected Topological Order in Gutzwiller Wave Functions
• Symmetry Enrichment in Three-Dimensional Topological Phases
• Topological Quantum Field Theory for Abelian Topological Phases and Loop Braiding Statistics in $(3+1)$-Dimensions
• Entanglement entropy of (3+1)D topological orders with excitations
• Fractional $S$-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
• Three-dimensional anomalous twisted gauge theories with global symmetry: Implications for quantum spin liquids
• Theory of oblique topological insulators
• A geometric interpretation of Milnor's triple linking numbers
• Compatible braidings with Hopf links, multi-loop, and Borromean rings in $(3+1)$-dimensional spacetime
• Topological Orders, Braiding Statistics, and Mixture of Two Types of Twisted $BF$ 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

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