A mathematical theory of gapless edges of 2d topological orders. Part II

Abstract

This is the second part of a two-part work on the unified mathematical theory of gapped and gapless edges of 2+1D topological orders. In Part I, we have developed the mathematical theory of chiral gapless edges. In Part II, we study boundary-bulk relation and non-chiral gapless edges. In particular, we explain how the notion of the center of an enriched monoidal category naturally emerges from the boundary-bulk relation. After the study of 0+1D gapless walls, we give the complete boundary-bulk relation for 2+1D topological orders with chiral gapless edges (including gapped edges) and 0d walls between edges. This relation is stated precisely and proved rigorously as a monoidal equivalence, which generalizes the functoriality of the usual Drinfeld center to an enriched setting. We also develop the mathematical theory of non-chiral gapless edges and 0+1D walls, and explain how to gap out certain non-chiral 1+1D gapless edges and 0+1D gapless walls categorically. In the end, we show that all anomaly-free 1+1D boundary-bulk rational CFT's can be recovered from 2d topological orders with chiral gapless edges via a dimensional reduction process. This provides physical meanings to some mysterious connections between mathematical results in fusion categories and those in rational CFT's.

54 pages. In Section 7, we add some discussion of the implication of our work to the study of gapless phases in all dimensions

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• Duality and defects in rational conformal field theory
• On the classification of topological orders
• Braided tensor categories and extensions of vertex operator algebras
• Cardy algebras and sewing constraints, I
• Classification of topological phases with finite internal symmetries in all dimensions
• Morita classes of algebras in modular tensor categories
• Pointed Drinfeld center functor

Cited by in corpus (28)

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• Bulk Anyons as Edge Symmetries: Boundary Phase Diagrams of Topologically Ordered States
• Holographic theory for continuous phase transitions -- the emergence and symmetry protection of gaplessness
• Categories of quantum liquids I
• Topological holography, quantum criticality, and boundary states
• Quantum Phases and Transitions in Spin Chains with Non-Invertible Symmetries
• One dimensional gapped quantum phases and enriched fusion categories
• Composing topological domain walls and anyon mobility
• Hasse Diagrams for Gapless SPT and SSB Phases with Non-Invertible Symmetries
• Towards Non-Invertible Anomalies from Generalized Ising Models
• Topological Orders, Braiding Statistics, and Mixture of Two Types of Twisted $BF$ Theories in Five Dimensions
• Categories of quantum liquids II
• Category of SET orders
• Fractional Hall Conductivity and Spin-c Structure in Solvable Lattice Hamiltonians
• Duality via Sequential Quantum Circuit in the Topological Holography Formalism
• Gauging or extending bulk and boundary conformal field theories: Application to bulk and domain wall problem in topological matter and their descriptions by (mock) modular covariant
• Categorical descriptions of one-dimensional gapped phases with Abelian onsite symmetries
• Edge Theories for Anyon Condensation Phase Transitions
• Theory of topological defects and textures in two-dimensional quantum orders with spontaneous symmetry breaking
• Chiral Virasoro algebra from a single wavefunction
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• Generalized Landau Paradigm for quantum phases and phase transitions
• Tube Category, Tensor Renormalization and Topological Holography
• The boundary phase transitions of the 2+1D ${\mathbb{Z}}_N$ topological order via topological Wick rotation
• Homomorphism, substructure, and ideal: Elementary but rigorous aspects of renormalization group or hierarchical structure of topological orders