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

Abstract

This is the first part of a two-part work on a unified mathematical theory of gapped and gapless edges of 2d topological orders. We analyze all the possible observables on the 1+1D world sheet of a chiral gapless edge of a 2d topological order, and show that these observables form an enriched unitary fusion category, the Drinfeld center of which is precisely the unitary modular tensor category associated to the bulk. This mathematical description of a chiral gapless edge automatically includes that of a gapped edge (i.e. a unitary fusion category) as a special case. Therefore, we obtain a unified mathematical description and a classification of both gapped and chiral gapless edges of a given 2d topological order. In the process of our analysis, we encounter an interesting and reoccurring phenomenon: spatial fusion anomaly, which leads us to propose the Principle of Universality at RG fixed points for all quantum field theories. Our theory also implies that all chiral gapless edges can be obtained from a so-called topological Wick rotations. This fact leads us to propose, at the end of this work, a surprising correspondence between gapped and gapless phases in all dimensions.

52 pages, 39 figures, we add some discussion on spatial fusion anomalies, and propose how to define a phase transition between two gapless phases via topological Wick rotation, and discuss some interesting implications of our theory to higher dimensional topological orders

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Cited by in corpus (43)

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• Higher central charges and topological boundaries in 2+1-dimensional TQFTs
• Topological Holography: Towards a Unification of Landau and Beyond-Landau Physics
• Symmetry as a shadow of topological order and a derivation of topological holographic principle
• Lecture Notes on Generalized Symmetries and Applications
• 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
• Classification of topological phases with finite internal symmetries in all dimensions
• A mathematical theory of gapless edges of 2d topological orders. Part II
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• Quantum Phases and Transitions in Spin Chains with Non-Invertible Symmetries
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• A topological phase transition on the edge of the 2d ${\mathbb{Z}}_2$ topological order
• Relative Anomaly in (1+1)d Rational Conformal Field Theory
• Composing topological domain walls and anyon mobility
• Hasse Diagrams for Gapless SPT and SSB Phases with Non-Invertible Symmetries
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• Towards Non-Invertible Anomalies from Generalized Ising Models
• Topological Orders, Braiding Statistics, and Mixture of Two Types of Twisted $BF$ Theories in Five Dimensions
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• Fractional Hall Conductivity and Spin-c Structure in Solvable Lattice Hamiltonians
• Category of SET orders
• Duality via Sequential Quantum Circuit in the Topological Holography Formalism
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• Fusion rule in conformal field theories and topological orders: A unified view of correspondence and (fractional) supersymmetry and their relation to topological holography
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• 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
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• Self-duality protected multi-criticality in deconfined quantum phase transitions
• Measuring the Unique Identifiers of Topological Order Based on Boundary-Bulk Duality and Anyon Condensation