Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions

Abstract

Gravitational anomalies can be realized on the boundary of topologically ordered states in one higher dimension and are described by topological orders in one higher dimension. In this paper, we try to develop a general theory for both topological order and gravitational anomaly in any dimensions. (1) We introduce the notion of BF category to describe the braiding and fusion properties of topological excitations that can be point-like, string-like, etc. A subset of BF categories -- closed BF categories -- classify topological orders in any dimensions, while generic BF categories classify (potentially) anomalous topological orders that can appear at a boundary of a gapped quantum liquid in one higher dimension. (2) We introduce topological path integral based on tensor network to realize those topological orders. (3) Bosonic topological orders have an important topological invariant: the vector bundles of the degenerate ground states over the moduli spaces of closed spaces with different metrics. They may fully characterize topological orders. (4) We conjecture that a topological order has a gappable boundary iff the above mentioned vector bundles are flat. (5) We find a holographic phenomenon that every topological order with a gappable boundary can be uniquely determined by the knowledge of the boundary. As a consequence, BF categories in different dimensions form a (monoid) cochain complex, that reveals the structure and relation of topological orders and gravitational anomalies in different dimensions. We also studied the simplest kind of bosonic topological orders that have no non-trivial topological excitations. We find that this kind of topological orders form a $\mathbb{Z}$ class in 2+1D (with gapless edge), a ${\mathbb{Z}}_2$ class in 4+1D (with gappable boundary), and a $\mathbb{Z}\oplus \mathbb{Z}$ class in 6+1D (with gapless boundary).

69 pages, 30 figures

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• Quantum Phases and Transitions in Spin Chains with Non-Invertible Symmetries
• One dimensional gapped quantum phases and enriched fusion categories
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• Foliated-Exotic Duality in Fractonic BF Theories
• String Condensations in 3+1D and Lagrangian Algebras
• Towards Non-Invertible Anomalies from Generalized Ising Models
• (3+1)D topological orders with only a ${\mathbb{Z}}_2$-charged particle
• Topological Orders, Braiding Statistics, and Mixture of Two Types of Twisted $BF$ Theories in Five Dimensions
• Quantum Current and Holographic Categorical Symmetry
• Non-Invertible T-duality at Any Radius via Non-Compact SymTFT
• Local Modules in Braided Monoidal 2-Categories
• Non-Invertible Gauss Law and Axions
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• Lieb-Schultz-Mattis Theorem and the Filling Constraint
• Remote detectability from entanglement bootstrap I: Kirby's torus trick
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• Generalized symmetry enriched criticality in (3+1)d