Category of SET orders

Abstract

We propose the representation principle to study physical systems with a given symmetry. In the context of symmetry enriched topological orders, we give the appropriate representation category, the category of SET orders, which include SPT orders and symmetry breaking orders as special cases. For fusion n-category symmetries, we show that the category of SET orders encodes almost all information about the interplay between symmetry and topological orders, in a natural and canonical way. These information include defects and boundaries of SET orders, symmetry charges, explicit and spontaneous symmetry breaking, stacking of SET orders, gauging of generalized symmetry, as well as quantum currents (SymTFT or symmetry TO). We also provide a detailed categorical algorithm to compute the generalized gauging. In particular, we proved that gauging is always reversible, as a special type of Morita-equivalence. The explicit data for ungauging, the inverse to gauging, is given.

44 pages, 11 figures, 1 table. Two new appendices and clarification on ordinary gauging, twisted gauging, and partial gauging are added

References in corpus (40)

• Anyons in an exactly solved model and beyond
• Generalized Global Symmetries
• Symmetry protected topological orders and the group cohomology of their symmetry group
• Classification of Gapped Symmetric Phases in 1D Spin Systems
• Classifying quantum phases using Matrix Product States and PEPS
• Symmetry Fractionalization, Defects, and Gauging of Topological Phases
• Complete classification of 1D gapped quantum phases in interacting spin systems
• On finite symmetries and their gauging in two dimensions
• A classification of symmetry enriched topological phases with exactly solvable models
• Higher Gauging and Non-invertible Condensation Defects
• Categorical symmetry and non-invertible anomaly in symmetry-breaking and topological phase transitions
• On the classification of topological orders
• Algebraic higher symmetry and categorical symmetry -- a holographic and entanglement view of symmetry
• A theory of 2+1D bosonic topological orders
• Symmetry enriched string-nets: Exactly solvable models for SET phases
• A theory of 2+1D fermionic topological orders and fermionic/bosonic topological orders with symmetries
• Exactly Solvable Models for Symmetry-Enriched Topological Phases
• Modular extensions of unitary braided fusion categories and 2+1D topological/SPT orders with symmetries
• Symmetry as a shadow of topological order and a derivation of topological holographic principle
• Non-Invertible Duality Transformation Between SPT and SSB Phases
• Classification of 2+1D topological orders and SPT orders for bosonic and fermionic systems with on-site symmetries
• Non-invertible anomalies and mapping-class-group transformation of anomalous partition functions
• A classification of 3+1D bosonic topological orders (II): the case when some point-like excitations are fermions
• Holographic theory for continuous phase transitions -- the emergence and symmetry protection of gaplessness
• Cluster state as a non-invertible symmetry protected topological phase
• Generalized Charges, Part I: Invertible Symmetries and Higher Representations
• Classification of topological phases with finite internal symmetries in all dimensions
• A mathematical theory of gapless edges of 2d topological orders. Part I
• Gapless edges of 2d topological orders and enriched monoidal categories
• A mathematical theory of gapless edges of 2d topological orders. Part II
• The balanced tensor product of module categories
• Categories of quantum liquids I
• The center functor is fully faithful
• A topological phase transition on the edge of the 2d ${\mathbb{Z}}_2$ topological order
• Symmetry-enriched topological order from partially gauging symmetry-protected topologically ordered states assisted by measurements
• Spontaneous symmetry breaking from anyon condensation
• The Morita Theory of Fusion 2-Categories
• Quantum Current and Holographic Categorical Symmetry
• Categories of quantum liquids II
• The boundary phase transitions of the 2+1D ${\mathbb{Z}}_N$ topological order via topological Wick rotation

Cited by in corpus (7)

• The Club Sandwich: Gapless Phases and Phase Transitions with Non-Invertible Symmetries
• Hasse Diagrams for Gapless SPT and SSB Phases with Non-Invertible Symmetries
• Fermionic Non-Invertible Symmetries in (1+1)d: Gapped and Gapless Phases, Transitions, and Symmetry TFTs
• Tube Category, Tensor Renormalization and Topological Holography
• Condensation Completion and Defects in 2+1D Topological Orders
• Systematic Construction of Interfaces and Anomalous Boundaries for Fermionic Symmetry-Protected Topological Phases
• Exactly solvable models for fermionic symmetry-enriched topological phases and fermionic 't Hooft anomaly