Two-dimensional topological order and operator algebras

Abstract

We review recent interactions between mathematical theory of two-dimensional topological order and operator algebras, particularly the Jones theory of subfactors. The role of representation theory in terms of tensor categories is emphasized. Connections to 2-dimensional conformal field theory are also presented. In particular, we discuss anyon condensation, gapped domain walls and matrix product operators in terms of operator algebras.

arXiv commentreview paper, 18 pages

References in corpus (24)

• Anyons in an exactly solved model and beyond
• Condensate induced transitions between topologically ordered phases
• Multi-interval Subfactors and Modularity of Representations in Conformal Field Theory
• Nets of Subfactors
• Correspondences of ribbon categories
• Anyons and matrix product operator algebras
• Chiral structure of modular invariants for subfactors
• Gapped Domain Walls, Gapped Boundaries and Topological Degeneracy
• On $\alpha$-induction, chiral generators and modular invariants for subfactors
• Exotic subfactors of finite depth with Jones indices (5+sqrt{13})/2 and (5+sqrt{17})/2
• Classification of Two-dimensional Local Conformal Nets with c<1 and 2-cohomology Vanishing for Tensor Categories
• Tensor categories and endomorphisms of von Neumann algebras (with applications to Quantum Field Theory)
• From vertex operator algebras to conformal nets and back
• A ${\mathbb{Z}}_2$-index of symmetry protected topological phases with time reversal symmetry for quantum spin chains
• Conformal Field Theory, Tensor Categories and Operator Algebras
• Canonical tensor product subfactors
• Classification of Local Conformal Nets. Case c < 1
• Spontaneous symmetry breaking from anyon condensation
• A remark on gapped domain walls between topological phases
• A remark on matrix product operator algebras, anyons and subfactors
• On Superselection Theory of Quantum Fields in Low Dimensions
• Unbounded field operators in categorical extensions of conformal nets
• Projector Matrix Product Operators, Anyons and Higher Relative Commutants of Subfactors
• A relative tensor product of subfactors over a modular tensor category

Cited by in corpus (8)

• Additivity, Haag duality, and non-invertible symmetries
• Projector Matrix Product Operators, Anyons and Higher Relative Commutants of Subfactors
• Operator-state correspondence in simple current extended conformal field theories: Toward a general understanding of chiral conformal field theories and topological orders
• 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
• Quantum cellular automata and categorical dualities of spin chains
• Fusion rule in conformal field theories and topological orders: A unified view of correspondence and (fractional) supersymmetry and their relation to topological holography
• Local topological order and boundary algebras
• A characterization of a finite-dimensional commuting square producing a subfactor of finite depth