On $\alpha$-induction, chiral generators and modular invariants for subfactors

Abstract

We consider a type III subfactor $N\subset M$ of finite index with a finite system of braided $N$-$N$ morphisms which includes the irreducible constituents of the dual canonical endomorphism. We apply $\alpha$-induction and, developing further some ideas of Ocneanu, we define chiral generators for the double triangle algebra. Using a new concept of intertwining braiding fusion relations, we show that the chiral generators can be naturally identified with the $\alpha$-induced sectors. A matrix $Z$ is defined and shown to commute with the S- and T-matrices arising from the braiding. If the braiding is non-degenerate, then $Z$ is a ``modular invariant mass matrix'' in the usual sense of conformal field theory. We show that in that case the fusion rule algebra of the dual system of $M$-$M$ morphisms is generated by the images of both kinds of $\alpha$-induction, and that the structural information about its irreducible representations is encoded in the mass matrix $Z$. Our analysis sheds further light on the connection between (the classifications of) modular invariants and subfactors, and we will construct and analyze modular invariants from $SU(n{)}_k$ loop group subfactors in a forthcoming publication, including the treatment of all $SU(2{)}_k$ modular invariants.

66 pages, latex, epic, eepic; minor changes, typos fixed, references added

References in corpus (1)

• Chiral structure of modular invariants for subfactors

Cited by in corpus (84)

• TFT construction of RCFT correlators I: Partition functions
• Duality and defects in rational conformal field theory
• Generalised twisted partition functions
• Chiral structure of modular invariants for subfactors
• Modular extensions of unitary braided fusion categories and 2+1D topological/SPT orders with symmetries
• The many faces of Ocneanu cells
• Tensor categories for vertex operator superalgebra extensions
• Projections in string theory and boundary states for Gepner models
• Topological Sectors and a Dichotomy in Conformal Field Theory
• Local fields in boundary conformal QFT
• A mathematical theory of gapless edges of 2d topological orders. Part I
• Generalized ADE Classification of Gapped Domain Walls
• Algebraic orbifold conformal field theories
• The exoticness and realisability of twisted Haagerup-Izumi modular data
• Local conformal nets arising from framed vertex operator algebras
• Subfactor realisation of modular invariants
• Gapless edges of 2d topological orders and enriched monoidal categories
• Solitons in Affine and Permutation Orbifolds
• Conformal Field Theory, Tensor Categories and Operator Algebras
• Integrable Lattice Realizations of Conformal Twisted Boundary Conditions
• Perturbed Defects and T-Systems in Conformal Field Theory
• The fusion algebra of bimodule categories
• A (Dummy's) Guide to Working with Gapped Boundaries via (Fermion) Condensation
• Generalized Orbifold Construction for Conformal Nets
• Ocneanu Cells and Boltzmann Weights for the SU(3) ADE Graphs
• The Nakayama automorphism of the almost Calabi-Yau algebras associated to SU(3) modular invariants
• Critical Phenomena, modular invariants and operator algebras
• How to remove the boundary in CFT - an operator algebraic procedure
• Haploid algebras in $C^{\ast }$-tensor categories and the Schellekens list
• Comments about quantum symmetries of SU(3) graphs
• A Remark on CFT Realization of Quantum Doubles of Subfactors. Case Index < 4
• Pivotal fusion categories of rank 3 (with an Appendix written jointly with Dmitri Nikshych)
• Fusion rules of modular invariants
• Phase boundaries in algebraic conformal QFT
• Locality and modular invariance in 2D conformal QFT
• Subfactors and Mathematical Physics
• A remark on gapped domain walls between topological phases
• A remark on matrix product operator algebras, anyons and subfactors
• Two-dimensional topological order and operator algebras
• On intermediate subfactors of Goodman-de la Harpe-Jones subfactors
• Algebraic conformal quantum field theory in perspective
• Racah - Wigner quantum 6j Symbols, Ocneanu Cells for AN diagrams, and quantum groupoids
• Higher Coxeter graphs associated to affine su(3) modular invariants
• Level bounds for exceptional quantum subgroups in rank two
• Spectral Measures and Generating Series for Nimrep Graphs in Subfactor Theory
• From Subfactors to Categories and Topology I. Frobenius algebras in and Morita equivalence of tensor categories
• Generalized Longo-Rehren subfactors and alpha-induction
• Galois Correspondence and Fourier Analysis on Local Discrete Subfactors
• On Discrete Symmetries in su(2) and su(3) Affine Theories and Related Graphs
• Algebraic Structures in Euclidean and Minkowskian Two-Dimensional Conformal Field Theory
• Quantum Operations on Conformal Nets
• 3-manifold invariants from cosets
• Modular invariants and subfactors
• On the Jones index values for conformal subnets
• Conformal Net Realizability of Tambara-Yamagami Categories and Generalized Metaplectic Modular Categories
• A2-Planar Algebras II: Planar Modules
• Bantay's trace in Unitary Modular Tensor Categories
• Spectral Measures and Generating Series for Nimrep Graphs in Subfactor Theory II: SU(3)
• Braided categories of endomorphisms as invariants for local quantum field theories
• Local Modules in Braided Monoidal 2-Categories
• Constructing non-semisimple modular categories with local modules
• Anyonic Chains -- $\alpha$-Induction -- CFT -- Defects -- Subfactors
• Fusion rule in conformal field theories and topological orders: A unified view of correspondence and (fractional) supersymmetry and their relation to topological holography
• Spectral Measures for $G_2$
• Modular Invariants and Twisted Equivariant K-theory II: Dynkin diagram symmetries
• Braided Subfactors, Spectral Measures, Planar algebras and Calabi-Yau algebras associated to SU(3) modular invariants
• From Operator Algebras to Superconformal Field Theory
• Modular Invariants and Twisted Equivariant K-theory
• A characterization of a finite-dimensional commuting square producing a subfactor of finite depth
• On a Conjecture of Kac-Wakimoto
• The rank of $G$-crossed braided extensions of modular tensor categories
• A relative tensor product of subfactors over a modular tensor category
• Spectral measures associated to rank two Lie groups and finite subgroups of $GL(2,\mathbb{Z})$
• An application of Mirror extensions
• A simple sufficient condition for triviality of obstruction in the orbifold construction for subfactors
• Topological Quantum Field Theories and Operator Algebras
• Frobenius algebras associated with the $\alpha$-induction for equivariantly braided tensor categories
• Classification of Module Categories for $SO(3{)}_{2m}$
• The relative Drinfeld commutant of a fusion category and $\alpha$-induction
• Interpolation categories for Conformal Embeddings
• Antisymmetric characters and Fourier duality
• On the homology of almost Calabi-Yau algebras associated to SU(3) modular invariants
• Modular invariants from subfactors
• On representing some lattices as lattices of intermediate subfactors of finite index