Generalized Orbifold Construction for Conformal Nets

Abstract

Let $\mathcal{B}$ be a conformal net. We give the notion of a proper action of a finite hypergroup acting by vacuum preserving unital completely positive (so-called stochastic) maps, which generalizes the proper actions of finite groups. Taking fixed points under such an action gives a finite index subnet ${\mathcal{B}}^K$ of $\mathcal{B}$, which generalizes the $G$-orbifold. Conversely, we show that if $\mathcal{A}\subset \mathcal{B}$ is a finite inclusion of conformal nets, then $\mathcal{A}$ is a generalized orbifold $\mathcal{A}={\mathcal{B}}^K$ of the conformal net $\mathcal{B}$ by a unique finite hypergroup $K$. There is a Galois correspondence between intermediate nets ${\mathcal{B}}^K\subset \mathcal{A}\subset \mathcal{B}$ and subhypergroups $L\subset K$ given by $\mathcal{A}={\mathcal{B}}^L$. In this case, the fixed point of ${\mathcal{B}}^K\subset \mathcal{A}$ is the generalized orbifold by the hypergroup of double cosets $L\backslash K/L$. If $\mathcal{A}\subset \mathcal{B}$ is an finite index inclusion of completely rational nets, we show that the inclusion $\mathcal{A}(I)\subset \mathcal{B}(I)$ is conjugate to a Longo--Rehren inclusion. This implies that if $\mathcal{B}$ is a holomorphic net, and $K$ acts properly on $\mathcal{B}$, then there is a unitary fusion category $\mathcal{F}$ which is a categorification of $K$ and $\mathrm{Rep}({\mathcal{B}}^K)$ is braided equivalent to the Drinfel'd center $Z(\mathcal{F})$. More generally, if $\mathcal{B}$ is completely rational conformal net and $K$ acts properly on $\mathcal{B}$, then there is a unitary fusion category $\mathcal{F}$ extending $\mathrm{Rep}(\mathcal{B})$, such that $K$ is given by the double cosets of the fusion ring of $\mathcal{F}$ by the Verlinde fusion ring of $\mathcal{B}$ and $\mathrm{Rep}({\mathcal{B}}^K)$ is braided equivalent to the Müger centralizer of $\mathrm{Rep}(\mathcal{B})$ in $Z(\mathcal{F})$.

40 pages, many TikZ figures. Comments are welcome

References in corpus (28)

• Multi-interval Subfactors and Modularity of Representations in Conformal Field Theory
• Modular Structure and Duality in Conformal Quantum Field Theory
• TFT construction of RCFT correlators III: Simple currents
• Nets of Subfactors
• Modular Invariants, Graphs and $\alpha$-Induction for Nets of Subfactors. III
• Applications of braided endomorphisms from conformal inclusions
• Defect lines, dualities, and generalised orbifolds
• On $\alpha$-induction, chiral generators and modular invariants for subfactors
• Chiral structure of modular invariants for subfactors
• Braided tensor categories and extensions of vertex operator algebras
• Modular Invariants, Graphs and $\alpha$-Induction for Nets of Subfactors I
• Conformal Subnets and Intermediate Subfactors
• The classification of subfactors of index at most 5
• Modular Invariants, Graphs and $\alpha$-Induction for Nets of Subfactors II
• Tensor categories and endomorphisms of von Neumann algebras (with applications to Quantum Field Theory)
• On the representation theory of Virasoro Nets
• Algebraic orbifold conformal field theories
• The exoticness and realisability of twisted Haagerup-Izumi modular data
• Modular Invariants from Subfactors: Type I Coupling Matrices and Intermediate Subfactors
• Conformal Orbifold Theories and Braided Crossed G-Categories
• News from the Virasoro algebra
• Canonical tensor product subfactors
• A Remark on CFT Realization of Quantum Doubles of Subfactors. Case Index < 4
• Models in Boundary Quantum Field Theory Associated with Lattices and Loop Group Models
• Phase boundaries in algebraic conformal QFT
• A Cuntz algebra approach to the classification of near-group categories
• Generalized Longo-Rehren subfactors and alpha-induction
• Spectral Measures for $G_2$

Cited by in corpus (16)

• Duality Defects in $E_8$
• On a class of selection rules without group actions in field theory and string theory
• Non-invertible symmetries act locally by quantum operations
• Bosonic Rational Conformal Field Theories in Small Genera, Chiral Fermionization, and Symmetry/Subalgebra Duality
• Spontaneous symmetry breaking from anyon condensation
• Haploid algebras in $C^{\ast }$-tensor categories and the Schellekens list
• Infinite index extensions of local nets and defects
• Compact Hypergroups from Discrete Subfactors
• Lorentzian 2d CFT from the pAQFT perspective
• Bayesian inversion and the Tomita-Takesaki modular group
• Galois Correspondence and Fourier Analysis on Local Discrete Subfactors
• Quantum Operations on Conformal Nets
• Hopf algebra actions in tensor categories
• Classification of unitary vertex subalgebras and conformal subnets for rank-one lattice chiral CFT models
• Fusion rules for permutation extensions of modular tensor categories
• von Neumann Subfactors and Non-invertible Symmetries