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• W2496297807 endingPage "1750002" @default.
• W2496297807 startingPage "1750002" @default.
• W2496297807 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 $mathcal{B}^Ksubset mathcal{A} subset mathcal{B}$ and subhypergroups $Lsubset K$ given by $mathcal{A}=mathcal{B}^L$. In this case, the fixed point of $mathcal{B}^Ksubset mathcal{A}$ is the generalized orbifold by the hypergroup of double cosets $Lbackslash K/ L$. 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 Muger centralizer of $mathrm{Rep}(mathcal{B})$ in $Z(mathcal{F})$." @default.
• W2496297807 created "2016-08-23" @default.
• W2496297807 creator A5065979252 @default.
• W2496297807 date "2017-01-10" @default.
• W2496297807 modified "2023-10-16" @default.
• W2496297807 title "Generalized orbifold construction for conformal nets" @default.
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• W2496297807 doi "https://doi.org/10.1142/s0129055x17500027" @default.
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