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 M\"uger centralizer of $\mathrm{Rep}(\mathcal{B})$ in $Z(\mathcal{F})$.