Generalized orbifold construction for conformal nets

Abstract

Let [Formula: see text] be a conformal net. We give the notion of a proper action of a finite hypergroup [Formula: see text] acting by vacuum preserving unital completely positive (so-called stochastic) maps on [Formula: see text] which generalizes the proper action of a finite group [Formula: see text]. Taking the fixed point under such an action gives a finite index subnet [Formula: see text] of [Formula: see text], which generalizes the [Formula: see text]-orbifold net. Conversely, we show that if [Formula: see text] is a finite inclusion of conformal nets, then [Formula: see text] is a generalized orbifold [Formula: see text] of the conformal net [Formula: see text] by a unique finite hypergroup [Formula: see text]. There is a Galois correspondence between intermediate nets [Formula: see text] and subhypergroups [Formula: see text] given by [Formula: see text]. In this case, the fixed point of [Formula: see text] is the generalized orbifold by the hypergroup of double cosets [Formula: see text]. If [Formula: see text] is a finite index inclusion of completely rational nets, we show that the inclusion [Formula: see text] is conjugate to an intermediate subfactor of a Longo–Rehren inclusion. This implies that if [Formula: see text] is a holomorphic net, and [Formula: see text] acts properly on [Formula: see text], then there is a unitary fusion category [Formula: see text] which is a categorification of [Formula: see text] and [Formula: see text] is braided equivalent to the Drinfel’d center [Formula: see text]. More generally, if [Formula: see text] is a completely rational conformal net and [Formula: see text] acts properly on [Formula: see text], then there is a unitary fusion category [Formula: see text] extending [Formula: see text], such that [Formula: see text] is given by the double cosets of the fusion ring of [Formula: see text] by the Verlinde fusion ring of [Formula: see text] and [Formula: see text] is braided equivalent to the Müger centralizer of [Formula: see text] in the Drinfel’d center [Formula: see text].