Quantum operations on conformal nets

Abstract

On a conformal net [Formula: see text], one can consider collections of unital completely positive maps on each local algebra [Formula: see text], subject to natural compatibility, vacuum preserving and conformal covariance conditions. We call quantum operations on [Formula: see text] the subset of extreme such maps. The usual automorphisms of [Formula: see text] (the vacuum preserving invertible unital *-algebra morphisms) are examples of quantum operations, and we show that the fixed point subnet of [Formula: see text] under all quantum operations is the Virasoro net generated by the stress-energy tensor of [Formula: see text]. Furthermore, we show that every irreducible conformal subnet [Formula: see text] is the fixed points under a subset of quantum operations. When [Formula: see text] is discrete (or with finite Jones index), we show that the set of quantum operations on [Formula: see text] that leave [Formula: see text] elementwise fixed has naturally the structure of a compact (or finite) hypergroup, thus extending some results of [M. Bischoff, Generalized orbifold construction for conformal nets, Rev. Math. Phys. 29 (2017) 1750002]. Under the same assumptions, we provide a Galois correspondence between intermediate conformal nets and closed subhypergroups. In particular, we show that intermediate conformal nets are in one-to-one correspondence with intermediate subfactors, extending a result of Longo in the finite index/completely rational conformal net setting [R. Longo, Conformal subnets and intermediate subfactors, Comm. Math. Phys. 237 (2003) 7–30].