Quantum operations on conformal nets

Author:

Bischoff Marcel1ORCID,Del Vecchio Simone2ORCID,Giorgetti Luca3ORCID

Affiliation:

1. Department of Mathematics, Morton Hall 321, 1 Ohio University, Athens, OH 45701, USA

2. Dipartimento di Matematica, Università degli Studi di Bari, Via E. Orabona, 4, Bari 70125, Italy

3. Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma I-00133, Italy

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].

Funder

Division of Mathematical Sciences

H2020 Marie Skodowska-Curie Actions

Ministero dell'Universita e della Ricerca

Publisher

World Scientific Pub Co Pte Ltd

Subject

Mathematical Physics,Statistical and Nonlinear Physics

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