Conformal inclusions of chiral conformal field theories, or more generally inclusions of quantum field theories, are described in the von Neumann algebraic setting by nets of subfactors, possibly with infinite Jones index if one takes non-rational theories into account. With this situation in mind, we study in a purely subfactor theoretical context a certain class of braided discrete subfactors with an additional commutativity constraint, that we call locality, and which corresponds to the commutation relations between field operators at space-like distance in quantum field theory. Examples of subfactors of this type come from taking a minimal action of a compact group on a factor and considering the fixed point subalgebra.We show that to every irreducible local discrete subfactor N subset of M of type III there is an associated canonical compact hypergroup (an invariant for the subfactor) which acts on M by unital completely positive (ucp) maps and which gives N as fixed points. To show this, we establish a duality pairing between the set of all N-bimodular ucp maps on M and a certain commutative unital C*-algebra, whose spectrum we identify with the compact hypergroup.If the subfactor has depth 2, the compact hypergroup turns out to be a compact group. This rules out the occurrence of compact quantum groups acting as global gauge symmetries in local conformal field theory.

Bischoff, M., Del Vecchio, S., Giorgetti, L. (2021). Compact hypergroups from discrete subfactors. JOURNAL OF FUNCTIONAL ANALYSIS, 281(1) [10.1016/j.jfa.2021.109004].

Compact hypergroups from discrete subfactors

Bischoff, M;Del Vecchio, S;Giorgetti, L
2021

Abstract

Conformal inclusions of chiral conformal field theories, or more generally inclusions of quantum field theories, are described in the von Neumann algebraic setting by nets of subfactors, possibly with infinite Jones index if one takes non-rational theories into account. With this situation in mind, we study in a purely subfactor theoretical context a certain class of braided discrete subfactors with an additional commutativity constraint, that we call locality, and which corresponds to the commutation relations between field operators at space-like distance in quantum field theory. Examples of subfactors of this type come from taking a minimal action of a compact group on a factor and considering the fixed point subalgebra.We show that to every irreducible local discrete subfactor N subset of M of type III there is an associated canonical compact hypergroup (an invariant for the subfactor) which acts on M by unital completely positive (ucp) maps and which gives N as fixed points. To show this, we establish a duality pairing between the set of all N-bimodular ucp maps on M and a certain commutative unital C*-algebra, whose spectrum we identify with the compact hypergroup.If the subfactor has depth 2, the compact hypergroup turns out to be a compact group. This rules out the occurrence of compact quantum groups acting as global gauge symmetries in local conformal field theory.
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/05
English
Discrete subfactors; Type III subfactors; Quantum groups; Conformal nets; Algebraic quantum field theory; Completely positive maps; Tensor categories; Compact hypergroups
Bischoff, M., Del Vecchio, S., Giorgetti, L. (2021). Compact hypergroups from discrete subfactors. JOURNAL OF FUNCTIONAL ANALYSIS, 281(1) [10.1016/j.jfa.2021.109004].
Bischoff, M; Del Vecchio, S; Giorgetti, L
Articolo su rivista
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2108/303681
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