Compact Hypergroups from Discrete Subfactors

TL;DR

This paper introduces a new class of braided discrete subfactors with locality constraints, associating them with canonical compact hypergroups that act on the larger algebra, revealing insights into symmetries in quantum field theory.

Contribution

It establishes a duality between subfactor bimodular ucp maps and a commutative $C^*$-algebra, linking subfactors to canonical hypergroups, and shows hypergroups are groups in depth 2 cases.

Findings

Every irreducible local discrete subfactor has an associated canonical compact hypergroup.

The hypergroup acts on the larger algebra via ucp maps, with fixed points being the subfactor.

Depth 2 subfactors correspond to compact groups, not quantum groups.

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 $\mathcal{N}\subset\mathcal{M}$ of type ${I\!I\!I}$ there is an associated canonical compact hypergroup (an invariant for the subfactor) which acts on $\mathcal{M}$ by unital completely positive (ucp) maps and which gives $\mathcal{N}$ as fixed points. To show this, we establish a duality pairing between the set of all $\mathcal{N}$-bimodular ucp maps on $\mathcal{M}$ 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 \emph{quantum} groups acting as global gauge symmetries in local conformal field theory.