# Compact Hypergroups from Discrete Subfactors

July 24, 2020

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.

Keywords:

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