Compact Hypergroups from Discrete Subfactors

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 ⊂ 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. 6. Representation theory 7. The case of depth 2: No compact quantum group orbifolds 8. A remark on non-local extensions 9. Examples 9.1. Group orbifolds 9.2. Double coset orbifolds 9.3. Quantum channels with infinite index 9.4. Local discrete subfactors in Conformal Field Theory 9.5. Constructions of discrete subfactors References