Galois Correspondence and Fourier Analysis on Local Discrete Subfactors

Discrete subfactors include a particular class of infinite index subfactors and all finite index ones. A discrete subfactor is called local when it is braided and it fulfills a commutativity condition motivated by the study of inclusion of Quantum Field Theories in the algebraic Haag-Kastler setting. In [BDG21], we proved that every irreducible local discrete subfactor arises as the fixed point subfactor under the action of a canonical compact hypergroup. In this work, we prove a Galois correspondence between intermediate von Neumann algebras and closed subhypergroups, and we study the subfactor theoretical Fourier transform in this context. Along the way, we extend the main results concerning $alpha$-induction and $sigma$-restriction for braided subfactors previously known in the finite index case.

Distortion for multifactor bimodules and representations of multifusion categories

We call a von Neumann algebra with finite dimensional center a multifactor. We introduce an invariant of bimodules over $rm II_1$ multifactors that we call modular distortion, and use it to formulate two classification results. We first classify finite depth finite index connected hyperfinite $rm II_1$ multifactor inclusions $Asubset B$ in terms of the standard invariant (a unitary planar algebra), together with the restriction to $A$ of the unique Markov trace on $B$. The latter determines the modular distortion of the associated bimodule. Three crucial ingredients are Popas uniqueness theorem for such inclusions which are also homogeneous, for which the standard invariant is a complete invariant, a generalized version of the Ocneanu Compactness Theorem, and the notion of Morita equivalence for inclusions. Second, we classify fully faithful representations of unitary multifusion categories into bimodules over hyperfinite $rm II_1$ multifactors in terms of the modular distortion. Every possible distortion arises from a representation, and we characterize the proper subset of distortions that arise from connected $rm II_1$ multifactor inclusions.

Compact Hypergroups from Discrete Subfactors

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 $mathcal{N}subsetmathcal{M}$ 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.

The hypergroupoid of boundary conditions for local quantum observables

We review the definition of hypergroups by Sunder, and we associate a hypergroup to a type III subfactor $Nsubset M$ of finite index, whose canonical endomorphism $gammainmathrm{End}(M)$ is multiplicity-free. It is realized by positive maps of $M$ that have $N$ as fixed points. If the depth is $>2$, this hypergroup is different from the hypergroup associated with the fusion algebra of $M$-$M$ bimodules that was Sunders original motivation to introduce hypergroups. We explain how the present hypergroup, associated with a suitable subfactor, controls the composition of transparent boundary conditions between two isomorphic quantum field theories, and that this generalizes to a hypergroupoid of boundary conditions between different quantum field theories sharing a common subtheory.