The hypergroupoid of boundary conditions for local quantum observables

TL;DR

This paper explores the association of hypergroups to type III subfactors in quantum field theory, revealing how these structures govern boundary conditions and generalize to hypergroupoids for different theories.

Contribution

It introduces a new hypergroup construction linked to subfactors that controls boundary condition composition and extends to hypergroupoids for different quantum field theories.

Findings

Hypergroups associated with subfactors control boundary conditions.

The hypergroup differs from the fusion algebra hypergroup for depth > 2.

Hypergroupoids generalize boundary conditions between different theories.

Abstract

We review the definition of hypergroups by Sunder, and we associate a hypergroup to a type III subfactor $N\subset M$ of finite index, whose canonical endomorphism $\gamma\in\mathrm{End}(M)$ 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 Sunder's 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.