Superselection sectors for posets of von Neumann algebras

TL;DR

This paper develops a framework for superselection sectors in von Neumann algebras indexed by posets with involution, constructing braided tensor categories and relating them to quantum spin systems.

Contribution

It introduces geometric axioms for posets to build braided tensor categories of superselection sectors, extending previous conformal net constructions to new settings.

Findings

Constructed braided tensor categories from poset-indexed von Neumann algebras.

Showed equivalence of categories for intertwined nets of algebras.

Extended results to cones in r^2 with weakened duality conditions.

Abstract

We study a commutant-closed collection of von Neumann algebras acting on a common Hilbert space indexed by a poset with an order-reversing involution. We give simple geometric axioms for the poset which allow us to construct a braided tensor category of superselection sectors analogous to the construction of Gabbiani and Fr\"ohlich for conformal nets. For cones in $\mathbb{R}^2$, we weaken our conditions to a bounded spread version of Haag duality and obtain similar results. We show that intertwined nets of algebras have isomorphic braided tensor categories of superselection sectors. Finally, we show that the categories constructed here are equivalent to those constructed by Naaijkens and Ogata for certain 2D quantum spin systems.