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Superselection sectors for posets of von Neumann algebras

Anupama Bhardwaj, Tristen Brisky, Chian Yeong Chuah, Kyle Kawagoe, Joseph Keslin, David Penneys, Daniel Wallick

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 Fröhlich 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.

Superselection sectors for posets of von Neumann algebras

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 Fröhlich for conformal nets. For cones in , 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.

Paper Structure

This paper contains 2 sections, 3 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Geometric poset axioms for a braided tensor category of superselection sectors

Key Result

Theorem 1

Given a poset ${\mathcal{P}}$ satisfying the geometric axioms geom:SelfDisjoint--geom:ZigZag and a ${\mathcal{P}}$-net of von Neumann algebras $A$, the superselection sectors $\mathsf{SSS}_p$ localized at $p$ carries a strict tensor product and a unitary braiding.

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3