Localized Endomorphisms of the Chiral Ising Model

TL;DR

This paper constructs localized endomorphisms in the chiral Ising model, demonstrating their equivalence to global superselection sectors, and explicitly deriving fusion rules and statistics operators within a rigorous algebraic framework.

Contribution

It introduces explicit examples of localized endomorphisms in the chiral Ising model and proves their equivalence to global sectors using Araki's formalism, including fusion rules and statistics operators.

Findings

Localized endomorphisms correspond to global superselection sectors

Fusion rules are explicitly derived within the algebraic framework

Statistics operators are explicitly calculated

Abstract

Based on the treatment of the chiral Ising model by Mack and Schomerus, we present examples of localized endomorphisms $\varrho_1^{\rm loc}$ and $\varrho_{1/2}^{\rm loc}$. It is shown that they lead to the same superselection sectors as the global ones in the sense that unitary equivalence $\pi_0\circ\varrho_1^{\rm loc}\cong\pi_1$ and $\pi_0\circ\varrho_{1/2}^{\rm loc}\cong\pi_{1/2}$ holds. Araki's formalism of the selfdual CAR algebra is used for the proof. We prove local normality and extend representations and localized endomorphisms to a global algebra of observables which is generated by local von Neumann algebras on the punctured circle. In this framework, we manifestly prove fusion rules and derive statistics operators.