Quantum Double Actions on Operator Algebras and Orbifold Quantum Field Theories

TL;DR

This paper explores the construction of nonlocal quantum field theories in 1+1 dimensions using disorder fields, revealing how quantum doubles and orbifold structures influence operator algebras and duality properties.

Contribution

It introduces a method to construct nonlocal theories with quantum double symmetries from local quantum field theories, clarifying Haag duality violations and extending to chiral theories.

Findings

Construction of disorder fields implementing gauge transformations.

Demonstration of Haag duality in extended fixpoint algebras.

Identification of quantum double actions and R-matrix relations.

Abstract

Starting from a local quantum field theory with an unbroken compact symmetry group $G$ in 1+1-dimensional spacetime we construct disorder fields implementing gauge transformations on the fields (order variables) localized in a wedge region. Enlarging the local algebras by these disorder fields we obtain a nonlocal field theory, the fixpoint algebras of which under the appropriately extended action of the group $G$ are shown to satisfy Haag duality in every simple sector. The specifically 1+1 dimensional phenomenon of violation of Haag duality of fixpoint nets is thereby clarified. In the case of a finite group $G$ the extended theory is acted upon in a completely canonical way by the quantum double $D(G)$ and satisfies R-matrix commutation relations as well as a Verlinde algebra. Furthermore, our methods are suitable for a concise and transparent approach to bosonization. The main technical ingredient is a strengthened version of the split property which should hold in all reasonable massive theories. In the appendices (part of) the results are extended to arbitrary locally compact groups and our methods are adapted to chiral theories on the circle.