On the extension of stringlike localised sectors in 2+1 dimensions

TL;DR

This paper investigates stringlike localized sectors in 2+1 dimensional algebraic quantum field theory, identifies obstructions to modularity caused by DHR sectors, and proposes a method to achieve modular tensor categories via the field net extension.

Contribution

It introduces a method to remove the non-modularity obstruction in stringlike sectors by extending to the Doplicher-Roberts field net, enabling modular tensor category structures.

Findings

DHR sectors create a non-trivial centre in the sector category.

Passing to the field net F(O) removes the modularity obstruction.

The sector category of F is characterized by a categorical crossed product.

Abstract

In the framework of algebraic quantum field theory, we study the category \Delta_BF^A of stringlike localised representations of a net of observables O \mapsto A(O) in three dimensions. It is shown that compactly localised (DHR) representations give rise to a non-trivial centre of \Delta_BF^A with respect to the braiding. This implies that \Delta_BF^A cannot be modular when non-trival DHR sectors exist. Modular tensor categories, however, are important for topological quantum computing. For this reason, we discuss a method to remove this obstruction to modularity. Indeed, the obstruction can be removed by passing from the observable net A(O) to the Doplicher-Roberts field net F(O). It is then shown that sectors of A can be extended to sectors of the field net that commute with the action of the corresponding symmetry group. Moreover, all such sectors are extensions of sectors of A. Finally, the category \Delta_BF^F of sectors of F is studied by investigating the relation with the categorical crossed product of \Delta_BF^A by the subcategory of DHR representations. Under appropriate conditions, this completely determines the category \Delta_BF^F.