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

In the framework of algebraic quantum field theory, we study the category ΔABF of stringlike localised representations of a net of observables O↦A(O) in three dimensions. It is shown that compactly localised (DHR) representations give rise to a non-trivial centre of ΔABF with respect to the braiding. This implies that ΔABF cannot be modular when non-trivial 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 ΔFBF of sectors of F is studied by investigating the relation with the categorical crossed product of ΔABF by the subcategory of DHR representations. Under appropriate conditions, this completely determines the category ΔFBF