On the Extension of Stringlike Localised Sectors in 2+1 Dimensions

In the framework of algebraic quantum field theory, we study the category Δ_ BFA 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 Δ_ BFA with respect to the braiding. This implies that Δ_ BFA 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 Δ_ BFF of sectors of F is studied by investigating the relation with the categorical crossed product of Δ_ BFA by the subcategory of DHR representations. Under appropriate conditions, this completely determines the category Δ_ BFF .