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

In the framework of algebraic quantum field theory, we study the category ({Delta_{{rm BF}}^{mathfrak{A}}}) of stringlike localised representations of a net of observables ({mathcal{O} mapsto mathfrak{A}(mathcal{O})}) in three dimensions. It is shown that compactly localised (DHR) representations give rise to a non-trivial centre of ({Delta_{{rm BF}}^{mathfrak{A}}}) with respect to the braiding. This implies that ({Delta_{{rm BF}}^{mathfrak{A}}}) 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 ({mathfrak{A}(mathcal{O})}) to the Doplicher-Roberts field net ({mathfrak{F}(mathcal{O})}). It is then shown that sectors of ({mathfrak{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 ({mathfrak{A}}). Finally, the category ({Delta_{{rm BF}}^{mathfrak{F}}}) of sectors of ({mathfrak{F}}) is studied by investigating the relation with the categorical crossed product of ({Delta_{{rm BF}}^{mathfrak{A}}}) by the subcategory of DHR representations. Under appropriate conditions, this completely determines the category ({Delta_{{rm BF}}^{mathfrak{F}}}).