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

Abstract

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 $${\Delta_{{\rm BF}}^{\mathfrak{F}}}$$ .