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