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Communications in Mathematical Physics

, Volume 303, Issue 2, pp 385–420 | Cite as

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

  • Pieter Naaijkens
Open Access
Article

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 \({\Delta_{{\rm BF}}^{\mathfrak{F}}}\).

Keywords

Tensor Category Double Cone Algebraic Quantum Superselection Sector Tensor Functor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

This research is funded by NWO grant no. 613.000.608, which is gratefully acknowledged. I would also like to thank Michael Müger for valuable discussions and suggestions, and Klaas Landsman for a critical reading of the manuscript.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Copyright information

© The Author(s) 2011

Authors and Affiliations

  1. 1.Radboud University Nijmegen, Institute for Mathematics, Astrophysics and Particle PhysicsNijmegenThe Netherlands

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