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 \({\Delta_{{\rm BF}}^{\mathfrak{F}}}\).
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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.
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Naaijkens, P. On the Extension of Stringlike Localised Sectors in 2+1 Dimensions. Commun. Math. Phys. 303, 385–420 (2011). https://doi.org/10.1007/s00220-011-1200-6
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DOI: https://doi.org/10.1007/s00220-011-1200-6
Keywords
- Tensor Category
- Double Cone
- Algebraic Quantum
- Superselection Sector
- Tensor Functor