Tube algebras, excitations statistics and compactification in gauge models of topological phases
Abstract We consider lattice Hamiltonian realizations of (d+1)-dimensional Dijkgraaf- Witten theory. In (2+1) d, it is well-known that the Hamiltonian yields point-like excita- tions classified by irreducible representations of the twisted quantum double. This can be confirmed using a tube algebra a...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2019-10-01
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| Series: | Journal of High Energy Physics |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1007/JHEP10(2019)216 |
| _version_ | 1811223433728491520 |
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| author | Alex Bullivant Clement Delcamp |
| author_facet | Alex Bullivant Clement Delcamp |
| author_sort | Alex Bullivant |
| collection | DOAJ |
| description | Abstract We consider lattice Hamiltonian realizations of (d+1)-dimensional Dijkgraaf- Witten theory. In (2+1) d, it is well-known that the Hamiltonian yields point-like excita- tions classified by irreducible representations of the twisted quantum double. This can be confirmed using a tube algebra approach. In this paper, we propose a generalisation of this strategy that is valid in any dimensions. We then apply this generalisation to derive the algebraic structure of loop-like excitations in (3+1) d, namely the twisted quantum triple. The irreducible representations of the twisted quantum triple algebra correspond to the simple loop-like excitations of the model. Similarly to its (2+1) d counterpart, the twisted quantum triple comes equipped with a compatible comultiplication map and an R-matrix that encode the fusion and the braiding statistics of the loop-like excitations, respectively. Moreover, we explain using the language of loop-groupoids how a model defined on a man- ifold that is n-times compactified can be expressed in terms of another model in n-lower dimensions. This can in turn be used to recast higher-dimensional tube algebras in terms of lower dimensional analogues. |
| first_indexed | 2024-04-12T08:32:54Z |
| format | Article |
| id | doaj.art-589acad2ee244e82a6928d3ce3133205 |
| institution | Directory Open Access Journal |
| issn | 1029-8479 |
| language | English |
| last_indexed | 2024-04-12T08:32:54Z |
| publishDate | 2019-10-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj.art-589acad2ee244e82a6928d3ce31332052022-12-22T03:40:07ZengSpringerOpenJournal of High Energy Physics1029-84792019-10-0120191017710.1007/JHEP10(2019)216Tube algebras, excitations statistics and compactification in gauge models of topological phasesAlex Bullivant0Clement Delcamp1Department of Pure Mathematics, University of LeedsMax-Planck-Institut für QuantenoptikAbstract We consider lattice Hamiltonian realizations of (d+1)-dimensional Dijkgraaf- Witten theory. In (2+1) d, it is well-known that the Hamiltonian yields point-like excita- tions classified by irreducible representations of the twisted quantum double. This can be confirmed using a tube algebra approach. In this paper, we propose a generalisation of this strategy that is valid in any dimensions. We then apply this generalisation to derive the algebraic structure of loop-like excitations in (3+1) d, namely the twisted quantum triple. The irreducible representations of the twisted quantum triple algebra correspond to the simple loop-like excitations of the model. Similarly to its (2+1) d counterpart, the twisted quantum triple comes equipped with a compatible comultiplication map and an R-matrix that encode the fusion and the braiding statistics of the loop-like excitations, respectively. Moreover, we explain using the language of loop-groupoids how a model defined on a man- ifold that is n-times compactified can be expressed in terms of another model in n-lower dimensions. This can in turn be used to recast higher-dimensional tube algebras in terms of lower dimensional analogues.http://link.springer.com/article/10.1007/JHEP10(2019)216Topological States of MatterAnyonsGauge Symmetry |
| spellingShingle | Alex Bullivant Clement Delcamp Tube algebras, excitations statistics and compactification in gauge models of topological phases Journal of High Energy Physics Topological States of Matter Anyons Gauge Symmetry |
| title | Tube algebras, excitations statistics and compactification in gauge models of topological phases |
| title_full | Tube algebras, excitations statistics and compactification in gauge models of topological phases |
| title_fullStr | Tube algebras, excitations statistics and compactification in gauge models of topological phases |
| title_full_unstemmed | Tube algebras, excitations statistics and compactification in gauge models of topological phases |
| title_short | Tube algebras, excitations statistics and compactification in gauge models of topological phases |
| title_sort | tube algebras excitations statistics and compactification in gauge models of topological phases |
| topic | Topological States of Matter Anyons Gauge Symmetry |
| url | http://link.springer.com/article/10.1007/JHEP10(2019)216 |
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