Oriented Closed Polyhedral Maps and the Kitaev Model
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866911910243336192 |
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| author | Szlachányi, Kornél |
| author_facet | Szlachányi, Kornél |
| contents | A kind of combinatorial map, called arrow presentation, is proposed to encode the data of the oriented closed polyhedral complexes $Σ$ on which the Hopf algebraic Kitaev model lives. We develop a theory of arrow presentations which underlines the role of the dual of the double $\mathcal{D}(Σ)^*$ of $Σ$ as being the Schreier coset graph of the arrow presentation, explains the ribbon structure behind curves on $\mathcal{D}(Σ)^*$ and facilitates computation of holonomy with values in the algebra of the Kitaev model. In this way, we can prove ribbon operator identities for arbitrary f.d. C$^*$-Hopf algebras and arbitrary oriented closed polyhedral maps. By means of a combinatorial notion of homotopy designed specially for ribbon curves, we can rigorously formulate ''topological invariance'' of states created by ribbon operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_08027 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Oriented Closed Polyhedral Maps and the Kitaev Model Szlachányi, Kornél Quantum Algebra Mathematical Physics A kind of combinatorial map, called arrow presentation, is proposed to encode the data of the oriented closed polyhedral complexes $Σ$ on which the Hopf algebraic Kitaev model lives. We develop a theory of arrow presentations which underlines the role of the dual of the double $\mathcal{D}(Σ)^*$ of $Σ$ as being the Schreier coset graph of the arrow presentation, explains the ribbon structure behind curves on $\mathcal{D}(Σ)^*$ and facilitates computation of holonomy with values in the algebra of the Kitaev model. In this way, we can prove ribbon operator identities for arbitrary f.d. C$^*$-Hopf algebras and arbitrary oriented closed polyhedral maps. By means of a combinatorial notion of homotopy designed specially for ribbon curves, we can rigorously formulate ''topological invariance'' of states created by ribbon operators. |
| title | Oriented Closed Polyhedral Maps and the Kitaev Model |
| topic | Quantum Algebra Mathematical Physics |
| url | https://arxiv.org/abs/2302.08027 |