Generalized Pentagon Equations
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914696875999232 |
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| author | Alekseev, Anton Naef, Florian Ren, Muze |
| author_facet | Alekseev, Anton Naef, Florian Ren, Muze |
| contents | Drinfeld defined the Knizhinik--Zamolodchikov (KZ) associator $Φ_{\rm KZ}$ by considering the regularized holonomy of the KZ connection along the {\em droit chemin} $[0,1]$. The KZ associator is a group-like element of the free associative algebra with two generators, and it satisfies the pentagon equation.
In this paper, we consider paths on $\mathbb{C}\backslash \{ z_1, \dots, z_n\}$ which start and end at tangential base points. These paths are not necessarily straight, and they may have a finite number of transversal self-intersections. We show that the regularized holonomy $H$ of the KZ connection associated to such a path satisfies a generalization of Drinfeld's pentagon equation. In this equation, we encounter $H$, $Φ_{\rm KZ}$, and new factors associated to self-intersections, to tangential base points, and to the rotation number of the path. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_19138 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalized Pentagon Equations Alekseev, Anton Naef, Florian Ren, Muze Quantum Algebra Drinfeld defined the Knizhinik--Zamolodchikov (KZ) associator $Φ_{\rm KZ}$ by considering the regularized holonomy of the KZ connection along the {\em droit chemin} $[0,1]$. The KZ associator is a group-like element of the free associative algebra with two generators, and it satisfies the pentagon equation. In this paper, we consider paths on $\mathbb{C}\backslash \{ z_1, \dots, z_n\}$ which start and end at tangential base points. These paths are not necessarily straight, and they may have a finite number of transversal self-intersections. We show that the regularized holonomy $H$ of the KZ connection associated to such a path satisfies a generalization of Drinfeld's pentagon equation. In this equation, we encounter $H$, $Φ_{\rm KZ}$, and new factors associated to self-intersections, to tangential base points, and to the rotation number of the path. |
| title | Generalized Pentagon Equations |
| topic | Quantum Algebra |
| url | https://arxiv.org/abs/2402.19138 |