Generalized Pentagon Equations
Abstract
Drinfeld defined the Knizhinik--Zamolodchikov (KZ) associator $\Phi_{\rm KZ}$ 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 $H$, $\Phi_{\rm KZ}$, and new factors associated to self-intersections, to tangential base points, and to the rotation number of the path.
- Publication:
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arXiv e-prints
- Pub Date:
- February 2024
- DOI:
- 10.48550/arXiv.2402.19138
- arXiv:
- arXiv:2402.19138
- Bibcode:
- 2024arXiv240219138A
- Keywords:
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- Mathematics - Quantum Algebra
- E-Print:
- 19 pages, 3 figures