Generalized Pentagon Equations
Abstract
Drinfeld defined the KnizhinikZamolodchikov (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 grouplike 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 selfintersections. 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 selfintersections, to tangential base points, and to the rotation number of the path.
 Publication:

arXiv eprints
 Pub Date:
 February 2024
 DOI:
 10.48550/arXiv.2402.19138
 arXiv:
 arXiv:2402.19138
 Bibcode:
 2024arXiv240219138A
 Keywords:

 Mathematics  Quantum Algebra
 EPrint:
 19 pages, 3 figures