Generalized Pentagon Equations

Drinfeld defined the Knizhnik–Zamolodchikov (KZ) associator \(\Phi _{\textrm{KZ}}\) ${\mathrm{\Phi }}_{\text{KZ}}$ by considering the regularized holonomy of the KZ connection along the 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 \({\mathbb {C}}\backslash \{ z_1, \dots , z_n\}\) $\mathbb{C}\backslash \{z_1,\cdots ,z_n\}$ 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 with such a path satisfies a generalization of Drinfeld’s pentagon equation. In this equation, we encounter H, \(\Phi _{\textrm{KZ}}\) ${\mathrm{\Phi }}_{\text{KZ}}$ , and new factors associated with self-intersections, tangential base points, and the rotation number of the path.