Year: 2024
Type: article
Abstract: Drinfeld defined the Knizhnik–Zamolodchikov (KZ) associator $$\Phi _{\textrm{KZ}}$$ Φ 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 $${\mathbb {C}}\backslash \{ z_1, \dots , z_n\}$$ C \ { z 1 , ⋯ , 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, $$\Phi _{\textrm{KZ}}$$ Φ KZ , and new factors associated with self-intersections, tangential base points, and the rotation number of the path.
Source: Annales Henri Poincaré
Authors: Anton Alekseev, Florian Naef, Muze Ren
Institutions: University of Geneva, Trinity College Dublin
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,990
OpenAlex ID: W4405412774