Generalized Pentagon Equations

2024/02/29 by Alekseev, Anton, Naef, Florian, Ren, Muze
#FOS: Mathematics #Quantum Algebra (math.QA)

paper · doi:10.48550/arxiv.2402.19138

Abstract

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 ℂ\backslash \ z₁, …, 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.

Related

• The pentagon equation and the confluence relations
• Connection Problem of Knizhnik-Zamolodchikov Equation on Moduli Space \mathcal M_0,5
• Poisson brackets and coaction maps of regularized holonomies of the KZ equation
• Explicit associator relations for multiple zeta values
• The Kashiwara-Vergne conjecture and Drinfeld's associators