Poisson brackets and coaction maps of regularized holonomies of the KZ equation

TL;DR

This paper derives explicit formulas for coaction maps and Poisson brackets related to the KZ equation's regularized holonomies, advancing the understanding of their algebraic structures.

Contribution

It provides new explicit formulas for coaction maps and Poisson brackets of KZ holonomies using a novel projection of the generalized pentagon equation.

Findings

Explicit formulas for KKS coaction maps

Poisson brackets for gl(N,C) holonomies

Use of projection of generalized pentagon equation

Abstract

We derive explicit closed formulas for the Kirillov-Kostant-Souriau (KKS) coaction maps of open path regularized holonomies of the Knizhnik-Zamolodchikov (KZ) equation, and the corresponding Poisson brackets for the Lie algebra ${\rm gl}(N, \mathbb{C})$. Our main technical tool is a certain projection of the generalized pentagon equation of \cite{AFR2024}.