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Generalized Pentagon Equations

Anton Alekseev, Florian Naef, Muze Ren

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 $\mathbb{C}\backslash \{ z_1, \dots, z_n\}$ which start and end at tangential base points. These paths are not necessarily straight, and they may have a finite number of transversal 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.

Generalized Pentagon Equations

Abstract

Drinfeld defined the Knizhinik--Zamolodchikov (KZ) associator by considering the regularized holonomy of the KZ connection along the {\em droit chemin} . 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 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 of the KZ connection associated to such a path satisfies a generalization of Drinfeld's pentagon equation. In this equation, we encounter , , and new factors associated to self-intersections, to tangential base points, and to the rotation number of the path.
Paper Structure (13 sections, 9 theorems, 92 equations, 3 figures)

This paper contains 13 sections, 9 theorems, 92 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Regularized holonomy
  3. Local solutions of the KZ equation
  4. Holonomy maps
  5. Composition of paths and rotation number
  6. Proof of the generalized pentagon equation
  7. Asymptotic regions and local solutions
  8. Regularized holonomies
  9. Self-intersection points
  10. $H_z$ and $H_w$
  11. $H_{zw}$
  12. $\Phi_{\rm KZ}$ contributions
  13. Properties of $C_l$

Key Result

Theorem 1.1

Let $X_1,\dots,X_n\in \mathcal{C}$, $P_1,P_2$ be two parenthesized products of $X_1,\dots,X_n$ (in any order order) possibly with insertions of the unit object $I$, and $f,g: P_1\to P_2$ be two isomorphisms obtained by composing braiding, associativity and unit isomorphism and their inverses possibl

Figures (3)

  • Figure 1: small region near the marked point
  • Figure 2: two ways of composing curves
  • Figure 3: $\Psi_1,\dots,\Psi_5$ are 5 solutions of the KZ equation and $\bullet_1,\bullet_2$ are two self intersection points of the curve in the (t,s) plane

Theorems & Definitions (20)

  • Theorem 1.1: Etingof2015
  • Theorem 1.2
  • Theorem 1.3: see Theorem \ref{['thm:C_l']}
  • Remark 2.1
  • Lemma 2.2
  • Remark 2.3
  • proof
  • Lemma 2.4
  • proof
  • Definition 2.5
  • ...and 10 more