Generalized Pentagon Equations

Drinfeld defined the Knizhnik-Zamolodchikov (KZ) associator <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>Φ</mml:mi> <mml:mtext>KZ</mml:mtext></mml:msub> </mml:math> by considering the regularized holonomy of the KZ connection along the <i>droit chemin</i> [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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>C</mml:mi> <mml:mo>\</mml:mo> <mml:mo>{</mml:mo> <mml:msub><mml:mi>z</mml:mi> <mml:mn>1</mml:mn></mml:msub> <mml:mo>,</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>,</mml:mo> <mml:msub><mml:mi>z</mml:mi> <mml:mi>n</mml:mi></mml:msub> <mml:mo>}</mml:mo></mml:mrow> </mml:math> 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 <i>H</i> of the KZ connection associated with such a path satisfies a generalization of Drinfeld's pentagon equation. In this equation, we encounter <i>H</i>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>Φ</mml:mi> <mml:mtext>KZ</mml:mtext></mml:msub> </mml:math> , and new factors associated with self-intersections, tangential base points, and the rotation number of the path.