Connection Problem of Knizhnik-Zamolodchikov Equation on Moduli Space ${\mathcal M}_{0,5}$

TL;DR

This paper investigates the connection problem of the KZ equation on the moduli space _{0,5}, showing that connection matrices relate to the Drinfel'd associator and deriving the five-term dilogarithm relation.

Contribution

It demonstrates that the connection matrices of the KZ equation on _{0,5} are expressed via the Drinfel'd associator and derives the five-term relation for dilogarithms.

Findings

Connection matrices are expressed in terms of the Drinfel'd associator.

The pentagon relation for associators is shown as a compatibility condition.

Derivation of the five-term relation for dilogarithms.

Abstract

In this article, we consider the connection problem of the KZ (Knizhnik-Zamolodchikov) equation on the moduli space $\cM_{0,5}$, and show that the connection matrices are expressed in terms of the Drinfel'd associator. As the compatibility condition on the connection problem, we have the pentagon relation for the Drinfeld associators. As an application of the connection problem, we derive the five term relation for dilogarithms.