Fundamental Solutions of the Knizhnik-Zamolodchikov Equation of One Variable and the Riemann-Hilbert Problem

Abstract

In this article, we show that the generalized inversion formulas of the multiple polylogarithms of one variable, which are generalizations of the inversion formula of the dilogarithm, characterize uniquely the multiple polylogarithms under certain conditions. This means that the multiple polylogarithms are constructed from the multiple zeta values. We call such a problem of determining certain functions a recursive Riemann-Hilbert problem of additive type. Furthermore we show that the fundamental solutions of the KZ equation of one variable are uniquely characterized by the connection relation between the fundamental solutions of the KZ equation normalized at $z=0$ and $z=1$ under some assumptions. Namely the fundamental solutions of the KZ equation are constructed from the Drinfel'd associator. We call this problem a Riemann-Hilbert problem of multiplicative type.