Fundamental solutions of the Knizhnik-Zamolodchikov equation of one variable and the Riemann-Hilbert problem

Abstract

In this article, we derive multiple polylogarithms from multiple zeta values by using a recursive Riemann-Hilbert problem of additive type. Furthermore we show that this Riemann-Hilbert problem is regarded as an inverse problem for the connection problem of the KZ equation of one variable, so that the fundamental solutions to the equation are derived from the Drinfel'd associator by using a Riemann-Hilbert problem of multiplicative type. These results say that the duality relation for the Drinfel'd associator can be interpreted as the solvability condition for this inverse problem.