Solutions to the extended WDW equations and the Painlevé VI equation

摘要

The notion of the extended WDVV equations is a generalization of the WDVV equations. Their solutions are vector-valued functions. In the three-dimensional case, there is a correspondence between the extended WDVV equations and the family of the Painlevé VI equations. It is expected that potential vector fields corresponding to algebraic solutions to the Painlevé VI equation can be written by using algebraic functions explicitly. The purpose of this paper is to establish a method of constructing potential vector fields corresponding to algebraic solutions. The idea is based on the argument by Jimbo and Miwa inducing the Painlevé VI equation from a Pfaffian system and on middle convolution.