Flat Structure on the Space of Isomonodromic Deformations

Abstract

Flat structure was introduced by K. Saito and his collaborators at the end of 1970's. Independently, the WDVV equation arose from the 2D topological field theory. B. Dubrovin unified these two notions as a Frobenius manifold structure. In this paper, we study isomonodromic deformations of an Okubo system, which is a special kind of system of linear differential equations. We show that the space of independent variables of such isomonodromic deformations can be equipped with a Saito structure (without a metric), which was introduced by C. Sabbah as a generalization of a Frobenius manifold. As a consequence, we introduce flat basic invariants of well-generated finite complex reflection groups and give explicit descriptions of Saito structures (without metrics) obtained from algebraic solutions to the sixth Painlevé equation.