Flat Structure on the Space of Isomonodromic Deformations

TL;DR

This paper explores the geometric structure of isomonodromic deformations of Okubo systems, establishing a flat Saito structure on their deformation space and applying it to complex reflection groups and Painlevé equations.

Contribution

It demonstrates that the space of isomonodromic deformations of Okubo systems admits a flat Saito structure, extending the concept beyond Frobenius manifolds.

Findings

Established a Saito structure on the deformation space

Introduced flat basic invariants for complex reflection groups

Explicitly described Saito structures from Painlevé VI solutions

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 Frobenius manifold structure. In this paper, we study isomonodromic deformations of an Okubo system, which is a special kind of systems 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 Frobenius manifold. As its 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 Painlev\'e equation.