Regular flat structure and generalized Okubo system

TL;DR

This paper explores the connection between regular flat structures and generalized Okubo systems, revealing how isomonodromic deformations induce flat structures and unifying Painlevé equations under a single framework.

Contribution

It introduces a novel geometric framework linking flat structures with generalized Okubo systems and unifies Painlevé equations PVI-PII as a single extended WDVV system.

Findings

Flat structures can be constructed on deformation spaces of generalized Okubo systems.

Painlevé equations PVI-PII are unified as a four-dimensional extended WDVV equation.

Degeneration of Jordan normal forms corresponds to Painlevé coalescence cascade.

Abstract

We study a relationship between regular flat structures and generalized Okubo systems. We show that the space of variables of isomonodromic deformations of a regular generalized Okubo system can be equipped with a flat structure. As its consequence, we introduce flat structures on the spaces of independent variables of generic solutions to (classical) Painlev\'e equations (except for PI). In our framework, the Painlev\'e equations PVI-PII can be treated uniformly as just one system of differential equations called the four-dimensional extended WDVV equation. Then the well-known coalescence cascade of the Painlev\'e equations corresponds to the degeneration scheme of the Jordan normal forms of a square matrix of rank four.