Generalization of Okamoto's equation to arbitrary $2\times 2$ Schlesinger systems

TL;DR

This paper generalizes Okamoto's equation to arbitrary 2x2 Schlesinger systems with multiple poles, providing a symmetric formulation and new scalar equations involving Virasoro algebra generators.

Contribution

It introduces a symmetric uniform formulation of Schlesinger systems using Virasoro generators and derives analogues of Okamoto's equation for systems with any number of poles.

Findings

Derived scalar equations for tau-function derivatives.

Expressed Virasoro generators in terms of singularity derivatives.

Unified symmetric formulation of Schlesinger systems.

Abstract

The $2\times 2$ Schlesinger system for the case of four regular singularities is equivalent to the Painlev\'e VI equation. The Painlev\'e VI equation can in turn be rewritten in the symmetric form of Okamoto's equation; the dependent variable in Okamoto's form of the PVI equation is the (slightly transformed) logarithmic derivative of the Jimbo-Miwa tau-function of the Schlesinger system. The goal of this note is twofold. First, we find a symmetric uniform formulation of an arbitrary Schlesinger system with regular singularities in terms of appropriately defined Virasoro generators. Second, we find analogues of Okamoto's equation for the case of the $2\times2 $ Schlesinger system with an arbitrary number of poles. A new set of scalar equations for the logarithmic derivatives of the Jimbo-Miwa tau-function is derived in terms of generators of the Virasoro algebra; these generators are expressed in terms of derivatives with respect to singularities of the Schlesinger system.