Okamoto Transformations relating Equivariant Instanton Bundles via Painleve VI

TL;DR

This paper explicitly constructs solutions to Painleve VI from equivariant instanton bundles with quadrupole symmetry, revealing transformations that relate these solutions and bundles, akin to creation operators.

Contribution

It generalizes Hitchin's logarithmic connection to vector bundles with an SL_2(C) action and identifies explicit Okamoto transformations linking instanton bundles and Painleve VI solutions.

Findings

Explicit solutions mbda_m^\u00b1 from instanton bundles E_m.

Identification of Okamoto transformations as creation operators.

Potential relation between instanton bundles E_m and the ground state E_0.

Abstract

We compute explicit solutions $\Lambda^\pm_m$ of the Painleve VI (PVI) differential equation from equivariant instanton bundles $E_m$ corresponding to Yang-Mills instantons with "quadrupole symmetry." This is based on a generalization of Hitchin's logarithmic connection to vector bundles with an $SL_2({\mathbb C})$ action. We then identify explicit Okamoto transformation which play the role of "creation operators" for construction $\Lambda^\pm_m$ from the "ground state" $\Lambda^\pm_0$, suggesting that the equivariant instanton bundles $E_m$ might similarly be related to the trivial "ground state" $E_0$.