Okamoto Transformations relating Equivariant Instanton Bundles via Painleve VI

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" $E_0$.