Isomonodromic deformations and SU2-invariant instantons on S4

Abstract

Anti-self-dual (ASD) solutions to the Yang-Mills equation (or instantons) over an anti-self-dual four manifold, which are invariant under an appropriate action of a three dimensional Lie group, give rise, via twistor construction, to isomonodromic deformations of connections on CP 1 having four simple singularities. As is well known this kind of deformations is governed by the sixth Painlevé equation Pvi(α, β, γ, δ). We work out the particular case of the SU2-action on S 4 , obtained from the irreducible representation on R 5 . In particular, we express the parameters (α, β, γ, δ) in terms of the instanton number. The present paper contains the proof of the result anounced in [16].