Isomonodromic deformations and SU2-invariant instantons on S4S4

Anti-self-dual (ASD) solutions to the Yang–Mills equation (or instantons) over an anti-self-dual 4-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 CP1CP1 having four simple singularities. As is well known, such deformations are governed by the sixth Painlevé equation Pvi(α,β,γ,δ)(α,β,γ,δ). We work out the particular case of the SU2-action on S4S4, obtained from the irreducible representation on R5R5. In particular, we express the parameters (α,β,γ,δ)(α,β,γ,δ) in terms of the instanton number. The present paper contains the proof of the result announced in [Richard Muñiz Manasliski, Painlevé VI equation from invariant instantons, in: Geometric and Topological Methods for Quantum field theory, Contemp. Math., vol. 434, Amer. Math. Soc., Providence, RI, 2007, pp. 215–222].