Isomonodromic deformations and SU2-invariant instantons on S4

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 CP1 having four simple singularities. As is well known, such deformations are governed by the sixth Painleve equation PVI (alpha, beta, gamma, delta). 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 (alpha, beta, gamma, delta) in terms of the instanton number. The present paper contains the proof of the result announced in [Richard Muniz Manasliski, Painleve 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]. (C) 2009 Elsevier B.V. All rights reserved.