Isomonodromic deformations and SU 2-invariant instantons on [formula omitted

Abstract

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 C P 1 having four simple singularities. As is well known, such deformations are governed by the sixth Painlevé equation P vi ( α , β , γ , δ ) . We work out the particular case of the SU 2 -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 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].