Isomonodromic deformations and SU2-invariant instantons on S4

TL;DR

This paper explores the relationship between SU2-invariant instantons on S4 and isomonodromic deformations governed by the sixth Painlevé equation, providing explicit parameter relations and a proof of a previously announced result.

Contribution

It explicitly connects SU2-invariant instantons on S4 with isomonodromic deformations and derives parameter relations in terms of instanton number.

Findings

Expressed Painlevé parameters in terms of instanton number

Derived explicit relations for SU2-invariant instantons on S4

Proved a previously announced theoretical result

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 C P 1 having four simple singularities. As is well known this kind of deformations is governed by the sixth Painlev\'e equation P vi ({\alpha}, \b{eta}, {\gamma}, {\delta}) . 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 pa- rameters ({\alpha}, \b{eta}, {\gamma}, {\delta}) in terms of the instanton number. The present paper contains the proof of the result anounced in [16].