Singular Instantons and Painlevé VI

Abstract

We consider a two parameter family of instantons, which is studied in [Sadun L., Comm. Math. Phys. 163 (1994), 257-291], invariant under the irreducible action of SU₂ on S⁴, but which are not globally defined. We will see that these instantons produce solutions to a one parameter family of Painlevé VI equations (PVI) and we will give an explicit expression of the map between instantons and solutions to PVI. The solutions are algebraic only for that values of the parameters which correspond to the instantons that can be extended to all of S⁴. This work is a generalization of [Muñiz Manasliski R., Contemp. Math., Vol. 434, Amer. Math. Soc., Providence, RI, 2007, 215-222] and [Muñiz Manasliski R., J. Geom. Phys. 59 (2009), 1036-1047, arXiv:1602.07221], where instantons without singularities are studied.