Singular Instantons and Painlev\'e VI

TL;DR

This paper explores a family of instantons on $S^4$ invariant under SU(2), linking them explicitly to solutions of Painlevé VI equations, and identifies conditions under which these solutions are algebraic.

Contribution

It provides an explicit map between a family of singular instantons and Painlevé VI solutions, extending previous work on non-singular instantons.

Findings

Instantons produce solutions to Painlevé VI equations.

Algebraic solutions correspond to extendable instantons.

Explicit expression of the instanton-Painlevé VI map.

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 ${\rm SU}_2$ on $S^4$, but which are not globally defined. We will see that these instantons produce solutions to a one parameter family of Painlev\'e VI equations ($\text{P}_{\text{VI}}$) and we will give an explicit expression of the map between instantons and solutions to $\text{P}_{\text{VI}}$. The solutions are algebraic only for that values of the parameters which correspond to the instantons that can be extended to all of $S^4$. This work is a generalization of [Mu\~niz Manasliski R., Contemp. Math., Vol. 434, Amer. Math. Soc., Providence, RI, 2007, 215-222] and [Mu\~niz Manasliski R., J. Geom. Phys. 59 (2009), 1036-1047, arXiv:1602.07221], where instantons without singularities are studied.