### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 057, 11 pages      arXiv:1602.07212      https://doi.org/10.3842/SIGMA.2016.057

### Singular Instantons and Painlevé VI

Richard Muñiz Manasliski
Centro de Matemática, Facultad de Ciencias, Iguá 4225 esq. Mataojo C.P. 11400, Montevideo, Uruguay

Received February 26, 2016, in final form June 09, 2016; Published online June 15, 2016

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 Painlevé 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 [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.

Key words: twistor theory; Yang-Mills instantons; isomonodromic deformations.

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