Painlevé VI equation from invariant instantons

Manasliski, Richard Muñiz

doi:10.1090/conm/434/08350

Cited by 2 publications

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“…The exceptional orbits are diffeomorphic to RP 2 and we denote them by RP + (t = 0) and RP − (t = 1). For more details and other descriptions of this action we refer to [3,19,20,22]. Any invariant object on S 4 is determined by its restriction to the curve c(t).…”

Section: The Action and Some Notationsmentioning

confidence: 99%

Singular Instantons and Painlevé VI

Manasliski¹

2016

SIGMA

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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 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 (P VI ) and we will give an explicit expression of the map between instantons and solutions to P 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

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Section: The Action and Some Notationsmentioning

confidence: 99%

Singular Instantons and Painlevé VI

Manasliski¹

2016

SIGMA

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Isomonodromic deformations and SU2-invariant instantons on S4

Manasliski

2009

Journal of Geometry and Physics

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a b s t r a c tAnti-self-dual (ASD) solutions to the Yang-Mills equation (or instantons) over an antiself-dual 4-manifold, which are invariant under an appropriate action of a threedimensional Lie group, give rise, via twistor construction, to isomonodromic deformations of connections on CP 1 having four simple singularities. As is well known, such deformations are governed by the sixth Painlevé equation Pvi (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

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Painlevé VI equation from invariant instantons

Cited by 2 publications

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Singular Instantons and Painlevé VI

Singular Instantons and Painlevé VI

Isomonodromic deformations and SU2-invariant instantons on S4

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