Cubic and Quartic Transformations of the Sixth Painlevé Equation in Terms of Riemann–Hilbert Correspondence

Mazzocco, Marta; Vidūnas, Raimundas

doi:10.1111/j.1467-9590.2012.00562.x

Cited by 8 publications

(21 citation statements)

References 28 publications

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“…The goal of the present paper was to provide the similar property for logarithmic connections, and therefore complete the whole picture for hyperelliptic curves. We note that similar constructions also hold within the class of connections on the 4-punctured sphere (see [20]).…”

Section: Related Workmentioning

confidence: 57%

A Map Between Moduli Spaces of Connections

Loray¹,

Ramírez²

2020

SIGMA

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We are interested in studying moduli spaces of rank 2 logarithmic connections on elliptic curves having two poles. To do so, we investigate certain logarithmic rank 2 connections defined on the Riemann sphere and a transformation rule to lift such connections to an elliptic curve. The transformation is as follows: given an elliptic curve C with elliptic quotient, and the logarithmic connection, we may pullback the connection to the elliptic curve to obtain a new connection on C. After suitable birational modifications we bring the connection to a particular normal form. The whole transformation is equivariant with respect to bundle automorphisms and therefore defines a map between the corresponding moduli spaces of connections. The aim of this paper is to describe the moduli spaces involved and compute explicit expressions for the above map in the case where the target space is the moduli space of rank 2 logarithmic connections on an elliptic curve C with two simple poles and trivial determinant.

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Section: Related Workmentioning

confidence: 57%

A Map Between Moduli Spaces of Connections

Loray¹,

Ramírez²

2020

SIGMA

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“…Finally, if we consider the Picard parameters θ 0 = θ 1 = θ t = θ ∞ = 1 2 for Painlevé VI equation, we can iterate arbitrary many times the quadratic tranformation. There is also a cubic transformation in this case (see [22]).…”

Section: Figure 1 Quadratic Transformation's Covermentioning

confidence: 90%

“…for Painlevé VI equation, we can iterate arbitrary many times the quadratic tranformation. There is also a cubic transformation in this case (see [22]).…”

Section: When Exponents Satisfy θmentioning

confidence: 90%

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Ramified covers and tame isomonodromic solutions on curves

Diarra

Loray²

2015

Trans. Moscow Math. Soc.

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“…the Painlevé sixth equation with parametersβ = γ = 0, δ = 1 2 and α = 2µ−appearing in the Frobenius manifold theory, via the change of coordinates G i,j = S 2 ij − 2. This change of coordinates actually corresponds to a quartic transformation on the sixth Painlevé equation[21]. Action of the braid group B 3 .…”

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confidence: 99%

Shear coordinate description of the quantized versal unfolding of aD₄singularity

Chekhov

Mazzocco

2010

J. Phys. A: Math. Theor.

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In this paper by using Teichmüller theory of a sphere with four holes/orbifold points, we obtain a system of flat coordinates on the general affine cubic surface having a D 4 singularity at the origin. We show that the Goldman bracket on the geodesic functions on the four-holed/orbifold sphere coincides with the Etingof-Ginzburg Poisson bracket on the affine D 4 cubic. We prove that this bracket is the image under the Riemann-Hilbert map of the Poisson Lie bracket on ⊕ 3 1 sl * (2, C). We realise the action of the mapping class group by the action of the braid group on the geodesic functions . This action coincides with the procedure of analytic continuation of solutions of the sixth Painlevé equation. Finally, we produce the explicit quantisation of the Goldman bracket on the geodesic functions on the four-holed/orbifold sphere and of the braid group action.

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Cubic and Quartic Transformations of the Sixth Painlevé Equation in Terms of Riemann–Hilbert Correspondence

Cited by 8 publications

References 28 publications

A Map Between Moduli Spaces of Connections

A Map Between Moduli Spaces of Connections

Ramified covers and tame isomonodromic solutions on curves

Shear coordinate description of the quantized versal unfolding of aD₄singularity

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Cubic and Quartic Transformations of the Sixth Painlevé Equation in Terms of Riemann–Hilbert Correspondence

Cited by 8 publications

References 28 publications

A Map Between Moduli Spaces of Connections

A Map Between Moduli Spaces of Connections

Ramified covers and tame isomonodromic solutions on curves

Shear coordinate description of the quantized versal unfolding of aD4singularity

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Product

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Shear coordinate description of the quantized versal unfolding of aD₄singularity