Cubic Pencils and Painlevé Hamiltonians

Kajiwara, Kenji; Masuda, Tetsu; Noumi, Masatoshi; Ohta, Yuichi; Yamada, Yasuhiko

doi:10.1619/fesi.48.147

Cited by 34 publications

(56 citation statements)

References 10 publications

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“…Such a conformal field theory with irregular singularities [40] are recently studied from the point of view of quantum Painlevé equations. From this point of view, the AGT relation may be considered as a quantization of the correspondence between the Seiberg-Witten theory and the Hitchin/Painlevé equations (see [41] for example). ), Λ .…”

Section: Summary and Discussionmentioning

confidence: 99%

Five-dimensional AGT conjecture and the deformed Virasoro algebra

Awata

Yamada

2010

J. High Energ. Phys.

199

208

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Section: Summary and Discussionmentioning

confidence: 99%

Five-dimensional AGT conjecture and the deformed Virasoro algebra

Awata

Yamada

2010

J. High Energ. Phys.

199

208

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“…Remark 8.4. Here we remark on the relation between infinitely near points and the multiple point [52]. For example, consider the degeneration of P a : 0, aε and P b : bε, 0 by the limit ε → 0, as in the degeneration of P 6 and P 7 .…”

Section: Degeneration Of Point Configurationsmentioning

confidence: 99%

Geometric aspects of Painlevé equations

Kajiwara¹,

Noumi²,

Yamada³

2017

J. Phys. A: Math. Theor.

124

250

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In this paper a comprehensive review is given on the current status of achievements in the geometric aspects of the Painlevé equations, with a particular emphasis on the discrete Painlevé equations. The theory is controlled by the geometry of certain rational surfaces called the spaces of initial values, which are characterized by eight point configuration on P 1 × P 1 and classified according to the degeneration of points. We give a systematic description of the equations and their various properties, such as affine Weyl group symmetries, hypergeometric solutions and Lax pairs under this framework, by using the language of Picard lattice and root systems. We also provide with a collection of basic data; equations, point configurations/root data, Weyl group representations, Lax pairs, and hypergeometric solutions of all possible cases.

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“…The fact that the Painlevé equations can be obtained by deautonomization and their relation to integrable systems is not new, see [38] and other references below, but the cluster nature of the integrable systems corresponding to the q-deformed situation has not yet been discussed. In the cluster language the deautonomization arises naturally and unambiguously, and this is one of the advantages of the cluster interpretation.…”

Section: Jhep02(2018)077mentioning

confidence: 99%

Cluster integrable systems, q-Painlevé equations and their quantization

Bershtein

Gavrylenko

Marshakov

2018

J. High Energ. Phys.

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We discuss the relation between the cluster integrable systems and q-difference Painlevé equations. The Newton polygons corresponding to these integrable systems are all 16 convex polygons with a single interior point. The Painlevé dynamics is interpreted as deautonomization of the discrete flows, generated by a sequence of the cluster quiver mutations, supplemented by permutations of quiver vertices.We also define quantum q-Painlevé systems by quantization of the corresponding cluster variety. We present formal solution of these equations for the case of pure gauge theory using q-deformed conformal blocks or 5-dimensional Nekrasov functions. We propose, that quantum cluster structure of the Painlevé system provides generalization of the isomonodromy/CFT correspondence for arbitrary central charge.

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Cubic Pencils and Painlevé Hamiltonians

Abstract: Abstract. We present a simple heuristic method to derive the Painlevé di¤erential equations from the corresponding geometry of rational surfaces.

Cited by 34 publications

References 10 publications

Five-dimensional AGT conjecture and the deformed Virasoro algebra

Five-dimensional AGT conjecture and the deformed Virasoro algebra

Geometric aspects of Painlevé equations

Cluster integrable systems, q-Painlevé equations and their quantization

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