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

SIGMA 19 (2023), 068, 26 pages      arXiv:2107.07204      https://doi.org/10.3842/SIGMA.2023.068

Moduli Spaces for the Fifth Painlevé Equation

Marius van der Put and Jaap Top
Bernoulli Institute, Nijenborgh 9, 9747 AG Groningen, The Netherlands

Received July 15, 2021, in final form September 07, 2023; Published online September 26, 2023

Abstract
Isomonodromy for the fifth Painlevé equation ${\rm P}_5$ is studied in detail in the context of certain moduli spaces for connections, monodromy, the Riemann-Hilbert morphism, and Okamoto-Painlevé spaces. This involves explicit formulas for Stokes matrices and parabolic structures. The rank 4 Lax pair for ${\rm P}_5$, introduced by Noumi-Yamada et al., is shown to be induced by a natural fine moduli space of connections of rank 4. As a by-product one obtains a polynomial Hamiltonian for ${\rm P}_5$, equivalent to the one of Okamoto.

Key words: moduli space for linear connections; irregular singularities; Stokes matrices; monodromy spaces; isomonodromic deformations; Painlevé equations.

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