Moduli Spaces for the Fifth Painlevé Equation

Abstract

Isomonodromy for the fifth Painlevé equation P₅ 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 P₅, 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 P₅, equivalent to the one of Okamoto.