Moduli Spaces for the Fifth Painlevé Equation

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.