Extension of Riemann–Hilbert maps over general fields

Determine how the Riemann–Hilbert maps for Painlevé initial value spaces and monodromy surfaces extend when the constructions are defined over fields more general than the complex numbers.

Background

The authors observe that the initial value spaces, monodromy surfaces, and their monodromy groups can be defined and studied over more general fields. However, the Riemann–Hilbert correspondence is an additional analytic construction, and the paper does not determine how to formulate or extend the corresponding maps in those settings.