On the Riemann-Hilbert problem for a $q$-difference Painlev\'e equation

TL;DR

This paper establishes a solvable Riemann-Hilbert problem for the $q$-difference Painlevé IV equation, linking its solutions to $q$-monodromy data and explicitly constructing its moduli space.

Contribution

It introduces a solvable Riemann-Hilbert framework for $q$-P_IV and constructs its moduli space, advancing the understanding of $q$-difference Painlevé equations.

Findings

Established solvability of the Riemann-Hilbert problem for $q$-P_IV

Created a bijective correspondence between solutions and $q$-monodromy data

Explicitly constructed the moduli space of $q$-P_IV

Abstract

A Riemann-Hilbert problem for a $q$-difference Painlev\'e equation, known as $q\textrm{P}_{\textrm{IV}}$, is shown to be solvable. This yields a bijective correspondence between the transcendental solutions of $q\textrm{P}_{\textrm{IV}}$ and corresponding data on an associated $q$-monodromy surface. We also construct the moduli space of $q\textrm{P}_{\textrm{IV}}$ explicitly.