On the Riemann-Hilbert problem for a $q$-difference Painlevé equation

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.