On the monodromy manifold of $q$-Painlev\'e VI and its Riemann-Hilbert problem

Abstract

We study the sixth $q$-difference Painlev\'e equation ($q{\textrm{P}_{\textrm{VI}}}$) through its associated Riemann-Hilbert problem (RHP) and show that the RHP is always solvable for irreducible monodromy data. This enables us to identify the solution space of $q{\textrm{P}_{\textrm{VI}}}$ with a monodromy manifold for generic parameter values. We deduce this manifold explicitly and show it is a smooth and affine algebraic surface when it does not contain reducible monodromy. Furthermore, we describe the RHP for reducible monodromy data and show that, when solvable, its solution is given explicitly in terms of certain orthogonal polynomials yielding special function solutions of $q{\textrm{P}_{\textrm{VI}}}$.