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

TL;DR

This paper analyzes the monodromy manifold of the $q$-Painlevé VI equation via its Riemann-Hilbert problem, establishing solvability conditions, explicit manifold description, and connections to special functions.

Contribution

It explicitly characterizes the monodromy manifold for $q$-PVI, proves RHP solvability for irreducible data, and links reducible cases to orthogonal polynomial solutions.

Findings

RHP is always solvable for irreducible monodromy data.

The monodromy manifold is a smooth affine algebraic surface under certain conditions.

Explicit solutions for reducible monodromy data involve orthogonal polynomials.

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 $q{\textrm{P}_{\textrm{VI}}}$.