The space of monodromy data for the Jimbo-Sakai family of $q$-difference equations

TL;DR

This paper develops a geometric Riemann-Hilbert correspondence for $q$-difference equations, specifically relating to Jimbo and Sakai's derivation of $q$-PVI, advancing the understanding of $q$-isomonodromy and $q$-Stokes phenomena.

Contribution

It introduces a geometric Riemann-Hilbert framework for $q$-difference equations, connecting isomonodromy conditions to $q$-Painlevé equations.

Findings

Formulation of a geometric Riemann-Hilbert correspondence for $q$-difference equations

Application to Jimbo and Sakai's derivation of $q$-PVI

Progress towards $q$-isomonodromy and $q$-Stokes analysis

Abstract

We formulate a geometric Riemann-Hilbert correspondence that applies to the derivation by Jimbo and Sakai of equation $q$-PVI from ``isomonodromy'' conditions. This is a step within work in progress towards the application of $q$-isomonodromy and $q$-isoStokes to $q$-Painlev\'e.