Non-Stationary Difference Equation and Affine Laumon Space: Quantization of Discrete Painlevé Equation

We show the relation of the non-stationary difference equation proposed by one of the authors and the quantized discrete Painlevé VI equation. The five-dimensional Seiberg-Witten curve associated with the difference equation has a consistent four-dimensional limit. We also show that the original equation can be factorized as a coupled system for a pair of functions $({\mathcal{F}}^{(1)},{\mathcal{F}}^{(2)})$, which is a consequence of the identification of the Hamiltonian as a translation element in the extended affine Weyl group. We conjecture that the instanton partition function coming from the affine Laumon space provides a solution to the coupled system.