Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 089, 47 pages      arXiv:2211.16772      https://doi.org/10.3842/SIGMA.2023.089

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

Hidetoshi Awata a, Koji Hasegawa b, Hiroaki Kanno ac, Ryo Ohkawa de, Shamil Shakirov fg, Jun'ichi Shiraishi h and Yasuhiko Yamada i
a) Graduate School of Mathematics, Nagoya University, Nagoya 464-8602, Japan
b) Mathematical Institute, Tohoku University, Sendai 980-8578, Japan
c) Kobayashi-Maskawa Institute, Nagoya University, Nagoya 464-8602, Japan
d) Osaka Central Advanced Mathematical Institute, Osaka Metropolitan University, Osaka 558-8585, Japan
e) Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
f) University of Geneva, Switzerland
g) Institute for Information Transmission Problems, Moscow, Russia
h) Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Tokyo 153-8914, Japan
i) Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan

Received December 06, 2022, in final form October 22, 2023; Published online November 09, 2023

Abstract
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 $\bigl(\mathcal{F}^{(1)},\mathcal{F}^{(2)}\bigr)$, 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.

Key words: affine Laumon space; affine Weyl group; deformed Virasoro algebra; non-stationary difference equation; quantum Painlevé equation.

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