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

Hidetoshi Awata   ; Koji Hasegawa   ; Hiroaki Kanno   ; Ryo Ohkawa   ; Shamil Shakirov   ; Jun'ichi Shiraishi   ; Yasuhiko Yamada

Symmetry, integrability and geometry: methods and applications, Tome 19 (2023)

Résumé

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.

Détail

Keywords: deformed Virasoro algebra, non-stationary difference equation
Mots-clés : affine Laumon space, affine Weyl group, quantum Painlevé equation.

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