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arXiv · 2211.16772

Non-Stationary Difference Equation and Affine Laumon Space: Quantization of Discrete Painlev\'e Equation

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

We show the relation of the non-stationary difference equation proposed by one of the authors and the quantized discrete Painlev\'e 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.

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Hidetoshi Awata, Koji Hasegawa, Hiroaki Kanno, Ryo Ohkawa, Shamil Shakirov, Jun'ichi Shiraishi, Yasuhiko Yamada. 2022-11-30. Non-Stationary Difference Equation and Affine Laumon Space: Quantization of Discrete Painlev\'e Equation. https://doi.org/10.3842/sigma.2023.089

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