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

SIGMA 17 (2021), 050, 21 pages      arXiv:2004.13916      https://doi.org/10.3842/SIGMA.2021.050

On $q$-Isomonodromic Deformations and $q$-Nekrasov Functions

Hajime Nagoya
School of Mathematics and Physics, Kanazawa University, Kanazawa, Ishikawa 920-1192, Japan

Received June 02, 2020, in final form May 04, 2021; Published online May 13, 2021

Abstract
We construct a fundamental system of a $q$-difference Lax pair of rank $N$ in terms of 5d Nekrasov functions with $q=t$. Our fundamental system degenerates by the limit $q\to 1$ to a fundamental system of a differential Lax pair, which yields the Fuji-Suzuki-Tsuda system. We introduce tau functions of our system as Fourier transforms of 5d Nekrasov functions. Using asymptotic expansions of the fundamental system at $0$ and $\infty$, we obtain several determinantal identities of the tau functions.

Key words: isomonodromic deformations; Nekrasov functions; Painlevé equations; determinantal identities.

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