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arXiv · 2512.25051

Bilinear tau forms of quantum Painlev\'e equations and $\mathbb{C}^2/\mathbb{Z}_2$ blowup relations in SUSY gauge theories

Giulio Bonelli·Anton Shchechkin·Alessandro Tanzini
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Abstract

We derive bilinear tau forms of the canonically quantized Painlev\'e equations, thereby relating them to those previously obtained from the $\mathbb{C}^2/\mathbb{Z}_2$ blowup relations for the $\mathcal{N}=2$ supersymmetric gauge theory partition functions on a general $\Omega$-background. We fully fix the refined Painlev\'e/gauge theory dictionary by formulating the proper equations for the quantum nonautonomous Painlev\'e Hamiltonians. We also describe the quantum Painlev\'e symmetries at the level of tau functions and, as a byproduct of this analysis, obtain several $\mathbb{C}^2/\mathbb{Z}_2$ blowup relations including those in nontrivial holonomy sectors of the gauge theory.

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Giulio Bonelli, Anton Shchechkin, Alessandro Tanzini. 2025-12-31. Bilinear tau forms of quantum Painlev\'e equations and $\mathbb{C}^2/\mathbb{Z}_2$ blowup relations in SUSY gauge theories. https://arxiv.org/abs/2512.25051

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