Bilinear tau forms of quantum Painlevé equations and $\mathbb{C}^2/\mathbb{Z}_2$ blowup relations in SUSY gauge theories
Giulio Bonelli, Anton Shchechkin, Alessandro Tanzini
TL;DR
This work provides a comprehensive quantum extension of Painlevé dynamics by deriving bilinear tau forms for canonically quantized Painlevé equations and linking them to the C^2/\mathbb{Z}_2 blowup relations that arise in SUSY gauge theories on general Omega backgrounds. It fixes the refined Painlevé/gauge dictionary through quantum Hamiltonians, introduces left/right tau functions, and establishes the equivalence of tau forms with Heisenberg dynamics across Painlevé VI, V, IV, III, II, I via a structured coalescence scheme. A central achievement is the elucidation of symmetry structures (extended affine Weyl groups) and the emergence of nontrivial blowup relations in nontrivial holonomy sectors, including Okamoto-like bilinear relations tied to Virasoro/AGT frameworks. The results deepen the connection between quantum isomonodromic deformations, gauge theory partition functions, and noncommutative Zak transforms, offering new tools for exploring quantum Painlevé hierarchies and their gauge-theory realizations with potential extensions to q-difference and cluster-integrable systems.
Abstract
We derive bilinear tau forms of the canonically quantized Painlevé 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 $Ω$-background. We fully fix the refined Painlevé/gauge theory dictionary by formulating the proper equations for the quantum nonautonomous Painlevé Hamiltonians. We also describe the symmetry structure of the quantum Painlevé tau functions and, as a byproduct of this analysis, obtain the $\mathbb{C}^2/\mathbb{Z}_2$ blowup relations in the nontrivial holonomy sector of the gauge theory.