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

TL;DR

This paper establishes bilinear tau forms for quantum Painlevé equations, connecting them to supersymmetric gauge theory blowup relations and clarifying their symmetry structures and Hamiltonian formulations.

Contribution

It introduces bilinear tau forms for quantum Painlevé equations and links them to $ abla$-blowup relations in SUSY gauge theories, enhancing the understanding of their symmetry and Hamiltonian structures.

Findings

Derived bilinear tau forms for quantum Painlevé equations.

Connected quantum Painlevé tau functions to $ abla$-blowup relations.

Clarified symmetry structures of quantum Painlevé tau functions.

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 symmetry structure of the quantum Painlev\'e 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.