Q-deformed Painleve tau function and q-deformed conformal blocks

TL;DR

This paper introduces a $q$-deformed formula for Painlevé tau functions, connecting $q$-difference Painlevé equations with $q$-Virasoro conformal blocks and Nekrasov partition functions, expanding the mathematical understanding of these integrable systems.

Contribution

It proposes a novel $q$-deformation of the Gamayun-Iorgov-Lisovyy formula for Painlevé tau functions, linking them to $q$-Virasoro conformal blocks and Nekrasov functions.

Findings

Derived a $q$-deformed tau function formula for $q$-difference Painlevé equations.

Established the connection between $q$-Painlevé tau functions and $q$-Virasoro conformal blocks.

Linked the tau function series to Nekrasov partition functions for 5d $SU(2)$ theory.

Abstract

We propose $q$-deformation of the Gamayun-Iorgov-Lisovyy formula for Painlev\'e $\tau$ function. Namely we propose formula for $\tau$ function for $q$-difference Painlev\'e equation corresponding to $A_7^{(1)}{}'$ surface (and $A_1^{(1)}$ symmetry) in Sakai's classification. In this formula $\tau$ function equals the series of $q$-Virasoro Whittaker conformal blocks (equivalently Nekrasov partition functions for pure $SU(2)$ 5d theory).