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

TL;DR

This work constructs a $q$-deformation of the Painlevé $\tau$-function framework by associating the $q$-difference Painlevé equation of surface type $A_7^{(1)\prime}/A_1^{(1)}$ with a quartet of $\tau$-functions that transform under the Weyl group $W$. It proposes a concrete $q$-tau function as a series over $q$-Virasoro Whittaker blocks, with a $q$-dependent prefactor determined by second-difference relations, and demonstrates that the $q\to1$ limit recovers the Gamayun–Iorgov–Lisovyy tau function for Painlevé III($D_8$). The paper also establishes bilinear relations at the level of $q$-blocks and explores algebraic $q$-solutions, linking the construction to Nekrasov partition functions for 5d $SU(2)$ gauge theory and suggesting broad generalizations to other Sakai surfaces. Overall, it provides a bridge between discrete Painlevé dynamics, $q$-deformed conformal blocks, and gauge-theoretic partition functions, with explicit proposals and conjectures that pave the way for further proof and extension.

Abstract

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

Figures (4)

• Figure 1: Sakai classification, surface type
• Figure 2: Sakai classification, symmetry type
• Figure 3: Blow-up scheme for $X$
• Figure 4: Action of $Dih_4$ generators

Theorems & Definitions (28)

• Theorem 2.1
• Remark 2.1
• Remark 2.2
• Proposition 2.1
• Proposition 2.2
• Proposition 3.1
• proof
• Conjecture 3.1
• Remark 3.1
• Definition 3.1