Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Riemann-Hilbert problems, Fredholm determinants, explicit combinatorial expansions, and connection formulas for the general $q$-Painlevé III$_3$ tau functions

Pavlo Gavrylenko

TL;DR

The paper develops a circle-based Riemann-Hilbert formulation of the $q$-Painlevé III$_3$ (type $A_7^{(1)'}$) isomonodromic problem and identifies its tau function as a Widom determinant with a rich Fredholm-determinant structure. It derives a complete bilinear (2-variable) framework for the tau function, constructs explicit minor expansions labeled by Maya diagrams, and rewrites the expansion in terms of Nekrasov functions for $q$-deformed conformal blocks and 5d $ ext{SU}(2)$ gauge theories. The work also connects the tau function to algebraic solutions, analyzes the connection problem with explicit elliptic gamma-based constants, and derives fusion kernels, suggesting deep links to elliptic cluster algebras and Arinkin–Borodin-type tau functions. Overall, it provides a self-contained, purely analytic Fredholm-determinant route to Kyiv-type formulas in the $q$-difference setting and broadens the bridge between isomonodromy, integrable systems, and gauge theory partition functions.

Abstract

We reformulate the $q$-difference linear system corresponding to the $q$-Painlevé equation of type $A_7^{(1)'}$ as a Riemann-Hilbert problem on a circle. Then, we consider the Fredholm determinant built from the jump of this Riemann-Hilbert problem and prove that it satisfies bilinear relations equivalent to $P(A_7^{(1)'})$. We also find the minor expansion of this Fredholm determinant in explicit factorized form and prove that it coincides with the Fourier series in $q$-deformed conformal blocks, or partition functions of the pure $5d$ $\mathcal{N}=1$ $SU(2)$ gauge theory, including the cases with the Chern-Simons term. Finally, we solve the connection problem for these isomonodromic tau functions, finding in this way their global behavior.

Riemann-Hilbert problems, Fredholm determinants, explicit combinatorial expansions, and connection formulas for the general $q$-Painlevé III$_3$ tau functions

TL;DR

The paper develops a circle-based Riemann-Hilbert formulation of the -Painlevé III (type ) isomonodromic problem and identifies its tau function as a Widom determinant with a rich Fredholm-determinant structure. It derives a complete bilinear (2-variable) framework for the tau function, constructs explicit minor expansions labeled by Maya diagrams, and rewrites the expansion in terms of Nekrasov functions for -deformed conformal blocks and 5d gauge theories. The work also connects the tau function to algebraic solutions, analyzes the connection problem with explicit elliptic gamma-based constants, and derives fusion kernels, suggesting deep links to elliptic cluster algebras and Arinkin–Borodin-type tau functions. Overall, it provides a self-contained, purely analytic Fredholm-determinant route to Kyiv-type formulas in the -difference setting and broadens the bridge between isomonodromy, integrable systems, and gauge theory partition functions.

Abstract

We reformulate the -difference linear system corresponding to the -Painlevé equation of type as a Riemann-Hilbert problem on a circle. Then, we consider the Fredholm determinant built from the jump of this Riemann-Hilbert problem and prove that it satisfies bilinear relations equivalent to . We also find the minor expansion of this Fredholm determinant in explicit factorized form and prove that it coincides with the Fourier series in -deformed conformal blocks, or partition functions of the pure gauge theory, including the cases with the Chern-Simons term. Finally, we solve the connection problem for these isomonodromic tau functions, finding in this way their global behavior.
Paper Structure (38 sections, 11 theorems, 312 equations, 4 figures)

This paper contains 38 sections, 11 theorems, 312 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Isomonodromic deformations and monodromy
  3. Linear system and isomonodromic deformations
  4. Bäcklund transformation
  5. Monodromy
  6. Parametrices
  7. Auxiliary systems
  8. Regularity conditions for $\Phi_{\pm}$
  9. Behavior of $\Phi_{\pm}$ around zero and infinity
  10. Isomonodromic evolution and Bäcklund transformations of $\Phi_{\pm}$
  11. Algebraic solution
  12. Determinants and their transformations
  13. Inversion formulas
  14. Bäcklund transformation of the determinant
  15. Isomonodromic evolution of the determinant
  16. ...and 23 more sections

Key Result

Theorem 3.1

For generic $q$-monodromy data, the following holds:

Figures (4)

  • Figure 1: Different domains of $\mathbb{C}\mathrm{P}^1$.
  • Figure 2: Example of the Young diagram. $x(s)=1$, $y(s)=2$, $a_Y(s)=2$, $l_Y(s)=2$.
  • Figure 3: Charged Frobenius coordinates.
  • Figure 4: Zeros (, ) and poles (, ) of different solutions.

Theorems & Definitions (18)

  • Theorem 3.1
  • Lemma 3.2
  • Definition 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Lemma 5.1
  • Theorem 5.2: Nakajima-Yoshioka
  • Remark 5.3
  • Theorem 5.4
  • Theorem 6.1
  • ...and 8 more