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How instanton combinatorics solves Painlevé VI, V and III's

O. Gamayun, N. Iorgov, O. Lisovyy

TL;DR

The paper proposes and tests a c=1 conformal-block framework for Painlevé transcendents, expressing the VI, V, and III tau functions as sums over Virasoro blocks and irregular blocks that are connected to Nekrasov instanton partition functions via the AGT correspondence. By exploiting coalescence limits and conformal perturbation theory, it provides combinatorial series representations that enable efficient finite-argument computations and establish links to Fredholm determinants of integrable kernels and to random-matrix theory, sine-Gordon correlators, and 2D polymers. The work offers explicit tau-function expansions around critical points, demonstrates consistency with Jimbo asymptotics, and connects to classical determinant formulas (Gessel) and Toeplitz determinants, while outlining several open problems such as connection coefficients and extensions to other Painlevé equations. Overall, the approach unifies isomonodromic painlevé dynamics with $c=1$ CFT, providing new computational tools and deep structural insights with broad applications across integrable systems and quantum field theory.

Abstract

We elaborate on a recently conjectured relation of Painlevé transcendents and 2D CFT. General solutions of Painlevé VI, V and III are expressed in terms of $c=1$ conformal blocks and their irregular limits, AGT-related to instanton partition functions in $\mathcal{N}=2$ supersymmetric gauge theories with $N_f=0,1,2,3,4$. Resulting combinatorial series representations of Painlevé functions provide an efficient tool for their numerical computation at finite values of the argument. The series involve sums over bipartitions which in the simplest cases coincide with Gessel expansions of certain Toeplitz determinants. Considered applications include Fredholm determinants of classical integrable kernels, scaled gap probability in the bulk of the GUE, and all-order conformal perturbation theory expansions of correlation functions in the sine-Gordon field theory at the free-fermion point.

How instanton combinatorics solves Painlevé VI, V and III's

TL;DR

The paper proposes and tests a c=1 conformal-block framework for Painlevé transcendents, expressing the VI, V, and III tau functions as sums over Virasoro blocks and irregular blocks that are connected to Nekrasov instanton partition functions via the AGT correspondence. By exploiting coalescence limits and conformal perturbation theory, it provides combinatorial series representations that enable efficient finite-argument computations and establish links to Fredholm determinants of integrable kernels and to random-matrix theory, sine-Gordon correlators, and 2D polymers. The work offers explicit tau-function expansions around critical points, demonstrates consistency with Jimbo asymptotics, and connects to classical determinant formulas (Gessel) and Toeplitz determinants, while outlining several open problems such as connection coefficients and extensions to other Painlevé equations. Overall, the approach unifies isomonodromic painlevé dynamics with CFT, providing new computational tools and deep structural insights with broad applications across integrable systems and quantum field theory.

Abstract

We elaborate on a recently conjectured relation of Painlevé transcendents and 2D CFT. General solutions of Painlevé VI, V and III are expressed in terms of conformal blocks and their irregular limits, AGT-related to instanton partition functions in supersymmetric gauge theories with . Resulting combinatorial series representations of Painlevé functions provide an efficient tool for their numerical computation at finite values of the argument. The series involve sums over bipartitions which in the simplest cases coincide with Gessel expansions of certain Toeplitz determinants. Considered applications include Fredholm determinants of classical integrable kernels, scaled gap probability in the bulk of the GUE, and all-order conformal perturbation theory expansions of correlation functions in the sine-Gordon field theory at the free-fermion point.

Paper Structure

This paper contains 28 sections, 145 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Painlevé equations
  3. Conventional form
  4. Parameterization
  5. Hamiltonian form
  6. Sigma form and tau functions
  7. Coalescence
  8. Analytic properties
  9. Conformal blocks and instanton partition functions
  10. Conformal blocks
  11. $\mathcal{N}=2$ SUSY theories
  12. Solutions
  13. Painlevé VI
  14. Painlevé V and III's
  15. Classical solutions: AGT vs Gessel's theorem
  16. ...and 13 more sections

Figures (3)

  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (7)

  • Conjecture 1
  • Remark 2
  • Conjecture 3
  • Conjecture 4
  • Conjecture 5
  • Conjecture 6
  • Remark 7