Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Monodromy dependence and connection formulae for isomonodromic tau functions

A. Its, O. Lisovyy, A. Prokhorov

TL;DR

This work advances the theory of isomonodromic tau functions by extending the Jimbo-Miwa-Ueno differential to a closed 1-form on the full extended space of isomonodromic data, enabling explicit computation of connection constants from monodromy data. The authors develop a Malgrange-Bertola-inspired construction to obtain a closed form for the extended tau function and apply it to two representative Painlev\\e equations: the Fuchsian Painlev\\e VI and the irregular Painlev\\e II, using Riemann-Hilbert techniques to relate asymptotics to monodromy. For Painlev\\e VI, they prove the conjectured constant formula for the ratio of leading asymptotic constants by expressing it through Barnes $G$-functions and monodromy parameters, and also derive the crossing behavior as $t\to1$. For Painlev\\e II, they compute the connection constant explicitly and determine its numerical value, together with a quasi-periodic Fourier representation of the tau function and a relation to the extended action form. The results provide a robust, determinant-free method to obtain constant factors in isomonodromic tau function asymptotics, with implications for conformal blocks and the Hamiltonian structure of isomonodromic deformations. The work highlights deep links between monodromy data, Riemann-Hilbert problems, and the symplectic geometry underlying isomonodromic systems.

Abstract

We discuss an extension of the Jimbo-Miwa-Ueno differential 1-form to a form closed on the full space of extended monodromy data of systems of linear ordinary differential equations with rational coefficients. This extension is based on the results of M. Bertola generalizing a previous construction by B. Malgrange. We show how this 1-form can be used to solve a long-standing problem of evaluation of the connection formulae for the isomonodromic tau functions which would include an explicit computation of the relevant constant factors. We explain how this scheme works for Fuchsian systems and, in particular, calculate the connection constant for generic Painlevé VI tau function. The result proves the conjectural formula for this constant proposed in \cite{ILT13}. We also apply the method to non-Fuchsian systems and evaluate constant factors in the asymptotics of Painlevé II tau function.

Monodromy dependence and connection formulae for isomonodromic tau functions

TL;DR

This work advances the theory of isomonodromic tau functions by extending the Jimbo-Miwa-Ueno differential to a closed 1-form on the full extended space of isomonodromic data, enabling explicit computation of connection constants from monodromy data. The authors develop a Malgrange-Bertola-inspired construction to obtain a closed form for the extended tau function and apply it to two representative Painlev\\e equations: the Fuchsian Painlev\\e VI and the irregular Painlev\\e II, using Riemann-Hilbert techniques to relate asymptotics to monodromy. For Painlev\\e VI, they prove the conjectured constant formula for the ratio of leading asymptotic constants by expressing it through Barnes -functions and monodromy parameters, and also derive the crossing behavior as . For Painlev\\e II, they compute the connection constant explicitly and determine its numerical value, together with a quasi-periodic Fourier representation of the tau function and a relation to the extended action form. The results provide a robust, determinant-free method to obtain constant factors in isomonodromic tau function asymptotics, with implications for conformal blocks and the Hamiltonian structure of isomonodromic deformations. The work highlights deep links between monodromy data, Riemann-Hilbert problems, and the symplectic geometry underlying isomonodromic systems.

Abstract

We discuss an extension of the Jimbo-Miwa-Ueno differential 1-form to a form closed on the full space of extended monodromy data of systems of linear ordinary differential equations with rational coefficients. This extension is based on the results of M. Bertola generalizing a previous construction by B. Malgrange. We show how this 1-form can be used to solve a long-standing problem of evaluation of the connection formulae for the isomonodromic tau functions which would include an explicit computation of the relevant constant factors. We explain how this scheme works for Fuchsian systems and, in particular, calculate the connection constant for generic Painlevé VI tau function. The result proves the conjectural formula for this constant proposed in \cite{ILT13}. We also apply the method to non-Fuchsian systems and evaluate constant factors in the asymptotics of Painlevé II tau function.

Paper Structure

This paper contains 25 sections, 22 theorems, 363 equations, 6 figures.

Table of Contents

  1. Introduction and statement of results
  2. Linear systems
  3. Tau function
  4. Connection problem
  5. Summary of results
  6. Outline of the paper
  7. Remarks
  8. Fuchsian systems
  9. Four-point tau function
  10. Riemann-Hilbert problem
  11. Parametrices
  12. Global approximation
  13. Asymptotics of $\omega$ and $\Omega$
  14. Painlevé VI monodromy data
  15. Inverse monodromy problem in rank $2$
  16. ...and 10 more sections

Key Result

Proposition 2.3

Let $\omega$ be a (possibly meromorphic in $a_{\nu}$) 1-form on $\widetilde{\mathcal{T}}\times\mathcal{M}$ locally defined by where $d_{\mathcal{M}}$ denotes the differential with respect to monodromy data. Its exterior differential $\Omega:=d\omega$ is a closed 2-form on $\mathcal{M}$ independent of $a_1,\ldots,a_n$.

Figures (6)

  • Figure 1: Contour $\Gamma_{\Psi}$ of the Riemann-Hilbert problem for $\Psi\left(z\right)$
  • Figure 2: Contours a) $\Gamma_{\Psi}^e$ and b) $\Gamma_{\Psi}^i$ for exterior and interior parametrix
  • Figure 3: Pants decomposition of the $4$-punctured Riemann sphere
  • Figure 4: Contour $\Gamma_{\overline\Psi}$ of the Riemann-Hilbert problem for $\overline\Psi\left(z\right)$
  • Figure 5: Contour and jump matrices of the PII-RH problem
  • ...and 1 more figures

Theorems & Definitions (38)

  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 3.2
  • Lemma 3.3
  • Remark 3.4
  • Corollary 3.5
  • Lemma 3.6
  • ...and 28 more