Monodromy dependence and connection formulae for isomonodromic tau functions

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.

arXiv comment54 pages, 6 figures; v4: rewritten Introduction and Subsection 3.3, added few refs to match published article

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Cited by in corpus (27)

• On Painlevé/gauge theory correspondence
• Fredholm determinant and Nekrasov sum representations of isomonodromic tau functions
• Irregular conformal blocks and connection formulae for Painlevé V functions
• On the analysis of incomplete spectra in random matrix theory through an extension of the Jimbo-Miwa-Ueno differential
• Argyres-Douglas theories, Painlevé II and quantum mechanics
• On some Hamiltonian properties of the isomonodromic tau functions
• On the connection problem for Painlevé I
• Tracy-Widom distributions in critical unitary random matrix ensembles and the coupled Painlevé II system
• Gap probability of the circular unitary ensemble with a Fisher-Hartwig singularity and the coupled Painlevé V system
• CORRIGENDUM: The dependence on the monodromy data of the isomonodromic tau function
• Resurgence, Painleve Equations and Conformal Blocks
• Higher rank isomonodromic deformations and W-algebras
• Hamiltonian structure of rational isomonodromic deformation systems
• Expansions for semiclassical conformal blocks
• Monodromy dependence and symplectic geometry of isomonodromic tau functions on the torus
• The $\tau$-function of the Ablowitz-Segur family of solutions to Painlevé II as a Widom constant
• Fredholm determinant representation of the Painlevé II $\tau$-function
• Remarks on holographic models of the Kerr-AdS${}_5$ geometry
• Short distance asymptotics for a generalized two-point scaling function in the two-dimensional Ising model
• The Malgrange Form and Fredholm Determinants
• Crossing invariant correlation functions at $c=1$ from isomonodromic $\tau$ functions
• Circular quiver gauge theories, isomonodromic deformations and $W_N$ fermions on the torus
• On the tau function of the hypergeometric equation
• QNMs of scalar fields on small Reissner-Nordström-AdS${}_5$ black holes
• Connection formulas for the lambda generalized Ising correlation functions
• Riemann-Hilbert Characterisation of Rational Functions with a General Distribution of Poles on the Extended Real Line Orthogonal with Respect to Varying Exponential Weights: Multi-Point Padé Approximants and Asymptotics
• On integrals of the tronquée solutions and the associated Hamiltonians for the Painlevé II equation