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 [ILT13]. We also apply the method to non-Fuchsian systems and evaluate constant factors in the asymptotics of Painlevé II tau function.

On integrals of the tronquée solutions and the associated Hamiltonians for the Painlevé II equation

“…Its connections with the Hamiltonian H and the integral of the Painlevé II transcedent w are established. To relate the integral of the Hamiltonian to Ψ, we derive several differential identities for the Hamiltonian with respect to the parameter α and the Stoke's multiplier ω, similar to those obtained in [9,34,37]. Based on the asymptotic analysis of the RH problem for Ψ carried out in [33,50], we obtain the asymptotics of w, H and other relevant functions as s → ±∞ in Section 3.…”

Irregular conformal blocks and connection formulae for Painlevé V functions

“…The evaluation of these connection constants constitutes one of the most subtle asymptotic problems in the Painlevé theory, relevant for many applications (see e.g. recent papers [ILP,BIP,ILT14,IP1,IP2,LR,Bot,DKV] and references to earlier works therein).…”

“…Even though both evaluations are conjectural, they are arrived at in different ways. The formula for Υ 0→i ∞ is obtained as a limit of the Painlevé VI connection constant [ILT13,ILP] under a prescription analogous to the one used above to define confluent CBs of the 2nd kind. Lacking a similar CFT interpretation of the real axis expansions, we found (1.19b) by employing instead the recurrence relations for Υ i ∞→+∞ with respect to monodromy parameters ν, ω.…”

Resurgence, Painlevé equations and conformal blocks

“…The PVI tau function has another form of resurgence, connecting different instanton sectors in the expansion about a given singular point. It has relatively recently been shown that Jimbo's small t expansion (2.2) may be extended to all orders, in a closed-form involving sums over c = 1 conformal blocks [21][22][23][34][35][36][37]. This remarkable all-orders expansion is a conformal block expansion for c = 1 conformal field theories.…”