Monodromy dependence and connection formulae for isomonodromic tau functions.

*(English)*Zbl 1396.33039The Jimbo-Miwa-Ueno tau function was introduced by M. Jimbo et al. in [Physica D 2, No. 2, 306–352 (1981; Zbl 1194.34167)]. The authors of the paper under review study the behavior of this function near the critical hyperplanes. They ‘study two nontrivial examples corresponding to the sixth and the second Painlevé equations’ and ‘express the parameters of the asymptotic behavior of the corresponding tau functions at the critical points explicitly in terms of the monodromy data of the associated linear systems’ given by the following system of linear ODEs with rational coefficients
\[
\frac{{d\Phi }} {{dz}} = A\left( z \right)\Phi\tag{1}
\]
They formulate a technique that is applicable to the general two-parameter families of Painlevé tau functions and which is independent of the determinant formulae. Here the authors invoke the same schema as used in a previous paper of the first and the third author [A. Its and A. Prokhorov, “Connection problem for the tau-function of the sine-Gordon reduction of Painlevé-III equation via the Riemann-Hilbert approach”, Int. Math. Res. Not. IMRN 2016, No. 22, 6856–6883 (2016)] for proving the connection formula for the Painlevé III (\(D_8\)) tau function which was conjectured by the first and the second authors in their previous study [A. Its et al., Int. Math. Res. Not. 2015, No. 18, 8903–8924 (2015; Zbl 1329.34140)]. Utilizing this technique the authors here now provide a solution to the “constant problem” for the sixth (a purely Fuchsian system) and second Painlevé (a system with irregular singularities) equations by introducing the 1-form
\[
\omega = \sum\limits_{\nu = 1, \ldots ,n,\infty } {{{\text{res} }_{z = {a_\nu }}}\text{Tr} \left( {{G^{\left( \nu \right)}}{{\left( z \right)}^{ - 1}}A\left( z \right)d{G^{\left( \nu \right)}}\left( z \right)} \right)} ,\quad d = {d_{\mathcal T}} + {d_{\mathcal M}} \tag{2}
\]
which extends the Jimbo-Miwa-Ueno form \(\omega_{JMU}\) introduced in [Jimbo et al., loc. cit.] and this work of the authors is motivated by the earlier works of B. Malgrange [Prog. Math. 37, 401–426 (1983; Zbl 0528.32017)] and M. Bertola [Comm. Math. Phys. 294, No. 2, 539–579 (2010; Zbl 1218.37099)]. Further, this extended 1-form is ‘closed on the full space of extended monodromy data of systems of linear ordinary differential equations with rational coefficients’ (1). While developing the theory for the Painlevé VI case the authors add ‘new conceptual and technical features’ to the scheme of their earlier paper [Its and Prokhorov, loc. cit.]. The landmark result of the present paper is the Theorem A in which the authors present ‘a rigorous derivation of the ratio \(\Upsilon \left( M \right) = \frac{{{{\mathcal C}_0}}} {{{{\mathcal C}_1}}}\)’ and thereby settle a longstanding open problem in this field ever since the 1980’s. This formula had earlier been conjectured in a previous paper co-authored by the second author in [N. Iorgov et al., J. High Energy Phys. 2013, No. 12, Paper No. 029, 26 p. (2013; Zbl 1342.81500)]. While analyzing the four-point Fuchsian tau function by the general Riemann-Hilbert analysis approach the authors also derive, as a byproduct, in Propositions 3.20 and 3.24 two asymptotic formulae which were earlier proved by M. Jimbo [Publ. Res. Inst. Math. Sci. 18, 1137–1161 (1982; Zbl 0535.34042)]. Next the authors discuss the tau function of the Painlevé II equation by dealing ‘with non-Fuchsian isomonodromic deformation of the \(2 \times 2\) linear system with a single irregular singular point of Poincaré rank 3 located at infinity’ by analyzing (1) in which now \(A\left( z \right) = {A_{ - 3}}{z^2} + {A_{ - 2}}z + {A_{ - 1}}.\) By imposing a symmetry condition on \(A\left( z \right)\) and utilizing simple gauge and affine transformations they reduce this equation to normal form and by identifying the space of monodromy data \(\mathcal M\) in this case with the set \({{\mathcal M}_{PII}} = \left\{ {s = \left( {{s_1},{s_2},{s_3}} \right) \in {{\mathbb C}^3}:{s_1} - {s_2} + {s_3} + {s_1}{s_2}{s_3} = 0} \right\}\) they deduce the ‘ratio \(\Upsilon \left( s \right) = \frac{{{{\mathcal C}_ + }}} {{{{\mathcal C}_ - }}}\) in terms of the monodromy data \(s \in {{\mathcal M}_{PII}}\)’ in Theorem B. Since the statements of the main results of this paper, i.e. Theorem A and Theorem B are somewhat involved and further they are dependent on a number of equations and elaborate mathematical statements made in the paper, therefore, only for the reasons of brevity alone, the reviewer restrains himself from stating these wonderful results here. The reviewer feels that this classic paper is bound to find many citations in the future publications in this and the allied fields of research concerning the Jimbo-Miwa-Ueno tau functions.

Reviewer: Lalit Mohan Upadhyaya (Mussoorie)

##### MSC:

33E17 | Painlevé-type functions |

34E05 | Asymptotic expansions of solutions to ordinary differential equations |

34E10 | Perturbations, asymptotics of solutions to ordinary differential equations |

34M35 | Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms |

34M40 | Stokes phenomena and connection problems (linear and nonlinear) for ordinary differential equations in the complex domain |

34M55 | Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies |

34M56 | Isomonodromic deformations for ordinary differential equations in the complex domain |

##### Keywords:

connection problem; tau function; isomonodromy; Painlevé equations; Riemann-Hilbert method; Fenchel-Nielsen coordinates; Teichmüller space; Fricke coordinates; Poisson bracket; Casimirs; symplectic leaf; Goldman bracket; skein relation; Barnes \(G\)-function; digamma function; nonresonant eigenvalues; Poincaré index; Stokes phenomenon; Stokes sectors; Lax pair; Birkhoff-Grothendieck-Malgrange theory; Hastings-McLeod solution; Glaisher-Kinkelin constant; Hamiltonian flows; Riemann-Hilbert representation; Jimbo-Fricke cubic hypersurface; Virasoro algebra; conformal block representations; Schlesinger system; Poisson bracket; canonical Darboux coordinates; Goldman bracket; Flaschka-Newell Lax pair; Ablowitz-Segur one-parameter family of solutions; nonlinear special functions; Riemann-Hilbert map; Schlesinger system; Malgrange-Miwa theorem; Jimbo-Miwa-Ueno tau function; ansatz
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\textit{A. R. Its} et al., Duke Math. J. 167, No. 7, 1347--1432 (2018; Zbl 1396.33039)

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