Elliptic Schlesinger system and Painlevé VI

Abstract

We construct an elliptic generalization of the Schlesinger system (ESS) with positions of marked points on an elliptic curve and its modular parameter as independent variables (the parameters in the moduli space of the complex structure). ESS is a non-autonomous Hamiltonian system with pair-wise commuting Hamiltonians. The system is bihamiltonian with respect to the linear and the quadratic Poisson brackets. The latter are the multi-color generalization of the Sklyanin-Feigin-Odeskii classical algebras. We give the Lax form of the ESS. The Lax matrix defines a connection of a flat bundle of degree one over the elliptic curve with first order poles at the marked points. The ESS is the monodromy independence condition on the complex structure for the linear systems related to the flat bundle. The case of four points for a special initial data is reduced to the Painlevé VI equation in the form of the Zhukovsky-Volterra gyrostat, proposed in our previous paper.

Fuchs versus Painlevé

“…Should we introduce higher order Painlevé ODE's (in analogy to the higher order KdV generalization of KdV) ? Should we consider Garnier systems [5], or even, more general Schlesinger systems [6,7] ? This paper is organized as follows.…”

Вырождение Эллиптической Системы Шлезингера

“…Настоящая работа посвящена изучению с помощью предельных процедур различных способов вырождения эллиптического обобщения [1]- [4] системы Шлезингера [5] и неавтономного эллиптического SL(N, C)-волчка. Рассматриваемые в настоящей работе методы построения предельных процедур могут быть обобщены на связанные с эллиптической системой Шлезингера уравнения Пенлеве и интегрируемые системы, возникающие в разных областях теоретической физики.…”

Hecke Transformations of Conformal Blocks in WZW Theory. I. KZB Equations for Non-Trivial Bundles

“…For arbitrary characteristic classes these type of models were described in [45]. Different aspects and applications of the Hecke transformations to integrable systems and related topics (such as Painlevé-Schlesinger equations [2,15,50,51,60,67,68,71], monopoles [14,25,31,37,39,49,58], quadratic Poisson structures [13,74], applications to AGT conjecture [53,54,55] etc.) can be found in wide range of literature.…”