Isomonodromic deformations of the sl(2) Fuchsian systems on the Riemann sphere

Details Isomonodromic deformations of the sl(2) Fuchsian systems on the Riemann sphere Isomonodromic deformations of the sl(2) Fuchsian systems on the Riemann sphere

2003-09-18

arxiv.org/abs/math-ph/0309048v5

Physics

Mathematical Physics

Revised version, 24 pages

Scientific paper

This paper is devoted to two geometric constructions related to the isomonodromic method. We follow the Drinfeld ideas and develop them in the case of the curve $X=\mathbb{P}^1\setminus\{a_1,...,a_n\}$. Thus we generalize the results of Arinkin and Lysenko to the case of arbitrary number $n$ of points. First, we construct separated Darboux coordinated in terms of the Hecke correspondences between moduli spaces. In this way we present a geometric interpretation of the Sklyanin formulas. In the second part of the paper, we construct Drinfeld's compactification of the initial data space and describe the compactifying divisor in terms of certain FH-sheaves. Finally, we give a geometric presentation of the dynamics of the isomonodromic system in terms of deformations of the compactifying divisor and explain the role of apparent singularities for Fuchsian equations. To illustrate the results and methods, we give an example of the simplest isomonodromic system with four marked points known as the Painlev´e-VI system.

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