Painlevé 单值流形、装饰特征标簇与簇代数
数学物理
2016-09-29 v4 代数几何
math.MP
可精确求解与可积系统
摘要
在本文中，我们引入装饰特征标簇的概念，用于 Painlevé 微分方程理论中出现的黎曼曲面。由于所有 Painlevé 微分方程（除第六个外）均展现斯托克斯现象，自然考虑带孔及此类孔上具边缘尖点的黎曼球面。装饰特征标被定义为 arXiv:1509.07044 中引入的带边缘尖点 Teichmüller 空间的复化。我们证明具有 s 个孔和 n>1 个带边缘尖点的黎曼球面的装饰特征标簇是维数为 3 s+ 2 n-6 的泊松流形，并显式计算出自然为簇类型的泊松括号。我们还展示了如何以几何项获得 Painlevé 微分方程的合流过程。
引用
@article{arxiv.1511.03851,
title = {Painlev\'e monodromy manifolds, decorated character varieties and cluster algebras},
author = {Leonid Chekhov and Marta Mazzocco and Vladimir Rubtsov},
journal= {arXiv preprint arXiv:1511.03851},
year = {2016}
}
备注
Parts of this paper overlap with arXiv:1212.6723, but it presents a much deeper geometric understanding and the new notion of decorated character variety which was not included there. 19 coloured pictures, 33 pages. To appear on Int. Mat. Res. Not