graded 流形与高阶 Lie 群oid 对称几何的讲义
摘要
本文研究在 graded 流形及其全局对应物——高阶 Lie 群oid 上的辛几何结构。我们首先介绍 graded 流形的概念，从度数为 1 的情况开始，将关键几何结构翻译为经典微分几何术语。随后扩展至度数为 2 的情况，给出几个示范性例子，特别强调等价上同调和 Lie 双诺导子。接着定义辛 Q 流形及其拉格朗日 Q 子流形，引入 graded 类 Weinstein 管状邻域定理的类比，并用于研究这些子流形的变形。转而关注高阶 Lie 群oid 和 Getzler 引入的移位辛结构。我们检讨其 Morita 不变性并提供若干文献中的例子。最后，我们引入移位拉格朗日结构并探讨其与矩元图和辛约化程序的联系。Throughout these notes, we illustrate the key constructions and results with concrete examples, highlighting their applications in mathematics and physics. These lecture notes are based on two mini-courses delivered by the first author at Geometry in Algebra and Algebra in Geometry VII (2023) in Belo Horizonte, Brazil, and at the INdAM Intensive Period: Poisson Geometry and Mathematical Physics (2024) in Napoli, Italy.
引用
@article{arxiv.2510.09448,
title = {Lecture notes on the symplectic geometry of graded manifolds and higher Lie groupoids},
author = {Miquel Cueca and Antonio Maglio and Fabricio Valencia},
journal= {arXiv preprint arXiv:2510.09448},
year = {2026}
}
备注
88 pages, V2: modifications from referee, V3: typos corrected