Elliptic partial differential equations in hyperkähler geometry and gauge theory: Moduli spaces of solutions and geometric invariants.

超克勒几何和规范理论中的椭圆偏微分方程：解的模空间和几何不变量。

• 批准号: 215771100
• 负责人: Privatdozent Dr. Jan Swoboda
• 金额: --
• 依托单位: Department of Mathematics
• 依托单位国家: 德国
• 项目类别: Research Fellowships
• 财政年份: 2012
• 资助国家: 德国
• 起止时间: 2011-12-31 至 2012-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/215771100?language=en
• 关键词: Elliptic; partial; differential; equations; hyperk

The study of nonlinear elliptic partial differential equations on manifolds is a central topic in Riemannian and symplectic geometry as well as in gauge theory. In many cases one considers moduli spaces of solutions to such equations with the aim of finding invariants of the underlying geometries. One particular instance are pseudoholomorphic curves (which are described by first order elliptic equations), resulting in Gromov-Witten invariants and Floer homology groups. This has lead to completely new insights into the geometry of symplectic manifolds throughout the past decades. Recently some of these elliptic methods have successfully been extended to the study of other geometries, like hyperkähler and quaternion kähler manifolds. Nonetheless, this approach raises many analytical issues which at present are only poorly understood. Two new types of first order elliptic partial differential equations with additional gauge symmetry, which naturally appear in hyperkähler geometry and in low dimensional gauge theory, shall in-depth be studied in this research endeavor. One the one hand, we aim to characterize the structure of solution spaces of these equations, like e.g. compactness and blow-up behaviour. On the other hand, through an investigation of the resulting moduli spaces, new geometric invariants shall be found and further be studied.

流形上非线性椭圆型偏微分方程的研究是黎曼几何、辛几何和规范理论的中心课题。在许多情况下，人们考虑这些方程的解的模空间，目的是找到基本几何的不变量。一个特殊的例子是伪全纯曲线（由一阶椭圆方程描述），导致Gromov-Witten不变量和Floer同调群。在过去的几十年里，这导致了对辛流形几何的全新见解。最近，这些椭圆方法中的一些已经成功地扩展到其他几何的研究，如超凯勒和四元数凯勒流形。尽管如此，这一方法提出了许多分析性问题，而目前对这些问题的了解还很有限。本奋进将深入研究自然出现在超凯勒几何和低维规范理论中的两类新的具有附加规范对称性的一阶椭圆型偏微分方程。 一方面，我们的目标是描述这些方程的解空间的结构，例如紧性和爆破行为。另一方面，通过对所得模空间的研究，可以发现新的几何不变量，并进一步研究。