Seiberg-Witten and Instanton Floer Homologies

Seiberg-Witten 和 Instanton Floer 同源性

• 批准号: 9802480
• 负责人: Tomasz Mrowka
• 金额: $ 6.32万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-01 至 2000-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9802480&HistoricalAwards=false
• 关键词: Seiberg; Witten; Instanton; Floer; Homologies

Abstract Proposal: DMS 9802480 Principal Investigators: Tomasz Mrowka and Matilde Marcolli The research project consists of three parts. The first goal is the construction of an equivariant version of Seiberg-Witten Floer homology, which is an invariant of the differentiable structure of the underlying three-manifold and avoids the problem of metric dependence that arises in the non-equivariant theory. The second part of the project consists of deriving the exact triangles formulae that detect how the Seiberg-Witten Floer homology changes when the three-manifold is modified by surgery and a suitable cutting and pasting technique that relates the Seiberg-Witten invariants of four-manifolds and three-manifolds. The remaining part of the project is dedicated to the investigation of the relation between the Seiberg-Witten Floer homology and the instanton Floer homology associated to Donaldson theory. The discovery of the Seiberg-Witten invariants, as an outcome of recent developments in string theory, has had a tremendous impact in the field of low dimensional topology. The rich interplay of geometry and theoretical physics has allowed a deeper understanding of the geometric structure of three and four-dimensional manifolds. The topology and geometry of three and four-dimensional manifolds is known to be especially rich of interesting phenomena and open problems: the failure of the classification methods used in higher dimensions makes it particularly important to construct computable invariants, hence the need to investigate the properties of the Seiberg-Witten invariants and their relation to the previously known Yang-Mills-Donaldson theory.

摘要建议：DMS 9802480主要研究人员：Tomasz Mrowka和Matilde Marcolli研究项目由三个部分组成。第一个目标是构建seiberg-witten的浮点同源性的模棱两可的版本，该版本是基础三元模型的可区分结构的不变结构，并避免了非等级理论中产生的度量依赖性问题。 该项目的第二部分包括得出确切的三角形公式，该公式检测到Seiberg-witten的浮子同源性在通过手术修改三个manifold以及与Seiberg-Witten不变性的四个manifolds和三个manifords的不变性相关的合适切割和粘贴技术时如何变化。该项目的其余部分致力于调查Seiberg-witten Floer同源性与与Donaldson理论相关的Instanton Floer同源性之间的关系。 作为弦理论最新发展的结果，Seiberg-witten不变性的发现对低维拓扑领域产生了巨大影响。几何和理论物理学的丰富相互作用使人们对三维流形的几何结构有了更深入的了解。 已知三维流形的拓扑和几何形状特别丰富了有趣的现象和开放问题：在较高维度中使用的分类方法的失败使得构建可计算的不变性材料尤为重要，因此需要研究Seiberg-Witten Inffortiants的属性及其与先前已知的Yang-Mills-Mills-Mills-donalds-nalds-donaldonaldsson的属性。