Seiberg-Witten and Instanton Floer Homologies

摘要提案：DMS 9802480主要研究员：Tomasz Mrowka和Matilde Marcolli研究项目包括三个部分。第一个目标是构造Seiberg-Witten Floer同调的等变版本，它是底层三流形的可微结构的不变量，避免了在非等变理论中出现的度量依赖问题。该项目的第二部分包括推导精确三角形公式，该公式检测当三流形被手术修改时Seiberg-Witten Floer同源性是如何变化的，并采用合适的剪切和粘贴技术，将四流形和三流形的Seiberg-Witten不变量联系起来。该项目的其余部分致力于研究Seiberg-Witten Floer同源性和与Donaldson理论相关的瞬时Floer同源性之间的关系。Seiberg-Witten不变量的发现，作为弦理论最新发展的结果，在低维拓扑领域产生了巨大的影响。几何学和理论物理学之间丰富的相互作用使得人们对三维和四维流形的几何结构有了更深的理解。众所周知，三维和四维流形的拓扑和几何结构特别丰富有趣的现象和开放的问题：在高维中使用的分类方法的失败使得构造可计算不变量变得特别重要，因此需要研究Seiberg-Witten不变量的性质及其与先前已知的Yang-Mills-Donaldson理论的关系。

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.