Floer Theories in Gauge Theory and Symplectic Geometry

基于提议者以前的工作中发展的思想和技术，该提议是进行与Floer理论有关的两个项目。第一个是将最近由Ozsvath和Szabo（OS）定义的3-和4-流形不变量与Seiberg-Witten不变量（SW）联系起来，基于Taubes对Seiberg等价性的证明所依据的几何图像的扩展，辛4-流形的维滕不变量和格罗莫夫不变量。由于这两个定义（OS vs SW）有着显著的不同，并且各有其优点，因此两者之间的关系在低维拓扑中应该会产生重要的结果。这些是辛不变量，一方面完善了Floer同调，另一方面与“1-loop”Gromov-Witten不变量有着有趣的关系。这种关系仍有待更好地理解，但它已经被用来获得辛拓扑的结果，这超出了thethe达到的弗洛尔同调。此外，这个不变量的拉格朗日交形式应该被看作是“开放的Gromov-维滕不变量”的一个最简单的例子，虽然它在一般意义上还没有严格的定义，但根据物理学家（Vafa，维滕等人）的观点，它应该与Chern-Simons不变量联系起来。规范理论不变量、全纯曲线的枚举不变量和Floer理论都起源于物理学，但在低维拓扑学和辛几何学领域产生了最重要的结果。该提案涉及迷人的relationsamong这些物理启发的不变量，以及它们的应用拓扑。

Building on ideas and techniques developed in proposer's previous works,the proposal is to pursue two projects related to aspects of Floer theory.The first is to relate the 3- and 4-manifold invariants recently definedby Ozsvath and Szabo (OS) with the Seiberg-Witten invariants (SW), based on an extension of the geometric picture underlying Taubes's proof of the equivalence of Seiberg-Witten invariants and Gromov invariants of symplectic 4-manifolds. Since the two definitions (OS vs SW) aredrastically different and each has its own advantage, a relation ofthe two should yield important consequences in low-dimensional topology.The second project is to extend and generalize the proposer's recentwork on Floer-theoretic torsions. These are symplectic invariants which on one hand refine the Floer homology, on the other hand haveinteresting relations with "1-loop" Gromov-Witten invariants. The relation remains to be better understood, yet it has already been used to obtain results in symplectic topology which are beyond thethe reach of Floer homologies. Moreover, the Lagrangian intersectionversion of this invariant should be regarded as a simplest example of"open Gromov-Witten invariants", which though still not rigorouslydefined in general, are expected to relate to the Chern-Simons invariants according to the physicists (Vafa, Witten et al).This proposal showcases the recent fertile interactions between mathematics and high energy physics. Gauge-theoretic invariants,enumeratitve invariants of holomorphic curves, and Floer theoryall have their origin in physics, yet have since yielded the mostimportant recent results in the fields of low dimensional topology andsymplectic geometry. The proposal deals with fascinating relationsamong these physics-inspired invariants, and their applications to topology.