Floer Theories in Gauge Theory and Symplectic Geometry

基于作者在前人工作中发展的思想和技术，该提议将继续与Floer理论相关的两个项目。第一个项目是将Ozsvath和Szabo(OS)最近定义的三维和四维流形不变量与Seiberg-Witten不变量(Sw)联系起来，基于Taubes证明Seiberg-Witten不变量和Gromov不变量等价的几何图解的扩展。由于这两个定义(OS和Sw)截然不同，且各有其优点，两者的关系在低维拓扑中应该会产生重要的结果。第二个项目是推广和推广作者最近关于Floer理论挠率的工作。这些辛不变量一方面提炼了Floer同调，另一方面与“1-圈”Gromov-Witten不变量有有趣的关系。这种关系还有待更好地理解，但它已经被用来得到超出Floer同调范围的辛拓扑中的结果。此外，这个不变量的拉格朗日相交形式应该被视为“开放的Gromov-Witten不变量”的一个最简单的例子，根据物理学家(Vafa，Witten等)的说法，它虽然在一般意义上仍没有严格的定义，但有望与Chern-Simons不变量有关。这一提议展示了数学和高能物理之间最近丰富的相互作用。规范理论不变量、全纯曲线的计数不变量和Floer理论都起源于物理，但后来在低维拓扑和辛几何领域产生了最重要的最新结果。该方案处理了这些受物理启发的不变量之间的有趣关系，以及它们在拓扑学中的应用。

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.