Topological Quantum Field Theories

摘要奖：DMS-0505735主要研究人员：阿尔伯特·施瓦茨七十年代末，派建立了第一批拓扑量子场论。这些理论的重要性在80年代末维滕的论文发表后变得清晰起来。Witten提出了一种更普遍的构建拓扑量子场论的方法，并用它来发现新的有趣的模型。其中最重要的拓扑理论是由Pi和Witten提出并由Witten解决的Chern-Simons理论，以及Witten的拓扑sigma模型(A模型和B模型)。Chern-Simons理论在纽结理论中是非常有用的，利用拓扑Sigma模型，特别是A-模型和B-模型之间的镜像对称性，在计数代数几何中得到了显著的结果。规范理论的拓扑版导致了四维和三维流形理论的显着发展。可以说，拓扑型量子场论为将借用物理学的思想应用于纯数学开辟了道路。然而，这些理论也被用作弦/M理论的有力工具。Witten指出，从任何N=2超共形理论出发，都可以通过所谓的扭曲来构造拓扑量子场论。这特别意味着，具有N=2超保角目标的弦理论具有拓扑扇区，这比完全理论更容易分析。拓扑区是许多重要猜想的试验场。另一方面，对于任何临界弦，我们都可以得到考虑物质场和鬼场在一起的二维拓扑理论。更一般地，满足一定条件的二维拓扑量子场论可以是“与引力耦合”的，由此产生的理论可以被认为是严格理论。特别是，我们希望将纽结理论的最新发展与拓扑弦理论联系起来；似乎这种关系应该导致纽结理论和弦理论的本质进步。我们正计划将数论的方法应用到拓扑弦理论中。我们的项目是跨学科的；它们的实现对数学和理论物理都将是重要的。

AbstractAward: DMS-0505735Principal Investigator: Albert SchwarzThe first examples of topological quantum field theories wereconstructed by PI in the late seventies. The importance of suchtheories became clear in late eighties after Witten'spapers. Witten suggested a more general way of construction oftopological quantum field theories and used it to find newinteresting models. Among the most important topological theoriesare Chern-Simons theory, suggested by PI and Witten and solved byWitten, and Witten's topological sigma-models (A-model andB-model). Chern-Simons theory was very useful in knot theory.Topological sigma-models, especially mirror symmetry betweenA-and B-models, were used to obtain striking results inenumerative algebraic geometry. Topological version of gaugetheories led to remarkable developments in the theory offour-dimensional and three-dimensional manifolds. One can saythat topological quantum field theories opened the way forapplications of ideas borrowed from physics to puremathematics. However, these theories were used also as a powerfulinstrument in string/M-theory. It was shown by Witten thatstarting with any N=2 superconformal theory one can constructtopological quantum field theories by means of so calledtwisting. This means, in particular,that string theory with N=2superconformal target has topological sectors, that can beanalyzed more easily than complete theory. Topological sectorsserved as a testing ground for many important conjectures. Fromthe other side, for any critical string we can obtaintwo-dimensional topological theory considering matter fieldstogether with ghosts. More generally, two-dimensional topologicalquantum field theory obeying certain conditions can be "coupledto gravity"; the resulting theory can be considered as stringtheory.The present proposal is devoted to some problems related totopological quantum field theories. In particular, we would liketo relate recent developments in knot theory to the theory oftopological strings; it seems that this relation should lead toessential progress both in knot theory and in string theory. Weare planning to apply methods of number theory to topologicalstring theory. Our projects are interdisciplinary; theirrealization will be important both for mathematics andtheoretical physics.