Topological Quantum Field Theories

拓扑量子场论

• 批准号: 0505735
• 负责人: Albert Schwarz
• 金额: --
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-08-01 至 2009-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0505735&HistoricalAwards=false
• 关键词: Topological; Quantum; Field; Theories

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.

AbstractAward：DMS-0505735首席研究员：阿尔伯特·施瓦茨拓扑量子场论的第一个例子是由PI在七十年代后期重建的。这些理论的重要性在维滕的论文发表后的80年代后期变得清晰起来。维滕提出了一种更普遍的方法来构造拓扑量子场论，并利用它来寻找新的有趣的模型。其中最重要的拓扑理论是由PI和维滕提出并由维滕解决的Chern-Simons理论，以及维滕的拓扑σ模型（A模型和B模型）。 Chern-Simons理论在纽结理论中非常有用，拓扑σ模型，特别是A-和B-模型之间的镜像对称，在计数代数几何中得到了惊人的结果。 拓扑版本的规范理论导致显着的发展理论四维和三维流形。 可以说，拓扑量子场论为将从物理学借用的思想应用到纯粹数学开辟了道路。然而，这些理论也被用作弦/M理论的有力工具。维滕证明，从任何N=2的超共形理论出发，人们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超共形目标的弦理论具有拓扑扇区，比完整理论更容易分析。 拓扑扇区是许多重要理论的试验场。另一方面，对于任意临界弦，我们都可以得到考虑物质场和鬼场的二维拓扑理论。更一般地说，二维拓扑量子场论在一定条件下可以“耦合到引力”，由此产生的理论可以被认为是弦理论。 特别是，我们想把纽结理论的最新发展与拓扑弦理论联系起来;似乎这种联系会使纽结理论和弦理论都取得实质性的进展。我们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很重要。