Gromov-Witten invariants and symplectic reduction

Gromov-Witten 不变量和辛约简

• 批准号: 0604705
• 负责人: Alexander Givental
• 金额: $ 30.56万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604705&HistoricalAwards=false
• 关键词: Gromov; Witten; invariants; symplectic; reduction

AbstractAward: DMS-0604705Principal Investigator: Alexander GiventalGromov -- Witten invariants characterize global properties ofphase spaces in Hamiltonian mechanics. It is our understandingthat Gromov--Witten theory has entered a very productive periodof its development. It is about to become a part of a moregeneral framework based on ideas of conformal field theory. Manykey outstanding problems of the theory such as, e.g. Virasoroconjecture, or Witten's W_n-gravity conjecture are likely toget either resolved on the basis of this framework or at leastadvanced to a substantial degree. Respectively, we are planningto continue our study of GW-invariants, their axiomaticstructure, their generalizations, their relationships withintegrable systems and symplectic field theory, methods of theircomputation including those associated with the mirrorconjecture, and try to utilize the advantages that may come fromconformal field theory. A particular problem that gave theproposal its name is a long-standing conjecture about thebehavior of Gromov-Witten invariants under symplectic reductionof target spaces by circle actions.From a more general prospective, problems we deal with in ourresearch lie on the crossroad of two major pathways inmathematics of the last two centuries. One of them is thein-depth pursuit of the intricate properties of algebraic curves--- in the form inherited from works of Gauss, Abel, Jacobi,Riemann, Klein and Poincare. The other is the broad conceptuallandscaping of mathematical physics dictated by the progress ofclassical and quantum mechanics and often associated with thenames of Hamilton, Maxwell, Gibbs, Poincare, Hilbert, Einsteinand Weyl. It is string theory that in the search for theultimate laws of nature places algebraic curves in the center ofthe modern landscape of fundamental physics, and generates newmathematical questions and points out plausible answers with anamazing pace and persistence. Some of the problems we work onare motivated by such questions, some others hopefully providethe answers that string theory did not really anticipate.

AbstractAward：DMS-0604705首席研究员：亚历山大GiventalGromov --维滕不变量描述了哈密顿力学中相空间的整体性质。我们认为格罗莫夫-维滕理论已经进入了一个非常富有成效的发展时期。它即将成为基于共形场论思想的更一般框架的一部分。 该理论的许多关键的未解决的问题，如Virasoro猜想，或维滕的W_n-引力猜想，都有可能在这个框架的基础上得到解决或至少在很大程度上得到推进。 我们计划分别继续研究GW-不变量，它们的公理结构，它们的推广，它们与可积系统和辛场论的关系，它们的计算方法，包括那些与镜像猜想有关的方法，并试图利用可能来自共形场论的优点。 一个特殊的问题，给它的名字的建议是一个长期存在的猜想下的Gromov-Witten不变量的行为辛约化的目标空间的圆行动。从更广泛的角度来看，我们在我们的研究中处理的问题在于在过去两个世纪的数学两个主要途径的十字路口。其中之一是对代数曲线的复杂性质的深入追求-另一个是数学物理的广泛概念和范围，由经典力学和量子力学的进展决定，并且经常与汉密尔顿、麦克斯韦、吉布斯、庞加莱、希尔伯特、爱因斯坦和外尔的名字联系在一起。 弦理论在探索自然界的终极法则时，把代数曲线置于现代基础物理学的中心位置，以惊人的速度和毅力提出新的数学问题，并指出合理的答案。 我们研究的一些问题就是基于这样的问题，而另一些问题则希望能提供弦理论没有预料到的答案。