Gromov-Witten invariants and symplectic reduction

摘要奖：DMS-0604705主要研究人员：Alexander GiventalGromov--Witten不变量表征哈密顿力学中相空间的整体性质。我们的理解是，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。它即将成为基于保形场理论思想的更一般框架的一部分。该理论的许多突出问题，例如Virasoro猜想或Witten的W_n引力猜想，都可能在这个框架的基础上得到解决，或者至少在很大程度上得到进展。我们分别计划继续研究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理论的关系，它们的计算方法，包括那些与镜像猜想相关的方法，并试图利用共形场理论可能带来的优势。这个提议得名的一个特殊问题是关于Gromov-Witten不变量在圆作用下目标空间的辛约化下的行为的长期猜想。从更广泛的角度来看，我们研究的问题位于过去两个世纪数学中两条主要路径的十字路口。其中之一是对代数曲线的复杂性质的深入追求-以继承自高斯、阿贝尔、雅各比、里曼、克莱因和庞加莱作品的形式。另一种是由经典和量子力学的进步决定的数学物理的广泛概念和图景，经常与汉密尔顿、麦克斯韦、吉布斯、庞加莱、希尔伯特、爱因斯坦和韦尔的名字联系在一起。弦理论在寻找自然界终极规律的过程中，将代数曲线置于现代基础物理学的中心，以惊人的速度和坚持不懈的速度产生新的数学问题并指出合理的答案。我们研究的一些问题是由这样的问题驱动的，另一些问题则希望提供弦理论没有真正预料到的答案。

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.