Gromov-Witten invariants and symplectic reduction

主要研究者：Alexander GiventalGromov—Witten不变量表征哈密顿力学中相空间的全局性质。我们的理解是，格罗莫夫-威滕理论已经进入了一个非常富有成效的发展时期。它将成为基于共形场论思想的更普遍框架的一部分。该理论的许多关键突出问题，如virasoro猜想，或Witten的w_n重力猜想，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。分别，我们计划继续研究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论的关系，它们的计算方法，包括与镜像猜想相关的方法，并尝试利用共形场论可能带来的优势。这个建议的名字来自于一个长期存在的关于目标空间被圆作用辛约化时Gromov-Witten不变量行为的猜想。从更广泛的角度来看，我们在研究中所处理的问题处于过去两个世纪数学发展的两条主要道路的十字路口。其中之一是对代数曲线复杂性质的深入追求——其形式继承了高斯、阿贝尔、雅可比、黎曼、克莱因和庞加莱的作品。另一个是由经典力学和量子力学的进步所决定的数学物理学的广义概念图景，通常与汉密尔顿、麦克斯韦、吉布斯、庞加莱、希尔伯特、爱因斯坦和韦尔等人的名字联系在一起。正是弦理论在探索自然的终极法则时，将代数曲线置于现代基础物理学的中心位置，产生了新的数学问题，并以惊人的速度和毅力指出了合理的答案。我们研究的一些问题是由这些问题引起的，而另一些问题则有望提供弦理论没有预料到的答案。

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.