Quantum Hirzebruch--Riemann--Roch Theory

量子希策布鲁赫--黎曼--罗赫理论

• 批准号: 1007164
• 负责人: Alexander Givental
• 金额: $ 27.7万
• 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: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1007164&HistoricalAwards=false
• 关键词: Quantum; Hirzebruch; Riemann; Roch; Theory

AbstractAward: DMS-1007164Principal Investigator: Alexander GiventalThe project will pursue various problems of Gromov-Witten theory, that is, the theory of topological invariants of phase spaces of Hamiltonian systems. Our research focuses on the axiomatic structure of Gromov-Witten invariants, their generalizations, their relationships with integrable systems and singularity theory, and methods of their computation, including those associated with Riemann-Roch theorems and the mirror conjecture. A central goal of this project is to resolve or advance a decade-old open problem of expressing Gromov-Witten invariants of Kahler manifolds defined as holomorphic Euler characteristics of complex vector bundles in terms of cohomological invariants of these bundles. Such expression would establish a true "quantum" analogue of the Hirzebruch-Riemann-Roch theorem. As a technical tool, the orbifold version of the classical Hirzebruch-Riemann-Roch theorem will be applied to Kontsevich's moduli spaces of stable maps. Applications of the theory to finite difference equations and integrable systems, representation theory of quantum groups, and the mirror symmetry phenomenon are expected.From a more general perspective, problems we deal with in our research lie on the crossroad of two major pathways in mathematics of the last two centuries. One of them is the in-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 conceptual landscaping of mathematical physics dictated by the progress of classical and quantum mechanics and often associated with the names of Hamilton, Maxwell, Gibbs, Poincare, Hilbert, Einstein and Weyl. It is string theory that in the search for the ultimate laws of nature places algebraic curves at the center of the modern landscape of fundamental physics, and generates new mathematical questions and points out plausible answers with an amazing pace and persistence. Some of the problems we work on are motivated by such questions, some others hopefully provide answers that string theory did not really anticipate.

AbstractAward：DMS-1007164主要研究员：亚历山大Givental该项目将追求Gromov-Witten理论的各种问题，即哈密顿系统相空间的拓扑不变量理论。我们的研究重点是Gromov-Witten不变量的公理结构，它们的推广，它们与可积系统和奇点理论的关系，以及它们的计算方法，包括与Riemann-Roch定理和镜像猜想相关的方法。该项目的一个中心目标是解决或推进一个十年之久的公开问题，即用复向量丛的上同调不变量来表示Kahler流形的Gromov-Witten不变量，定义为复向量丛的全纯欧拉特征。这样的表达将建立一个真正的“量子”类似的赫泽布鲁赫-黎曼-罗克定理。作为一种技术工具，经典的Hirzebruch-Riemann-Roch定理的轨道版本将被应用于稳定映射的Kontsevich模空间。该理论的应用有限差分方程和可积系统，表示理论的量子群，和镜像对称现象的expected.From一个更一般的角度来看，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。其中之一是深入追求的复杂性质的代数曲线-在形式继承的作品高斯，阿贝尔，雅可比，黎曼，克莱因和庞加莱。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泛概念景观，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规律时，把代数曲线置于现代基础物理学的中心，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由这些问题激发的，另一些问题则希望提供弦理论没有真正预料到的答案。