Quantum Hirzebruch--Riemann--Roch Theory

摘要奖：DMS-1007164首席研究员：Alexander Givental该项目将探讨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。我们的研究集中在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。这个项目的一个中心目标是解决或推进一个十年来公开的问题，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Euler特征。这样的表述将建立一个真正的“量子”类比的赫兹布鲁赫-黎曼-罗赫定理。作为一种技术工具，经典的Hirzebruch-Riemann-Roch定理的orbilold形式将被应用于稳定映射的Kontsevich模空间。从更广泛的角度来看，我们所研究的问题位于过去两个世纪数学中两条主要道路的十字路口。其中之一是对代数曲线的复杂性质的深入追求--这种形式继承了高斯、阿贝尔、雅各比、里曼、克莱因和庞加莱的作品。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。弦理论在寻找自然终极法则的过程中，将代数曲线置于基础物理的现代版图的中心，并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-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.