MODULI SPACES, MOTIVES, PERIODS and SCATTERING AMPLITUDES

模空间、动机、周期和散射幅度

• 批准号: 1301776
• 负责人: Alexander Goncharov
• 金额: $ 30.64万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1301776&HistoricalAwards=false
• 关键词: MODULI; SPACES; MOTIVES; PERIODS; SCATTERING

The PI would like to study scattering amplitudes in quantum field theory by using two recently developed mathematical theories: theory of mixed motives and theory of cluster varieties. The PI wants to develop further the on-shell approach to scattering amplitudes related to the geometry of Grassmannians, and to find effective ways to calculate the scattering amplitudes. The PI wants to study a Feynman integral description of the derived category of mixed real Hodge sheaves. The PI wants to study canonical bases in representation theory and geometry, and relate them to the mirror symmetry. He wants to continue his work on moduli spaces of local systems on 2D-surfaces and its quantization, and relationship with representation theory and mirror symmetry. The theory of hyperbolic 3D-manifolds can be viewed as the study of certain local systems on 3D-manifolds with values in one of the simplest complex Lie group. He wants to develop a similar theory for all complex reductive Lie groups, as well as its quantum analog.During the last years several new ideas in pure mathematics had a big impact on theoretical physics, and vice verse, many ideas coming from physics had a tremendous impact on pure mathematics. In particular, the ideas of one of the most sophisticated areas of mathematics, theory of motives, found its applications in the problem of calculation of scattering amplitudes in quantum field theory - the data observed in experimenters. On the other hand, the general idea of quantization found its concrete realizations in many of areas of pure mathematics. The PI wants to investigate several concrete problems of number theory, algebraic geometry, and representation theory by using quantum dilogarithms, quantization, quantum cohomology and Feynman integrals, and other tools inspired by physics. The PI wants to pursue the newly found links between the quantum field theory and algebraic geometry to find, in particular, effective ways to calculate the scattering amplitudes.

PI希望通过使用两个最近发展的数学理论来研究量子场论中的散射振幅：混合动机理论和簇变体理论。PI希望进一步发展与格拉斯曼几何相关的散射振幅的壳上方法，并找到有效的方法来计算散射振幅。PI想要研究混合真实的霍奇层的衍生范畴的费曼积分描述。PI希望研究表示论和几何中的正则基，并将它们与镜像对称联系起来。他想继续他的工作模空间的局部系统的二维表面和它的量化，并与代表性理论和镜像对称的关系。双曲三维流形理论可以看作是研究三维流形上的某些局部系统，其值在一个最简单的复李群中。他想发展一个类似的理论，所有复杂的还原李群，以及其量子模拟。在过去几年中，一些新的想法在纯数学产生了很大的影响，理论物理学，反之亦然，许多想法来自物理学产生了巨大的影响，纯数学。特别是，思想的一个最复杂的领域的数学，理论的动机，发现其应用问题的计算散射振幅在量子场论-数据观察到的实验。另一方面，量子化的一般思想在纯数学的许多领域都有具体的实现。PI希望通过使用量子微分、量子化、量子上同调和费曼积分以及其他受物理学启发的工具来研究数论、代数几何和表示论的几个具体问题。PI希望追求量子场论和代数几何之间的新发现的联系，特别是找到计算散射振幅的有效方法。