Moduli Spaces, Motives, Periods, and Scattering Amplitudes

近年来，纯数学中的新思想对理论物理产生了强烈的影响，反之亦然。特别是数学中最复杂的领域之一，即所谓的动机理论，它是代数几何的一部分，在根据实验中观察到的数据计算散射振幅时得到了应用。另一方面，起源于理论物理学的量子化的一般概念在纯数学的许多领域得到了具体的实现。利用这些新的见解，本项目将研究数论、代数几何和表示理论中的几个具体问题。这个项目涉及几个相关主题的研究。一方面，本项目将利用混合动机理论和簇变理论研究量子场论中的散射振幅。在第二个方向上，该项目将进一步发展散射振幅的on-shell方法，目标是找到计算散射振幅的有效方法。第三个方向，研究混合霍奇理论的量子场论方法，发展量子霍奇场论。在第四个方向上，该项目将研究与表示理论和几何相关的三维Calabi-Yau类别的Donaldson-Thomas不变量。在第五个方向上，研究者将继续研究二维表面上局部系统的模空间及其量化，以及与表示理论和镜像对称的关系。特别地，研究者将研究拓扑曲面上局部系统的模空间的hyperkähler结构，以及曲面上非交换局部系统的模空间。在最后一个方向上，双曲三维流形理论可以看作是对三维流形上的某些局部系统的研究，其值在最简单的复李群之一上。该项目旨在为所有复杂约化李群及其量子模拟发展一个类似的理论。

Recently, new ideas in pure mathematics have had a strong impact on theoretical physics, and vice versa. In particular, the ideas of one of the most sophisticated areas of mathematics, the so-called theory of motives, which is part of algebraic geometry, found application in the calculation of scattering amplitudes from the data observed in experiments. On the other hand, the general notion of quantization that originated in theoretical physics has found concrete realizations in many of areas of pure mathematics. Using these new insights, this project will investigate several concrete questions in number theory, algebraic geometry, and representation theory. This project involves research in several related topics. In one direction, the project will study scattering amplitudes in quantum field theory by using theory of mixed motives and theory of cluster varieties. In a second direction, the project will further develop the on-shell approach to scattering amplitudes, with the goal of finding effective ways to calculate scattering amplitudes. In a third direction, the project will study a quantum field theory approach to mixed Hodge theory and develop quantum Hodge field theory. In a fourth direction, the project will study Donaldson-Thomas invariants of three-dimensional Calabi-Yau categories relevant to representation theory and geometry. In a fifth direction the investigator will continue work on moduli spaces of local systems on two-dimensional surfaces and its quantization, and on the relationship with representation theory and mirror symmetry. In particular, the investigator will study the hyperkähler structure of the moduli spaces of local systems on topological surfaces, and moduli spaces of non-commutative local systems on surfaces. In a final direction, the theory of hyperbolic three-dimensional manifolds can be viewed as the study of certain local systems on three-dimensional manifolds with values in one of the simplest complex Lie groups. The project aims to develop a similar theory for all complex reductive Lie groups, as well as its quantum analog.