Moduli Spaces, Motives, Periods, and Scattering Amplitudes

最近，纯数学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的思想，即所谓的动机理论，是代数几何的一部分，在从实验中观察到的数据计算散射振幅中找到了应用。另一方面，起源于理论物理学的量子化的一般概念已经在纯数学的许多领域中找到了具体的实现。利用这些新的见解，这个项目将研究数论，代数几何和表示论中的几个具体问题。 该项目涉及几个相关主题的研究。在一个方向上，该项目将通过使用混合动机理论和簇变体理论来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壳上方法，目标是找到计算散射振幅的有效方法。在第三个方向，该项目将研究混合霍奇理论的量子场论方法，并发展量子霍奇场论。在第四个方向，该项目将研究与表示论和几何相关的三维卡-丘范畴的唐纳森-托马斯不变量。在第五个方向的调查将继续工作的模空间的局部系统的二维表面及其量化，并与代表性理论和镜像对称的关系。特别是，研究者将研究拓扑曲面上局部系统的模空间的超凯勒结构，以及曲面上非交换局部系统的模空间。在最后一个方向，双曲三维流形理论可以被看作是研究三维流形上的某些局部系统，其值在最简单的复李群之一中。该项目旨在为所有复杂的还原李群及其量子类似物开发类似的理论。

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.