Moduli Spaces, Motives, Periods, and Scattering Amplitudes

最近，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，作为代数几何的一部分，数学中最复杂的领域之一--所谓的动机理论--的思想被应用于从实验中观察到的数据计算散射幅度。另一方面，起源于理论物理的量子化的一般概念已经在纯数学的许多领域找到了具体的实现。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。这个项目涉及几个相关主题的研究。在一个方向上，该项目将利用混合动机理论和团簇变种理论来研究量子场论中的散射幅度。在第二个方向上，该项目将进一步发展散射幅度的壳上方法，目标是找到计算散射幅度的有效方法。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，并发展量子霍奇场理论。在第四个方向，该项目将研究与表示理论和几何有关的三维Calabi-Yau范畴的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子化，以及与表象理论和镜像对称性的关系。特别是研究了拓扑曲面上局部系统的模空间和曲面上非对易局部系统的模空间的超Kähler结构。在最后一个方向上，双曲三维流形理论可以被视为研究三维流形上的某些局部系统，这些流形的值在最简单的复李群之一。该项目旨在为所有复约化李群及其量子类比开发类似的理论。

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.