Motivic structures in physics

由Grothendieck首创的动机理论是现代代数几何的亮点之一。它最初的动机是代数簇上同调的几个不同概念的存在，以及对潜在普遍结构的追求。自那以后，它已经发展成为一门非常复杂和迷人的纯数学学科，目前仍处于许多正在进行的研究的中心，多年来，它在算术和代数几何方面产生了许多深刻的结果。考虑到动机理论的最初动机和历史发展，动机理论似乎与理论物理学世界相去甚远。然而，近年来，从事微扰量子场论工作的物理学家进行的显式计算，以及随后越来越多的精确数学结果，揭示了量子场论中动机(在适当意义上衡量动机的复杂性的数值不变量)的周期与费曼图的幅度之间的深刻联系。这个项目关注于动机在理论物理的各个分支中的作用，从费曼振幅开始，通过研究不同模型之间的关系(一边是参数费曼积分，另一边是基于扭矩的费曼振幅计算)，而且还利用量子场论中出现的代数变体作为代数几何中新理论的试验场，例如所谓的“场上的一元几何”。此外，这个项目的一部分，通过伊辛模型配分函数的代数几何性质及其推广--Potts模型，研究了统计物理中动机和周期的发生。这个项目的另一个部分涉及“非对易动机”新理论的发展及其在弦理论和量子场论中的作用。这是一个跨学科的项目，它连接了非常抽象的纯数学领域、动机理论，以及高能物理、统计物理和弦理论中的具体数学模型。上述研究项目将起到将新的数学工具引入一些快速发展的物理领域的效果，同时也将通过来自物理理论的新的思想、动机和直觉的输入，在代数和算术几何的研究领域内产生新的数学结果，并进一步发展一些新的纯数学分支。该项目有很强的教育成分，有几个研究生和本科生直接参与。

The theory of motives, initiated by Grothendieck, is one of the highlights of modern algebraic geometry. It was initially motivated by the existence of several different notions of cohomology for algebraic varieties and the quest for an underlying universal structure. It has since developed into a very complex and fascinating subject of pure mathematics, which is currently still at the center of much ongoing investigation and which, over the years, gave rise to many deep results in arithmetic and algebraic geometry. Given its original motivation and its historic development, the theory of motives appears very far remote from the world of theoretical physics. However, in recent years, explicit computations carried out by physicists working in perturbative quantum field theory, followed by an increasing number of precise mathematical results, have uncovered a deep connection between periods of motives (numerical invariant that, in an appropriate sense, measure the complexity of a motive) and amplitudes of Feynman diagrams in quantum field theory. This project is focused on the role of motives in various branches of theoretical physics, starting from the Feynman amplitudes, by investigating the relations between different models (the parametric Feynman integrals on one side, and the twistor based computations of Feynman amplitudes on the other), but also by using the algebraic varieties that occur in quantum field theory as a testing ground for new theories in algebraic geometry such as the so-called "geometry over the field with one element". Moreover, part of this project investigates the occurrence of motives and periods in statistical physics, through the algebro-geometric properties of the partition function of Ising models and their generalizations, the Potts models. Another part of this project deals with the development of a new theory of "noncommutative motives" and its role in string theory and quantum field theory.This is an interdisciplinary project that bridges between a very abstract field of pure mathematics, the theory of motives, and concrete mathematical models in high-energy physics, statistical physics, and string theory. The research project described above will have the effect of importing new mathematical tools into some fast developing areas of physics, and will also, at the same time, lead to new mathematical results, and the further development of some new branched of pure mathematics, within the research areas of algebraic and arithmetic geometry, through a new input of ideas, motivation, and intuition from physical theories. The project has a strong educational component, with the direct involvement of several graduate and undergraduate students.