Motivic structures in physics

物理学中的动机结构

• 批准号: 1201512
• 负责人: Matilde Marcolli
• 金额: $ 18.77万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-06-01 至 2016-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1201512&HistoricalAwards=false
• 关键词: Motivic; structures; physics

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.

由Grothendieck发起的动机理论是现代代数几何形状的亮点之一。最初，它是由于代数品种的几种不同的共同体学概念以及寻求潜在的普遍结构的概念的动机。 此后，它已发展为一个非常复杂和引人入胜的纯数学主题，该主题目前仍处于许多正在进行的研究的中心，并且多年来，它在算术和代数几何形状中产生了许多深刻的结果。鉴于其最初的动机及其历史发展，动机理论似乎远离理论物理世界。然而，近年来，从事扰动量子场理论的物理学家进行的明确计算，其次是越来越多的精确数学结果，已经发现了动机时期之间的深厚联系（数值不变性，在适当的含义上，衡量动机的复杂性）和Feynman feynman filegraume in量子现场理论中的Feynman图。 该项目的重点是通过调查不同模型之间的关系（一方面参数Feynman积分的参数，基于扭曲的feynman amplitudes a amplienties a amplieties a implaig a amplieties to ander ofermbraic coreby ofermbraic corepar in n pressient ofermbraic aeldies contormers在Quantermbra nepard eartial neperion n otermbraim aperion，就可以将动机在理论物理学的各个分支中的作用（从Feynman振幅开始）开始，但也将基于代数的地位上的地理位置上的地理位置进行了探索。例如所谓的“带有一个元素的田地上的几何形状”。此外，该项目的一部分通过ISING模型的分区函数的代数几何特性及其概括（POTTS模型）研究了统计物理学中的动机和周期。该项目的另一部分介绍了一种新的“非交互动机”理论的发展及其在弦理论和量子场理论中的作用。这是一个跨学科的项目，该项目在纯数学的非常抽象的领域，动机理论，在高增强物理学，统计物理学和弦乐理论中桥接了一个非常抽象的领域。上述研究项目将具有将新的数学工具导入到一些快速发展的物理领域中，同时还将导致新的数学结果，并在代数和算术几何学的研究领域内进一步发展一些新的纯数学分支，并通过从物理和直觉中获得的新投入，以及从物理上进行思想的新投入。该项目具有强大的教育成分，几位研究生和本科生的直接参与。