Arithmetic Aspects of Electric-Magnetic Duality

电磁二象性的算术方面

• 批准号: 2001398
• 负责人: David Ben-Zvi
• 金额: $ 29.61万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-09-01 至 2024-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2001398&HistoricalAwards=false
• 关键词: Arithmetic; Aspects; Electric; Magnetic; Duality

This mathematics research project aims to apply insights from physics to questions in number theory. A central theme in theoretical physics is electric-magnetic duality, the symmetry between electricity and magnetism in Maxwell's equations and its generalization to idealized (supersymmetric) variants of the equations that govern the weak and strong nuclear forces. One of the central themes in number theory is the Langlands program, a profound link between arithmetic and geometry that counts among its successes the proof of Fermat's Last Theorem. A surprisingly close analogy between these two themes has been developed in the last decade. This project employs structures that are natural in physics to provide crucial insights missing in the number theory -- specifically, the behavior of boundaries or edges in physics suggests a new understanding of the use of integral calculus in arithmetic. This project provides research training opportunities for graduate students.The goal of this project is to develop new connections between physics and number theory. Electric-magnetic duality, the symmetry between electricity and magnetism (a variant of the Fourier transform), and its generalization (S-duality) to supersymmetric variants of the nonabelian gauge theories that describe fundamental interactions, can be considered analogous to the Langlands program, a profound nonabelian analog of the Fourier transform linking arithmetic questions to the algebra and analysis of symmetry groups. This project will take insights from physics (the understanding of electric-magnetic duality for boundary conditions) and apply them to fundamental problems in number theory (the connection between period integrals and the arithmetic of L-functions). The investigator will work to disseminate physics ideas to number theorists and vice versa, bridging two intellectually distant communities.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该数学研究项目旨在将物理学的见解应用于数字理论中的问题。理论物理学中的一个核心主题是电磁双重性，麦克斯韦方程中的电和磁性之间的对称性及其对控制弱核力量的理想化（超对称性）变体的概括。数字理论中的中心主题之一是兰兰兹计划，这是算术和几何形状之间的深刻联系，其成功是Fermat的最后一个定理的证明。在过去的十年中，这两个主题之间已经开发了一个令人惊讶的紧密比喻。该项目采用物理学自然的结构来提供数字理论中缺少的关键见解 - 特别是，物理学中边界或边缘的行为表明，对在算术中使用积分的使用有了新的了解。该项目为研究生提供了研究培训机会。该项目的目的是建立物理学和数字理论之间的新联系。 Electric-magnetic duality, the symmetry between electricity and magnetism (a variant of the Fourier transform), and its generalization (S-duality) to supersymmetric variants of the nonabelian gauge theories that describe fundamental interactions, can be considered analogous to the Langlands program, a profound nonabelian analog of the Fourier transform linking arithmetic questions to the algebra and analysis of symmetry groups.该项目将从物理学（对边界条件的电磁二元性的理解）中获得见解，并将其应用于数字理论的基本问题（周期积分与L功能的算术之间的联系）。研究者将努力将物理思想传播到编号理论家，反之亦然，弥合了两个智力远处的社区。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的智力优点和更广泛影响的评估评估标准的评估值得支持的。