Arithmetic Aspects of Electric-Magnetic Duality

这个数学研究项目的目的是将物理学的见解应用于数论中的问题。理论物理中的一个中心主题是电磁二元性，即麦克斯韦方程中电和磁之间的对称性，以及它对支配弱核力和强核力的方程的理想化(超对称)变体的推广。数论的中心主题之一是朗兰兹程序，它是算术和几何之间的深刻联系，它的成功之一是费马大定理的证明。在过去的十年里，这两个主题之间出现了惊人的相似之处。这个项目使用了物理学中自然的结构来提供数论中缺失的重要见解--具体地说，物理学中边界或边的行为表明了对积分在算术中的使用的新理解。这个项目为研究生提供了研究训练的机会。这个项目的目标是在物理学和数论之间建立新的联系。电磁对偶性，即电和磁之间的对称性(傅立叶变换的一种变体)，以及它对描述基本相互作用的非阿贝尔规范理论的超对称变体的推广(S对偶)，可以被认为类似于朗兰兹计划，后者是傅里叶变换的深刻的非阿贝尔模拟，将算术问题与对称群的代数和分析联系起来。这个项目将从物理学(对边界条件的电磁对偶的理解)中获得见解，并将它们应用于数论中的基本问题(周期积分与L函数的算术之间的联系)。这位研究人员将致力于向数字理论家传播物理思想，反之亦然，架起两个智力上相距遥远的社区的桥梁。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.