Arithmetic Aspects of Electric-Magnetic Duality

这个数学研究项目旨在将物理学的见解应用于数论问题。理论物理学的一个中心主题是电磁对偶性，即麦克斯韦方程中的电和磁之间的对称性，以及它对控制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纲领，这是算术和几何之间的一个深刻联系，其成功之处包括费马大定理的证明。在过去的十年里，这两个主题之间出现了惊人的相似之处。该项目采用物理学中自然的结构来提供数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算术中积分的使用有了新的理解。本计画为研究生提供研究训练的机会。本计画的目标是发展物理学与数论之间的新连结。电磁对偶性，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交换规范理论的超对称变体的推广（S对偶性），可以被认为类似于朗兰兹纲领，一个深刻的非交换傅立叶变换的模拟，将算术问题与对称群的代数和分析联系起来。该项目将从物理学（对边界条件的电磁对偶性的理解）中获得见解，并将其应用于数论中的基本问题（周期积分与L函数算术之间的联系）。该研究员将致力于传播物理学思想的数论家，反之亦然，弥合两个智力上遥远的社区。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.