L-functions via geometric quantization

通过几何量化的 L 函数

• 批准号: 2302346
• 负责人: David Ben-Zvi
• 金额: $ 38万
• 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: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2302346&HistoricalAwards=false
• 关键词: functions; via; geometric; quantization

The project explores and deepens an unexpected link between theoretical high energy physics and number theory. Some of the most important and mysterious quantities in number theory, in particular in the grand unified vision of the subject known as the Langlands program, are known as L-functions. In the physics of quantum field theory the fundamental quantities one measures, the partition functions, have a similarly indirect and elusive definition. The PI believes that the study of L-functions can be significantly enhanced by thinking of them as the output of a quantum mechanical system. This perspective makes evident surprising new symmetries and unifying structures in the subject. This project demonstrates the utility of an emerging physics point of view on arithmetic whose broad dissemination the PI is spearheading, including through a planned book. In addition the PI is actively involved in training graduate students and giving expository lectures to broad scientific audiences.The object of this project is to develop and disseminate new connections between physics and number theory. The PI and collaborators recently constructed a bridge from electric-magnetic duality to the arithmetic of L-functions, and this project provides a central pillar supporting this bridge. In the physics of quantum field theory, partition functions are the fundamental invariants. In number theory and arithmetic geometry, L-functions are the fundamental invariants and at the heart of the overarching vision provided by the Langlands program. The investigator will develop a new geometric theory of L-functions (in the setting of number fields) directly modeled on partition functions in gauge theory. This reveals a hidden quantum mechanical structure to the subject, situating L-functions as part of the theory of geometric quantization. Conversely, the investigator will develop a new theory of L-functions for 3-manifolds, suggesting new structure in topology inspired by number theory. The investigator will work to communicate physics ideas to number theorists and vice versa, connecting two intellectually distant communities, in particular through a book aimed at an advanced graduate audience disseminating the shockingly effective dictionary between gauge theory and automorphic forms.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.

该项目探索并加深了理论高能物理学与数量理论之间的意外联系。数量理论中最重要，最神秘的数量，尤其是在称为兰兰兹计划的大统一愿景中，被称为L功能。在量子场理论的物理学中，基本数量的测量值，分区功能具有类似的间接和难以捉摸的定义。 PI认为，通过将其视为量子机械系统的输出，可以显着增强对L功能的研究。这种观点使该主题中的新对称性和统一结构显而易见。该项目展示了新兴物理学的观点在算术上的效用，该观点广泛传播PI是率领的，包括通过计划的书。此外，PI还积极参与培训研究生，并向广泛的科学受众提供说明性讲座。该项目的目的是发展和传播物理学与数字理论之间的新联系。 PI和合作者最近建造了一座从电磁二元性到L功能算术的桥梁，该项目提供了一个支撑这座桥梁的中央支柱。在量子场理论的物理学中，分区函数是基本不变的。在数字理论和算术几何形状中，L功能是兰兰兹计划提供的总体视野的基本不变式。研究者将开发一种直接建立在仪表理论中的分区函数的L功能（在数字场的环境中）的新几何理论。这揭示了对受试者的隐藏量子机械结构，将L功能作为几何量化理论的一部分。相反，研究者将针对3个manifolds开发一种新的L功能理论，这表明拓扑中的新结构受数字理论的启发。调查人员将努力将物理思想传达给数字理论家，反之亦然，尤其是通过一本针对高级研究生受众的书，该书籍传播了仪表理论和自称形式之间令人震惊的有效词典。该奖项反映了NSF的法定任务，并反映了通过评估范围的支持者，并通过评估了范围的范围。