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.

该项目探索并深化了理论高能物理和数论之间意想不到的联系。数论中一些最重要和最神秘的量，特别是在被称为朗兰兹计划的学科的大统一愿景中，被称为L函数。在量子场论的物理学中，基本量一度量，配分函数，有一个类似的间接和难以捉摸的定义。国际物理学家认为，把L函数看作量子力学系统的输出，可以大大加强对它们的研究。这种视角使主题中明显出现了令人惊讶的新对称性和统一的结构。这个项目展示了一种新兴的关于算术的物理学观点的实用性，其广泛传播是PI带头的，包括通过一本计划中的书。此外，国际物理学会还积极参与培养研究生和向广大科学听众进行说明性演讲。这个项目的目的是发展和传播物理学和数论之间的新联系。最近，Pi和他的合作者搭建了一座从电磁对偶到L函数运算的桥梁，这个项目提供了支撑这座桥梁的中心支柱。在量子场论的物理学中，配分函数是基本不变量。在数论和算术几何中，L函数是基本的不变量，也是朗兰兹程序提供的总体愿景的核心。研究人员将发展一种新的L函数几何理论(在数域环境中)，直接以规范理论中的配分函数为模型。这揭示了对象隐藏的量子力学结构，位于L函数中，作为几何量子化理论的一部分。反之，作者将发展三维流形的L函数理论，在数论的启发下提出新的拓扑学结构。这位研究人员将致力于与数字理论家交流物理思想，反之亦然，将两个智力上相距遥远的社区联系起来，特别是通过一本针对高级研究生受众的书，传播规范理论和自同构形式之间令人震惊的有效词典。该奖项反映了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。