L-functions via geometric quantization

该项目探索并深化了理论高能物理与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大统一视野中，被称为l函数。在量子场论的物理学中，我们测量的基本量，配分函数，也有类似的间接和难以捉摸的定义。PI认为，将l函数视为量子力学系统的输出，可以显著增强对它们的研究。这一视角使得令人惊讶的新的对称和统一的结构在主题中变得明显。该项目展示了一种新兴的物理观点对算术的效用，PI正在率先广泛传播，包括通过计划中的书籍。此外，PI还积极参与研究生培训，并为广大科学听众提供说明性讲座。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。PI和合作者最近构建了一个从电磁对偶到l函数算法的桥梁，这个项目提供了一个支撑这个桥梁的中心支柱。在量子场论的物理学中，配分函数是基本的不变量。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。研究者将发展一种新的l函数几何理论（在数域的设置中），直接以规范理论中的配分函数为模型。这揭示了一个隐藏的量子力学结构，将l函数定位为几何量子化理论的一部分。相反，研究者将发展一种新的3流形的l函数理论，在数论的启发下提出新的拓扑结构。研究者将致力于与数论学家交流物理思想，反之亦然，连接两个智力上遥远的群体，特别是通过一本针对高级研究生读者的书，传播规范论和自同构形式之间令人震惊的有效词典。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.