L-functions via geometric quantization

University Of Texas At Austin, Austin TX

Investigators

Abstract

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.