Arithmetic Aspects of Electric-Magnetic Duality

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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.