Langlands Duality and Quantum Physics

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The Langlands Program was launched by Robert Langlands in the late 1960s with the goal of bringing together number theory and harmonic analysis. These ideas have since propagated to other areas of mathematics, such as algebraic geometry and the representation theory of infinite-dimensional algebras. And in recent works by Witten and others, it was shown that the Langlands duality patterns are closely related to the electro-magnetic duality of Quantum Field Theory. The goals of this project are two-fold. First, it is to develop further the connection between the Langlands Program and the electro-magnetic duality, generalize it to a broader context, and use it to obtain new mathematical results. Second, to study further the geometric trace formulas developed by the PI in collaboration with R. Langlands and B.C. Ngo, and to apply them to the Langlands functoriality conjecture. The two parts of the proposal are interconnected, because these trace formulas are most naturally interpreted in the framework of the categorical Langlands correspondence and electro-magnetic duality.The interdisciplinary nature of this proposal will serve to advance discovery and understanding of models of Quantum Field Theory and at the same time will stimulate the development of the Langlands Program by bringing in new insights from geometry and physics.