Langlands Duality and Quantum Physics

朗兰兹计划是由罗伯特·朗兰兹在20世纪60年代末发起的，目的是将数论和调和分析结合起来。自那以后，这些思想已经传播到数学的其他领域，如代数几何和无限维代数的表示理论。Witten等人最近的工作表明，朗兰兹对偶图案与量子场论中的电磁对偶密切相关。这个项目的目标有两个。首先，进一步发展朗兰兹程序和电磁对偶之间的联系，将其推广到更广泛的背景下，并利用它来获得新的数学结果。第二，进一步研究PI与R.Langland和B.C.NGO合作发展的几何迹公式，并将它们应用于Langland函数猜想。该提案的两个部分是相互关联的，因为这些迹公式是最自然地在范畴朗兰兹对应和电磁对偶的框架下解释的。该提案的跨学科性质将有助于促进对量子场论模型的发现和理解，同时将通过引入几何和物理的新见解来刺激朗兰兹计划的发展。

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.