Langlands Duality and Quantum Physics

朗兰兹对偶性和量子物理

• 批准号: 1201335
• 负责人: Edward Frenkel
• 金额: $ 20.55万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1201335&HistoricalAwards=false
• 关键词: Langlands; Duality; Quantum; Physics

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.

朗兰兹计划是由罗伯特·朗兰兹在20世纪60年代末发起的，目的是将数论和谐波分析结合起来。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。在Witten等人最近的工作中，证明了朗兰兹对偶模式与量子场论的电磁对偶密切相关。这个项目有两个目标。一是进一步发展朗兰兹纲领与电磁对偶之间的联系，将其推广到更广泛的范围，并利用它来获得新的数学结果。其次，进一步研究PI与R. Langlands和bc . Ngo合作开发的几何迹公式，并将其应用于Langlands泛函猜想。这两部分的建议是相互联系的，因为这些迹公式是最自然的解释在范畴朗兰兹对应和电磁对偶的框架。该提议的跨学科性质将有助于推进量子场论模型的发现和理解，同时将通过从几何和物理学中引入新的见解来刺激朗兰兹计划的发展。