Hyperbolic Kac-Moody Group Symmetry and Applications

双曲 Kac-Moody 群对称性及其应用

• 批准号: 1101282
• 负责人: Lisa Carbone
• 金额: $ 13.8万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-15 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101282&HistoricalAwards=false
• 关键词: Hyperbolic; Kac; Moody; Group; Symmetry

This project concerns hyperbolic Kac-Moody groups, which are infinite dimensional Lie groups that have yet been extensively studied. We are interested in the Lie groups of infinite dimensional hyperbolic Kac-Moody algebras which contain affine Kac-Moody subalgebras. Some of the proposed problems are physically motivated. Our study of physical theories such as supergravity,a theory that incorporates both supersymmetry and general relativity, has revealed a number of compelling and intriguing mathematical problems, consistent with open problems in Kac-Moody group theory. The proposed work is the first mathematical initiative that aims to apply the symmetry properties of hyperbolic Kac-Moody groups to the study of supergravity models in theoretical physics. While some of these correspondences of hyperbolic Kac-Moody symmetry with supergravity theories are conjectural, developing a full mathematical theory of hyperbolic Kac-Moody groups and their symmetric spaces amenable to computation will have a significant impact on the understanding of open problems concerning the symmetry groups of supergravity.The objective of this project is to advance understanding in the study of algebraic symmetries underlying high energy theoretic physics. Almost all finite dimensional semisimple Lie groups and Lie algebras occur in space-time symmetries and the development of the Standard Model of particle physics, which could not have progressed without an understanding of symmetries and group transformations. Infinite dimensional generalizations, known as Kac-Moody algebras and their associated groups, naturally form two distinct classes, namely affine and hyperbolic. By the 1980's the class of affine Kac-Moody algebras was shown to have wide applications in physical theories such as elementary particle theory, quantum field theory, gauge theory, conformal field theory, gravity and string theory. This project concerns Lie groups of infinite dimensional hyperbolic Kac-Moody algebras which contain affine Kac-Moody subalgebras.

该项目涉及双曲 Kac-Moody 群，它们是尚未得到广泛研究的无限维李群。我们对无限维双曲 Kac-Moody 代数的李群感兴趣，其中包含仿射 Kac-Moody 子代数。提出的一些问题是出于物理动机。我们对超引力（一种结合超对称性和广义相对论的理论）等物理理论的研究揭示了许多引人注目且有趣的数学问题，与卡克-穆迪群论中的开放问题一致。这项工作是第一个数学倡议，旨在将双曲 Kac-Moody 群的对称性应用于理论物理中超重力模型的研究。虽然双曲 Kac-Moody 对称性与超引力理论的一些对应关系是推测性的，但发展双曲 Kac-Moody 群及其可计算的对称空间的完整数学理论将对理解有关超引力对称群的开放问题产生重大影响。该项目的目标是促进对高能理论物理基础代数对称性研究的理解。几乎所有有限维半单李群和李代数都发生在时空对称性和粒子物理标准模型的发展中，如果不了解对称性和群变换，这些标准模型就不可能取得进展。无限维推广，称为 Kac-Moody 代数及其相关群，自然地形成两个不同的类别，即仿射和双曲。到 20 世纪 80 年代，仿射 Kac-Moody 代数类被证明在物理理论中具有广泛的应用，例如基本粒子论、量子场论、规范场论、共形场论、引力和弦理论。该项目涉及无限维双曲 Kac-Moody 代数的李群，其中包含仿射 Kac-Moody 子代数。