CAREER: Homotopical representation theory and TQFTs

职业：同伦表示理论和 TQFT

• 批准号: 2239698
• 负责人: Cris Negron
• 金额: $ 42.5万
• 依托单位: University of Southern California
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-15 至 2028-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2239698&HistoricalAwards=false
• 关键词: CAREER; Homotopical; representation; theory; TQFTs

This project will develop novel connections between modern mathematics and quantum field theory. Quantum field theories are an effective tool in modern physics. The project will develop the mathematics needed to rigorously describe and analyze such theories by leveraging recent innovations in abstract algebra and geometry to explicitly describe quantum field theories in mathematical terms. In addition to scientific advances, the project will both develop text-based resources for mathematicians that will advance researchers' abilities to access and deploy cutting edge methods in algebra and geometry and provide support to community-based education programs in the Los Angeles area. More precisely, the PI will use homotopical approaches in representation theory to provide rigorous mathematical realizations of topological quantum field theories whose associated moduli of vacua are non-compact. Topological field theories of this type arise naturally as twists of supersymmetric quantum field theories. In producing the aforementioned realizations, the PI will develop new relationships between representations of quantum groups and the geometry of Springer resolutions.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.

该项目将在现代数学和量子场理论之间发展新的联系。量子场理论是现代物理学的有效工具。该项目将通过利用抽象代数和几何形状的最新创新来开发严格描述和分析此类理论所需的数学，以用数学术语明确描述量子场理论。除了科学进步外，该项目还将为数学家开发基于文本的资源，这些资源将促进研究人员在代数和几何形状中访问和部署尖端方法的能力，并为洛杉矶地区的基于社区的教育计划提供支持。更确切地说，PI将使用表示理论中的同位方法来提供对拓扑量子场理论的严格数学实现，其相关的真空模态是非紧凑的。随着超对称量子场理论的扭曲，这种类型的拓扑场理论自然出现。在产生上述实现时，PI将在量子群体的表示与Springer分辨率的几何形状之间建立新的关系。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的智力优点和更广泛影响的评估标准通过评估来支持的。