CAREER: Elliptic cohomology and quantum field theory

职业：椭圆上同调和量子场论

• 批准号: 2340239
• 负责人: Daniel Berwick Evans
• 金额: $ 55万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-06-15 至 2029-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2340239&HistoricalAwards=false
• 关键词: CAREER; Elliptic; cohomology; quantum; field

The research of this award lies at the interface between theoretical physics and geometry. An unsolved conjecture posits a deep connection between the geometry of supersymmetric quantum field theories and certain structures in algebraic topology. Resolving this conjecture would provide new insight into the mathematical foundations of quantum field theory, while also providing several long-anticipated applications of algebraic topology in physics. The projects the PI will work on leverage higher categorical symmetries to gain new insights into this 30-year-old conjecture. The award supports graduate students working with the PI whose research will contribute to this area. The PI will also continue his involvement in mathematics education for incarcerated people through the Education Justice Project in Illinois.The proposed research is centered on an equivariant refinement of Stolz and Teichner’s conjectured geometric model for elliptic cohomology from 2-dimensional supersymmetric field theories. The overarching goal is to link structures in Lurie’s 2-equivariant elliptic cohomology with the geometry of supersymmetric gauge theories. Some of the projects are natural extensions of prior work at heights zero and one, focusing on height 2 generalizations of specific quantum field theories that are expected to construct elliptic Thom classes. Other projects will initiate the study of 2-equivariant geometry, interfacing with topics in string geometry, loop group representation theory, and elliptic power operations.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.

该奖项的研究涉及理论物理学和几何学之间的一个未解决的猜想，它提出了超对称量子场论的几何学与代数拓扑中的某些结构之间的深刻联系，解决这个猜想将为量子的数学基础提供新的见解。场论，同时还提供代数拓扑在物理学中的几个期待已久的应用，PI 将致力于利用更高的分类对称性来获得对这个 30 岁的人的新见解。该奖项支持与 PI 一起工作的研究生，他们的研究将为这一领域做出贡献。PI 还将通过伊利诺伊州的教育正义项目继续参与囚犯的数学教育。拟议的研究集中于对Stolz 和 Teichner 从二维超对称场论猜想的椭圆上同调几何模型的首要目标是连接 Lurie 的 2 等变椭圆中的结构。一些项目是先前在高度 0 和 1 上的工作的自然延伸，重点关注预期构建椭圆 Thom 类的特定量子场论的高度 2 推广。 2-等变几何，与弦几何、环群表示理论和椭圆幂运算等主题相结合。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值进行评估，被认为值得支持以及更广泛的影响审查标准。