CAREER: Elliptic cohomology and quantum field theory

本奖项的研究处于理论物理与几何的交叉领域。一个未解决的猜想假设了超对称量子场论的几何与代数拓扑中的某些结构之间的深刻联系。解决这一猜想将为量子场论的数学基础提供新的见解，同时也提供了代数拓扑在物理学中几个期待已久的应用。PI的项目将利用更高的分类对称性来获得这个30年猜想的新见解。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还将通过伊利诺斯州的教育正义项目继续参与囚犯的数学教育。本文的研究重点是对Stolz和Teichner在二维超对称场论中提出的椭圆上同调猜想几何模型的等变改进。总体目标是将Lurie的2等变椭圆上同调中的结构与超对称规范理论的几何结构联系起来。一些项目是先前高度0和高度1工作的自然延伸，重点是高度2的特定量子场论的推广，这些理论有望构建椭圆Thom类。其他项目将开始研究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。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.