CAREER: Elliptic cohomology and quantum field theory

该奖项的研究处于理论物理和几何之间的交界处。一个尚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存在着深刻的联系。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学在物理学中期待已久的几个应用。PI将致力于的项目利用更高的分类对称性，以获得对这一30年来的猜想的新见解。该奖项支持在国际和平研究所工作的研究生，他们的研究将对这一领域做出贡献。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。拟议的研究集中在Stolz和Teichner的椭圆上同调猜想几何模型的等变精化上，该模型来自二维超对称场论。首要目标是将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联系起来。其中一些项目是之前在高度0和1的工作的自然延伸，专注于特定量子场论的高度2概括，预计将构建椭圆形Thom类。其他项目将启动2等变几何的研究，与弦几何、循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优势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.