CAREER: Elliptic cohomology and quantum field theory

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.