Elliptic Cohomology, Geometry, and Physics

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

In the last forty years, ideas from theoretical physics have frequently driven important advances in pure mathematics. The story often begins with calculations in seemingly unrelated physical and mathematical theories that by coincidence give the same answer. Careful investigations then reveal a deep structure linking the disciplines. Such discoveries open new vistas and prompt novel lines of research. A nascent example of this phenomenon relates a certain type of quantum field theory to an object in algebraic topology called elliptic cohomology. The deep structure explaining the relationship between these areas remains elusive. However, the expectation is that the answer will illuminate important objects in several branches of mathematics and physics. In particular, a resolution should provide insight into foundational questions in string theory, while also revealing the geometric meaning of elliptic cohomology. The projects will develop new tools to study this problem. The award supports graduate students working with the PI whose research will contribute to this area. The PI will continue mentoring and advising activities along with his involvement in mathematics education for incarcerated people through the Education Justice Project in Illinois. The main goal of the work is to leverage 2-equivariant elliptic cohomology to reduce the key questions to the study of a few structures in geometry and quantum field theory. The methods utilize a filtration on field theories mimicking the chromatic filtration, allowing one to break the larger problem into smaller and more manageable pieces. Success in lower stages of the filtration already offers several long-anticipated geometric applications of elliptic cohomology, as well as new deformation invariants of quantum field theories. Prior work shows that the first step of the filtration affords a geometric model for equivariant elliptic cohomology over the complex numbers. The proposed research builds upon this existing model and moves beyond characteristic zero. 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.