LEAPS-MPS: Quantum Field Theories and Elliptic Cohomology

Providence College, Providence RI

Investigators

Abstract

The dialog between mathematics and physics has a long history of yielding insights in both disciplines, often revealing deep, unifying structure, such as in general relativity and Riemannian geometry. Topology is the study of shapes; in particular, topology asks about properties of shapes that remain unchanged as the shape is smoothly transformed. Over the past century, advances in quantum physics, especially condensed matter physics and superstring theory, depended on innovations in topology. One such innovation is factorization algebras, a mathematical model for the observables of a (classical or quantum) field theory. This project uses factorization algebras to explore a conjectured relationship between a certain type of quantum field theory and an object in topology called elliptic cohomology. The precise structure of this relationship has remained elusive for decades; a resolution to this conjecture would provide insight into both a geometric interpretation of elliptic cohomology and foundational questions in quantum physics related to string theory. This award also supports a regional conference on topology and mathematical physics, a distinguished lecture series, and research opportunities for undergraduates at the PI’s institution, both increasing participation of underrepresented minorities in mathematics and enhancing the research environment of the PI’s institution. More specifically, this project explores the relationship between equivariant factorization algebras, supersymmetric twisted field theories, and elliptic cohomology, with the goal of giving insight into a geometric understanding of the latter. The first component of the project involves analyzing the relationship between smoothly equivariant factorization algebras and supersymmetric twisted functorial field theories, using tools of higher operads. Supersymmetric twisted field theories have been used to bridge mathematical physics and cohomology theories; in low dimensions there are known differential geometric descriptions of these field theories using deRham cohomology and K-theory. In the next supersymmetric dimension, twisted field theories are conjecturally related to a generalized elliptic cohomology theory, topological modular forms (TMF). The conjectured relationship has been investigated for over thirty years but remains unresolved. Having a description of these supersymmetric field theories as equivariant factorization algebras would reframe the conjecture, allowing more direct use of examples of field theories from physics. The second component of the project involves looking at a categorification of principal bundles, where the symmetries are given by a smooth 2-group. Principal bundles for smooth 2-groups are also related to questions in elliptic cohomology, since the string group (an example of a smooth 2-group arising as a categorical central extension) gives orientation data for TMF. This project is jointly funded by the LEAPS-MPS program and the Established Program to Stimulate Competitive Research (EPSCoR). 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.