Topological Quantum Field Theory

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

In the 1980s, mathematics received an incredible influx of ideas coming from physics that had tremendous mathematical interest and applications. Topology is a field of mathematics that studies properties of shapes which are preserved under deformations such as stretching, crumpling, and bending, but not tearing or gluing. Quantum field theories that only depend on topological properties of space, and not geometric ones such as length and angle, became a field of mathematics, known as topological field theory (TFT). This project constitutes a multifaceted investigation of TFTs in dimensions three, four, and higher, using new techniques which have only been fully developed in the past decade. The problems considered are of a foundational nature, and answers to these are expected to lead to new techniques and interactions among different fields of mathematics and have potential applications in physics. Graduate students are an important part of this research program; graduate education, mentoring as well as dissemination of the results to the broader community of scientists via lectures and workshops is a key broader impact of the project. The PI's program will initially focus on recent advances relating TFT to the classification of smooth manifolds up to stable diffeomorphism. Special emphasis will be placed on 4-dimensional aspects and obtaining new examples of TFTs that distinguish homotopy equivalent manifolds. Another goal is to construct non-semisimple TFTs. Unlike the currently known examples, these have the potential to distinguish exotic smooth structures. A long-term goal of the program is to give a proof of the relative tangle hypothesis. This is the main missing part of Lurie’s proof of the cobordism hypothesis and fills a major gap in the proof of this important result. 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.