Quantum Topology Beyond Semi-Simplicity

Utah State University, Logan UT

Investigators

Abstract

Physics-inspired mathematics has effectively laid the foundation and provided the terminology for the most advanced areas of modern physics. This project aims to create new algebraic and geometric tools to help formulate ideas and methods of quantum physics in a precise mathematical way. Low-dimensional topology is an area of mathematics that studies three and four dimensional spaces. Theory of quantum groups has been productively used in low-dimensional topology, particularly with the creation of quantum invariants. Within this context, the Principal Investigator (PI) and his collaborators have developed new systematic strategies. The focus of this project is to further investigate and develop these strategies, aiming to exploit the powerful properties of the “renormalized” quantum invariants in several areas of mathematics. The unique attributes of this work open the door to novel research avenues in algebra, topology, geometry, and mathematical physics. The broader impacts are through STEM education, mentoring, and outreach. The PI will advise graduate students and postdocs on projects related to the main objectives of the project. The PI has co-organized many conferences and will continue such outreach activities to develop communication and collaborative research with other mathematicians, as well as to foster broader applications of the work supported by this award. The past thirty years have witnessed a transformative influx of quantum field theory into low-dimensional topology, leading to a novel perspective on link and three-manifold invariants. The discovery of the Jones polynomial by V. Jones in 1984 and its interpretation through three-dimensional quantum field theory by E. Witten in 1989 have paved the way for the application of new algebraic methods to study topology. These advancements have given rise to a new field of mathematics known as "quantum topology”. Most of the theory of quantum invariants involves monoidal categories with certain additional properties, such as being semi-simple. The PI and his collaborators have created a theory of re-normalized quantum invariants (RQIs) of low-dimensional manifolds arising from categories that are not semi-simple. The RQIs have their own unique features and provide mathematical interpretations of Topological Quantum Field Theories (TQFTs) with categories of line operators that are non-semi-simple; furthermore, the RQIs are more powerful than their standard counterparts. This project will explore the nature and physical meaning of the re-normalized Reshetikhin-Turaev three-manifold invariants and their associated TQFTs via certain classes of examples appearing in the context of Chern-Simons theory and vertex operator algebras. It also proposes to enhance and generalize recently defined non-compact skein TQFTs. 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.