Quantum Topology beyond Semi-Simplicity

Inspired by physics, mathematics has successfully provided the background and language for most of the sophisticated areas of modern physics. This project aims to create new algebraic and geometric tools to formulate ideas and methods of quantum physics in a precise mathematical way. Specifically, the theory of quantum groups associated with Lie algebras has been widely and productively used in low-dimensional topology, and in particular, with the creation of quantum invariants.

Within this context, the Principal Investigator (PI) and his collaborators have offered new systematic strategies to define re-normalized quantum invariants arising from non-semi-simple categories. The focus of this grant is to develop a framework for many of the quantum invariants coming from both semi-simple and non-semi-simple categories. The unique properties of such a framework opens the door to new research avenues in algebra, topology, geometry, mathematical physics, and related areas of mathematics.

The broader impacts of this grant address two main categories: mentoring and outreach. Throughout the project, the PI will advise graduate students and postdocs on projects related to the main objectives of the grant. The PI has co-organized many conferences and a workshop and will continue such outreach with the aim of developing communication and collaborative research with other mathematicians, as well as fostering broader applications of the work of this grant.

This grant considers four types of invariants arising from non-semi-simple categories: Hennings, Kuperberg, Reshetikhin-Turaev re-normalized Quantum Invariants (RQIs) and Turaev-Viro RQIs. All of these invariants have unique features and strengths. In particular, Hennings and Kuperberg invariants are defined using the structure of Hopf algebras (and thus have many examples) but have certain vanishings which do not allow extensions to full TQFTs; the RQIs are quite powerful and their definitions are based on representation theory making their existence for some significant examples elusive.

This project will generalize and re-normalize both Hennings and Kuperberg invariants and show that the new invariants lead to TQFTs with additional data (overcoming the vanishing obstruction). Furthermore, the PI will construct a general framework which relates the Hennings and Kuperberg type invariants to the previously defined Reshetikhin-Turaev and Turaev-Viro RQIs, allowing a unification of the strengths of all four invariants.

The PI will use this framework to advance and draw connections with Chern-Simons theory with complex (super) groups and Levin-Wen models.

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.