Unifying TQFTs for gl(1,1) and the Alexander polynomial

This project will investigate the math behind certain quantum-mechanical phenomena that are topological in nature. These phenomena are in the same general ballpark as topological insulators (states of matter with exotic electronic properties that are of modern interest in materials science) and the topological framework for quantum computing. The specific quantum theories in question connect directly to a classical object in the mathematical study of knotted loops of string, called the Alexander polynomial and first introduced in the 1920s. Quantum theories connected to the Alexander polynomial are expected to be easier to understand and to shed light on related quantum theories, e.g. the Chern-Simons theories that are used for topological quantum computing. Preliminary work leading up to this project has indicated that a well-known 1990s-era quantum theory for the Alexander polynomial should satisfy better structural properties than has previously been supposed. This project will elucidate these structural properties and use them to advance our understanding of related quantum theories. This project provides research training opportunities for graduate students.

In more technical detail, the project will study the Frohman-Nicas topological quantum field theory (TQFT) for the Alexander polynomial from the point of view of decategorified Heegaard Floer homology. On the Heegaard Floer side, the most general formulations of the theory involve a topological construction called sutured 3-manifolds, but the connection between sutured 3-manifolds and the Frohman-Nicas TQFT, or related Chern-Simons TQFTs, has been largely unexplored up to this point. This project will generalize and reinterpret the Frohman-Nicas TQFT in the setting of sutured 3-manifolds, establishing better functoriality properties than the ones that hold in the non-sutured setting. It will also develop an elaboration of this sutured Frohman-Nicas TQFT that is sensitive to Spin-c structures on 3-manifolds, and work out how the result relates to the type of Spin-c decorated TQFT that plays a prominent role in modern work of Gukov, Putrov, Vafa, and others on nonsemisimple 3d TQFT in the Chern-Simons context. Going beyond the sutured setting, it will show how the sutured Frohman-Nicas TQFT arises from a more fundamental construction assigning a certain representation category to a point.

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.

Award Number: 2502205
Principal Investigator: Andrew Manion
Funds Obligated: $120,250
State: NC