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QFT's for non-semisimple TQFT's

Banff International Research Station for Mathematical Innovation and Discovery

Thirty years ago, work of Witten and Reshetikhin-Turaev activated the study of quantum invariants of links and three-manifolds. A cornerstone of subsequent developments, leading up to our current knot-homology conference, was a three-pronged approach involving 1) quantum field theory (Chern-Simons); 2) rational VOA's (WZW); and 3) semisimple representation theory of quantum groups. The second and third perspectives have since been extended, to logarithmic VOA's and related non-semisimple quantum-group categories. I will propose a family of 3d quantum field theories that similarly extend the first perspective to a non-semisimple (and more so, derived) regime. The 3d QFT's combine Chern-Simons theory with a topologically twisted supersymmetric theory. They support boundary VOA's whose module categories are dual to modules for Feigin-Tipunin algebras and (correspondingly) to modules for small quantum groups at even roots of unity. The QFT is also compatible with deformations by flat connections, related to the Frobenius center of quantum groups at roots of unity. This is joint work with T. Creutzig, N. Garner, and N. Geer. I will mention potential connections to related recent work of Gukov-Hsin-Nakajima-Park-Pei-Sopenko and promising routes to categorification, from a physics perspective.

Mathematics; Category Theory; Homological Algebra; Quantum Theory; Algebraic Topology

video/mp4

Author affiliation: University of Edinburgh

2023-10-23

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/86280

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/