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Braided tensor categories and the cobordism hypothesis

Banff International Research Station for Mathematical Innovation and Discovery

In work with David Ben-Zvi and Adrien Brochier, we introduced a (would-be) 4-D topological field theory which relates to N=4 d=4 SYM in the same way that the Reshetikhin-Turaev 3-D theory relates to Chern-Simons theory. On surfaces it assigns certain explicit categories quantizing quasi-coherent sheaves on the character variety of the surface (along the Atiyah-Bott/ Goldman/Fock-Rosly Poisson bracket), and these in turn relate to many well-known constructions in quantum algebra. The parenthetical "would be" above means that, while the theory had an a priori definition on *surfaces* via factorization homology -- due to work of Ayala-Francis, Lurie, and Scheimbauer, these techniques do not apply to 3- and 4-manifolds. In this talk I'll explain work with Adrien Brochier and Noah Snyder, which constructs the 3-manifold invariants following the prescription of the cobordism hypothesis. This is in the spirit of Douglas-Schommer-Pries-Snyder's work on finite tensor categories -- but in the infinite setting -- and also echoes early ideas of Lurie and Walker. The resulting 3-manifold invariants quantize Lagrangians in the character variety of the boundary. They are not at all well-understood or computed explicitly in general, but they appear phenomenologically to relate to many emerging structures, such as quantum A-polyonomials, DAHA-Jones polynomials, and Khovanov-Rozansky knot homologies.

Mathematics; Category theory; homological algebra; Quantum theory; Mathematical physics

video/mp4

Author affiliation: University of Edinburgh

2019-03-12

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/68656

Unreviewed

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