Categorical aspects of the Fueter Equation

Abstract: The (3d) Fueter equation is the three-dimensional analog of the pseudoholomorphic map equation, and as such underlies a three-dimensional topological quantum field theory. This PDE underlies the mathematics of the A-type twist of the 3D N=4 sigma model, which has a hyperkahler manifold as its target. One can think of this topological quantum field theory as a simultaneous complexification and categorification of the Fukaya category; in particular, it assigns to a ("weak") hyperkahler manifold a 2-category with objects holomorphic Lagrangians, which in an appropriate sense categorifies the Fukaya category. Certain basic open problems remain about the analysis of the Fueter equation, but this categorical viewpoint suggests new tractable directions in the differential geometry of this equation. In particular, just as holomorphic strips between nearby Lagrangans are in bijection with Morse trajectories of a real morse function, Fueter maps between nearby holomorphic Lagrangians are in bijection with complex gradient trajectories of a holomorphic morse function, also known as zeta-instantons. Thus, in the (A-twist) Fueter 2-category, hom-categories are locally modeled on Fukaya-Seidel categories, just as in the B-twist Kapustin-Rozansky-Saulinas category, hom-categories are locally modeled on matrix factorization categories. Categorical 3D mirror symmetry should exchange these pairs of 2-categories associated to pairs of 3d mirror manifolds. I will survey these ideas and describe interesting directions and puzzles in this story. This is based on joint work with Aleksander Doan, as well as on discussions with Justin Hilburn and Benjamin Gammage.

algebraic geometrysymplectic geometry

Audience: researchers in the topic

M-seminar