Holomorphic Floer Theory and the Fueter Equation

We outline a proposal for a $2$-category $\mathrm{Fuet}_M$ associated to a hyperk\"ahler manifold $M$, which categorifies the subcategory of the Fukaya category of $M$ generated by complex Lagrangians. Morphisms in this $2$-category are formally the Fukaya--Seidel categories of holomorphic symplectic action functionals. As such, $\mathrm{Fuet}_M$ is based on counting maps to $M$ satisfying the Fueter equation with boundary values on holomorphic Lagrangians. We make the first step towards constructing this category by establishing some basic analytic results about Fueter maps, such as the energy bound and maximum principle. When $M=T^*X$ is the cotangent bundle of a K\"ahler manifold $X$ and $(L_0, L_1)$ are the zero section and the graph of the differential of a holomorphic function $F: X \to \mathbb{C}$, we prove that all Fueter maps correspond to the complex gradient trajectories of $F$ in $X$, which relates our proposal to the Fukaya--Seidel category of $F$. This is a complexification of Floer's theorem on pseudo-holomorphic strips in cotangent bundles. Throughout the paper, we suggest problems and research directions for analysts and geometers that may be interested in the subject.