Homological mirror symmetry for the 4-torus

Abstract

We use the quilt formalism of Mau, Wehrheim, and Woodward to give a sufficient condition for a finite collection of Lagrangian submanifolds to split-generate the Fukaya category, and deduce homological mirror symmetry for the standard $4$-torus. As an application, we study Lagrangian genus $2$ surfaces ${\Sigma }_2\subset T^4$ of Maslov class zero, deriving numerical restrictions on the intersections of ${\Sigma }_2$ with linear Lagrangian $2$-tori in $T^4$