Homological mirror symmetry for the four-torus

TL;DR

This paper proves homological mirror symmetry for the 4-torus by establishing conditions for Lagrangian submanifolds to generate the Fukaya category, and explores restrictions on genus two Lagrangian surfaces within the torus.

Contribution

It introduces a sufficient condition for generating the Fukaya category and applies it to prove mirror symmetry for the 4-torus, also analyzing genus two Lagrangian surfaces.

Findings

Homological mirror symmetry established for the 4-torus.

Derived numerical restrictions on intersections of genus two Lagrangian surfaces.

Provided a framework for Lagrangian generation in Fukaya categories.

Abstract

We use the quilt formalism of Mau-Wehrheim-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 two surfaces of Maslov class zero, deriving numerical restrictions on the intersections of such a surface with linear Lagrangian 2-tori in in the 4-torus.