Homological mirror symmetry for the four-torus

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.

56 pages, 5 figures. The new version has been completely reorganised and includes longer explanations in the algebraic sections

References in corpus (2)

• Remarks on A-branes, Mirror Symmetry, and the Fukaya category
• The Z-graded symplectic Floer cohomology of monotone Lagrangian sub-manifolds

Cited by in corpus (10)

• Homological Mirror Symmetry for Calabi-Yau hypersurfaces in projective space
• Fukaya categories of the torus and Dehn surgery
• Homological mirror symmetry for the quintic 3-fold
• The Lagrangian cobordism group of $T^2$
• The closed-open string map for $S^1$-invariant Lagrangians
• Categorical mirror symmetry on cohomology for a complex genus 2 curve
• An extension of the Siegel space of complex abelian varieties and conjectures on stability structures
• Versality in mirror symmetry
• Homological mirror symmetry for local Calabi-Yau manifolds via SYZ
• Rigidity of Lagrangian embeddings into symplectic tori and K3 surfaces