On homological mirror symmetry of toric Calabi-Yau three-folds

TL;DR

This paper explores the homological mirror symmetry for toric Calabi-Yau threefolds by constructing Lagrangian objects on the mirror and proposing a categorical correspondence with coherent sheaves.

Contribution

It provides explicit constructions of Lagrangian sections and spheres on the mirror and conjectures their categorical equivalence with sheaves on the original Calabi-Yau.

Findings

Constructed Lagrangian sections and spheres on the mirror.

Proposed explicit correspondence with line bundles and sheaves.

Conjectured embedding of Fukaya category into derived category.

Abstract

We use Lagrangian torus fibrations on the mirror $X$ of a toric Calabi-Yau threefold $\check X$ to construct Lagrangian sections and various Lagrangian spheres on $X$. We then propose an explicit correspondence between the sections and line bundles on $\check X$ and between spheres and sheaves supported on the toric divisors of $\check X$. We conjecture that these correspondences induce an embedding of the relevant derived Fukaya category of $X$ inside the derived category of coherent sheaves on $\check X$.