On exceptional collections of line bundles and mirror symmetry for toric Del-Pezzo surfaces

TL;DR

This paper explores the connection between exceptional collections of line bundles on toric Del-Pezzo surfaces and mirror symmetry, using Landau-Ginzburg models and tropical Fukaya categories to establish a correspondence.

Contribution

It constructs a map from the critical points of the Landau-Ginzburg potential to the tropical Fukaya category, linking algebraic and symplectic geometry in mirror symmetry for toric Del-Pezzo surfaces.

Findings

Established a correspondence between critical points and exceptional collections.

Described a map linking Landau-Ginzburg critical points to tropical Fukaya categories.

Provided evidence for homological mirror symmetry in this setting.

Abstract

Let $X$ be a toric Del-Pezzo surface and let $Crit(W) \subset (\mathbb{C}^{\ast})^n$ be the solution scheme of the Landau-Ginzburg system of equations. Denote by $X^{\circ}$ the polar variety of $X$. Our aim in this work is to describe a map $L : Crit(W) \rightarrow Fuk_{trop}(X^{\circ})$ whose image under homological mirror symmetry corresponds to a full strongly exceptional collection of line bundles.