Morse Homology, Tropical Geometry, and Homological Mirror Symmetry for Toric Varieties

TL;DR

This paper constructs an A-infinity category for a toric variety's mirror and proves its equivalence to the category of line bundles, advancing the Homological Mirror Symmetry conjecture for toric varieties.

Contribution

It establishes a quasi-equivalence between the Fukaya category of Lagrangians and the derived category of line bundles for toric varieties.

Findings

Constructed an A-infinity category for the Landau-Ginzburg mirror

Proved the category is quasi-equivalent to line bundles on the toric variety

Advances the proof of the Homological Mirror Symmetry conjecture for toric varieties

Abstract

Given a smooth projective toric variety X, we construct an A-infinity category of Lagrangians with boundary on a level set of the Landau-Ginzburg mirror of X. We prove that this category is quasi-equivalent to the DG category of line bundles on X. This establishes part of the Homological Mirror Conjecture for toric varieties.