Homological mirror symmetry of $\mathbb{F}_1$ via Morse homotopy

TL;DR

This paper extends the homological mirror symmetry framework for toric manifolds by utilizing Morse homotopy of moment polytopes, specifically applying it to the Hirzebruch surface _1, building on previous work for projective spaces.

Contribution

It introduces an extension of Morse homotopy-based homological mirror symmetry to the Hirzebruch surface _1, broadening the class of toric manifolds covered by this approach.

Findings

Established homological mirror symmetry for _1 using Morse homotopy.

Extended the Morse homotopy framework from projective spaces to Hirzebruch surfaces.

Validated the approach via Strominger-Yau-Zaslow construction.

Abstract

This is a sequel to our paper arXiv:2008.13462, where we proposed a definition of the Morse homotopy of the moment polytope of toric manifolds. Using this as the substitute of the Fukaya category of the toric manifolds, we proved a version of homological mirror symmetry for the projective spaces and their products via Strominger-Yau-Zaslow construction of the mirror dual Landau-Ginzburg model. In this paper we go this way further and extend our previous result to the case of the Hirzebruch surface $\mathbb{F}_1$.