Homological mirror symmetry for punctured spheres

TL;DR

This paper proves a case of homological mirror symmetry by establishing an equivalence between the wrapped Fukaya category of punctured spheres and the category of singularities of a mirror Landau-Ginzburg model, revealing deep geometric correspondences.

Contribution

It demonstrates homological mirror symmetry for punctured spheres, linking symplectic and algebraic categories through explicit equivalences and fractional grading analysis.

Findings

Wrapped Fukaya category is equivalent to the category of singularities.

Cyclic covers correspond to orbifold quotients on the mirror side.

Supports the homological mirror symmetry conjecture for punctured spheres.

Abstract

We prove that the wrapped Fukaya category of a punctured sphere ($S^2$ with an arbitrary number of points removed) is equivalent to the triangulated category of singularities of a mirror Landau-Ginzburg model, proving one side of the homological mirror symmetry conjecture in this case. By investigating fractional gradings on these categories, we conclude that cyclic covers on the symplectic side are mirror to orbifold quotients of the Landau-Ginzburg model.