Topological Fukaya category and mirror symmetry for punctured surfaces

TL;DR

This paper proves a version of homological mirror symmetry for punctured Riemann surfaces by showing the topological Fukaya category is equivalent to matrix factorizations of the mirror Landau-Ginzburg model, with new gluing results.

Contribution

It establishes the equivalence between the topological Fukaya category and matrix factorizations for punctured surfaces, advancing mirror symmetry understanding.

Findings

Topological Fukaya category of punctured surfaces is equivalent to matrix factorizations of the mirror LG model.

Develops new gluing techniques for the topological Fukaya category.

Provides a framework for homological mirror symmetry in punctured Riemann surfaces.

Abstract

In this paper we establish a version of homological mirror symmetry for punctured Riemann surfaces. Following a proposal of Kontsevich we model A-branes on a punctured surface $\Sigma$ via the topological Fukaya category. We prove that the topological Fukaya category of $\Sigma$ is equivalent to the category of matrix factorizations of the mirror LG model $(X,W)$. Along the way we establish new gluing results for the topological Fukaya category of punctured surfaces which might be of independent interest.