Topological Fukaya category and mirror symmetry for punctured surfaces

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 Σ via the topological Fukaya category. We prove that the topological Fukaya category of σ is equivalent to the category of matrix factorizations of a certain mirror LG model (X,W). Along the way we establish new gluing results for the topological Fukaya category of punctured surfaces which are of independent interest.