Topological Fukaya category and mirror symmetry for punctured 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$\unicode[STIX]{x1D6F4}$via the topological Fukaya category. We prove that the topological Fukaya category of$\unicode[STIX]{x1D6F4}$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.