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 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. a punctured surface Σ, there is an equivalenceWe will refer to F top (X) as the topological Fukaya category of Σ, and we denote it F uk top (Σ). In this paper we take F uk top (Σ) as a model for the category of A-branes on Σ. We prove homological mirror symmetry for punctured Riemann surfaces by showing that F uk top (Σ) is equivalent to the category of B-branes on a mirror geometry LG model. Hori-Vafa homological mirror symmetryLet us review the setting of Hori-Vafa mirror symmetry for LG models [HV, GKR]. Let X be a toric threefold with trivial canonical bundle. The fan of X can be realized as a smooth subdivision of the cone over a two-dimensional lattice polytope, see Section 3.1.1 for more details. The height function on the fan of X gives rise to a regular mapwhich is called the superpotential. The category of B-branes for the LG model (X, W ) is the Z 2 -graded category of matrix factorizations M F (X, W ). The mirror of the LG-model (X, W ) is a smooth algebraic curve Σ W in C * × C * , called the mirror curve (see Section 3). The following is our main result.Theorem 1.1 Hori-Vafa homological mirror symmetry. There is an equivalenceTheorem 1.1 provides a proof of homological mirror symmetry for punctured surfaces, provided that we model the category of A-branes via the topological Fukaya category. This extends to all genera earlier results for curves of genus zero and one which were obtained in [STZ] and [DK]. We also mention work of Nadler, who studies both directions of Hori-Vafa mirror symmetry for higher dimensional pairs of pants [N3, N4, N5].We learnt the statement of Hori-Vafa homological mirror symmetry for punctured surfaces from the inspiring paper [AAEKO]. In [AAEKO] the authors prove homological mirror symmetry for punctured spheres. Their main theorem is parallel to our own (in genus zero) with the important difference that they work with the wrapped Fukaya category, rather than with its topological model. See also related work of Bocklandt [B]. Mirror symmetry for higher-dimensional pairs of pants was studied by Sheridan in [Sh].Denote F uk wr (Σ) the wrapped Fukaya category of a punctured surface Σ. Our main result combined with the main result of [AAEKO] gives equivalencesfor all Riemann surfaces Σ W which can be realized as unramified cyclic covers of punctured spheres. Thus, for this class of examples, the topological Fukaya category captures the wrapped Fukaya category, corroborating Kontsevich's proposal. We also remark that a proof of the equivalence between topological and wrapped Fukaya categor...

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