Deformed mirror symmetry for punctured surfaces

Mirror symmetry originally envisions a correspondence between deformations of the A-side and deformations of the B-side. In this paper, we achieve an explicit correspondence in the case of punctured surfaces. The starting point is the noncommutative mirror equivalence Gtl Q≅mf(Jac Qˇ,ℓ) for a punctured surface Q. We pick a deformation Gtl_q Q which captures a large part of the deformation theory and includes the relative Fukaya category. To find the corresponding deformation of mf(Jac Qˇ,ℓ), we deform work of Cho-Hong-Lau which interprets mirror symmetry as Koszul duality. As result we explicitly obtain the corresponding deformation mf(Jac_q Qˇ,ℓ_q) together with a deformed mirror functor Gtl_q Q→∼mf(Jac_q Qˇ,ℓ_q). 

The bottleneck is to verify that the algebra Jac_q Qˇ is indeed a (flat) deformation of Jac Qˇ. We achieve this by deploying a result of Berger-Ginzburg-Taillefer on deformations of CY3 algebras, which however requires the relations to be homogeneous. We show how to replace this homogeneity requirement by a simple boundedness condition and obtain flatness of Jac_q Qˇ for almost all Q. 

We finish the paper with examples, including a full treatment of the 3-punctured sphere and 4-punctured torus. With the help of our computations in [24], we describe Jac_qQˇ explicitly. It turns out that the deformed potential ℓ_q is still central in Jac_q Qˇ, in contrast to the popular slogan that central elements do not survive under deformation.