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 $\operatorname{Gtl} Q \cong \operatorname{mf} (\operatorname{Jac} \check{Q}, \ell)$ for a punctured surface $Q$. We pick a deformation $\operatorname{Gtl}_q Q$ which captures a large part of the deformation theory and includes the relative Fukaya category. To find the corresponding deformation of $\operatorname{mf} (\operatorname{Jac} \check{Q}, \ell)$, we deform work of Cho-Hong-Lau which interprets mirror symmetry as Koszul duality. As result we explicitly obtain the corresponding deformation $\operatorname{mf} (\operatorname{Jac}_q \check{Q}, \ell_q)$ together with a deformed mirror functor $\operatorname{Gtl}_q Q \xrightarrow{\sim} \operatorname{mf} (\operatorname{Jac}_q \check{Q}, \ell_q)$. The bottleneck is to verify that the algebra $\operatorname{Jac}_q \check{Q}$ is indeed a (flat) deformation of $\operatorname{Jac} \check{Q}$. 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 $\operatorname{Jac}_q \check{Q}$ 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 arXiv:2305.09112, we describe $\operatorname{Jac}_q \check{Q}$ explicitly. It turns out that the deformed potential $\ell_q$ is still central in $\operatorname{Jac}_q \check{Q}$, in contrast to the popular slogan that central elements do not survive under deformation.