Deformed Mirror Symmetry for Punctured Surfaces

TL;DR

This work establishes explicit noncommutative mirror symmetry for punctured surfaces by connecting the A-side deformation $\operatorname{Gtl}_q Q$ with a corresponding B-side deformation $\operatorname{mf}(\operatorname{Jac}_q \check{Q}, \ell_q)$ via a deformed Cho–Hong–Lau construction. It extends Koszul duality machinery to produce a deformed mirror functor $F_q: \operatorname{Gtl}_q Q \to \operatorname{mf}(\operatorname{Jac}_q \check{Q}, \ell_q)$ and proves that $\operatorname{Jac}_q \check{Q}$ is a flat deformation of $\operatorname{Jac} \check{Q}$ under a mild boundedness condition that replaces the usual homogeneity requirement. The paper provides explicit descriptions of the deformed Jacobi algebra and central element for key examples (3-punctured sphere and 4-punctured torus), building on prior deformation-theory work and CHL computations. By demonstrating flatness and constructing the deformed mirror functor, the results realize noncommutative mirror symmetry for punctured surfaces and illustrate persistence of central elements under deformation, with potential implications for relative Fukaya-type structures on the A-side.

Abstract

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.

Figures (31)

• Figure 4.1: Illustration for $\operatorname{Gtl} Q$ and $\operatorname{Gtl}_q Q$
• Figure 4.2: A zigzag path $L$
• Figure 4.3: On consistency
• Figure 4.4: F-term equivalence
• Figure 4.5: Three-punctured sphere and its mirror dimer

Theorems & Definitions (392)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Remark 2.1
• Definition 2.8
• Remark 2.2