Homological mirror symmetry for open Riemann surfaces from pair-of-pants decompositions

TL;DR

This paper establishes a homological mirror symmetry correspondence for punctured Riemann surfaces by reconstructing their wrapped Fukaya categories from pair-of-pants decompositions, linking symplectic and algebraic geometry.

Contribution

It provides a method to compute the wrapped Fukaya category of punctured Riemann surfaces using pair-of-pants decompositions and constructs the mirror Landau-Ginzburg models from local affine pieces.

Findings

Wrapped Fukaya categories reconstructed from pairs of pants.

Matching of $A_ Infty$-structures across decompositions.

Mirror Landau-Ginzburg models built from local affine mirrors.

Abstract

Given a punctured Riemann surface with a pair-of-pants decomposition, we compute its wrapped Fukaya category in a suitable model by reconstructing it from those of various pairs of pants. The pieces are glued together in the sense that the restrictions of the wrapped Floer complexes from two adjacent pairs of pants to their adjoining cylindrical piece agree. The $A_\infty$-structures are given by those in the pairs of pants. The category of singularities of the mirror Landau-Ginzburg model can also be constructed in the same way from local affine pieces that are mirrors of the pairs of pants.