Noncommutative mirror symmetry for punctured surfaces

TL;DR

This paper extends noncommutative mirror symmetry results from punctured spheres to general punctured Riemann surfaces, linking dimer models with wrapped Fukaya categories through noncommutative algebraic structures.

Contribution

It generalizes the mirror symmetry correspondence to broader surfaces and introduces a duality on dimer models connecting algebraic and symplectic categories.

Findings

Wrapped Fukaya categories are equivalent to categories of noncommutative matrix factorizations.

A duality on dimer models explains the connection between algebraic and geometric categories.

The approach applies to any consistent dimer model and associated punctured Riemann surface.

Abstract

Recently Abouzaid, Auroux, Efimov, Katzarkov and Orlov showed that the wrapped Fukaya Categories of punctured spheres and finite unbranched covers of punctured spheres are derived equivalent to the categories of singularities of a superpotential on certain crepant resolutions of toric 3 dimensional singularities. We generalize this result to other punctured Riemann surfaces and reformulate it in terms of certain noncommutative algebras coming from dimer models. In particular, given any consistent dimer model we can look at a subcategory of noncommutative matrix factorizations and show that this category is $A_\infty$-isomorphic to a subcategory of the wrapped Fukaya category of a punctured Riemann surface. The connection between the dimer model and the punctured Riemann surface then has a nice interpretation in terms of a duality on dimer models.