Noncommutative mirror symmetry for punctured surfaces
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 -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.
Cite
@article{arxiv.1111.3392,
title = {Noncommutative mirror symmetry for punctured surfaces},
author = {Raf Bocklandt},
journal= {arXiv preprint arXiv:1111.3392},
year = {2013}
}
Comments
34 pages. In the new version circumvents lemma 4.3 from the old version (which did not hold for all dimers) by using covers and suitable Z-gradings. It also discusses the relation with the wrapped Fukaya category in more detail and has an extra appendix on wrapped Fukaya category by Mohammed Abouzaid on this topic