Noncommutative mirror symmetry for punctured surfaces

Details Noncommutative mirror symmetry for punctured surfaces Noncommutative mirror symmetry for punctured surfaces

2011-11-14

arxiv.org/abs/1111.3392v1

Mathematics

Algebraic Geometry

25 pages

Scientific paper

Recently Abouzaid, Auroux, Efimov, Katzarkov and Orlov showed that the wrapped Fukaya Categories 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 $\cA_\infty$-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. Finally, we tie this result to the classical commutative mirror symmetry.

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