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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Noncommutative mirror symmetry for punctured surfaces

Author: Raf Bocklandt; With an appendix by Mohammed Abouzaid
Journal: Trans. Amer. Math. Soc. 368 (2016), 429-469
MSC (2010): Primary 16G20, 14J33
Published electronically: April 3, 2015
MathSciNet review: 3413869
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Abstract: In 2013, 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 $\mathtt {A}_\infty$-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.

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Additional Information

Raf Bocklandt
Affiliation: Korteweg de Vries institute, University of Amsterdam (UvA), Science Park 904, 1098 XH Amsterdam, The Netherlands

Received by editor(s): December 20, 2011
Received by editor(s) in revised form: February 4, 2013, and November 12, 2013
Published electronically: April 3, 2015
Article copyright: © Copyright 2015 American Mathematical Society