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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Noncommutative mirror symmetry for punctured surfaces
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by Raf Bocklandt; With an appendix by Mohammed Abouzaid PDF
Trans. Amer. Math. Soc. 368 (2016), 429-469 Request permission

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.
References
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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
  • Email: raf.bocklandt@gmail.com
  • 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
  • © Copyright 2015 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 368 (2016), 429-469
  • MSC (2010): Primary 16G20, 14J33
  • DOI: https://doi.org/10.1090/tran/6375
  • MathSciNet review: 3413869