Selecta Mathematica

, Volume 20, Issue 4, pp 979–1002 | Cite as

Ribbon graphs and mirror symmetry



Given a ribbon graph \(\Gamma \) with some extra structure, we define, using constructible sheaves, a dg category \(\mathrm {CPM}(\Gamma )\) meant to model the Fukaya category of a Riemann surface in the cell of Teichmüller space described by \(\Gamma .\) When \(\Gamma \) is appropriately decorated and admits a combinatorial “torus fibration with section,” we construct from \(\Gamma \) a one-dimensional algebraic stack \(\widetilde{X}_\Gamma \) with toric components. We prove that our model is equivalent to \(\mathcal {P}\mathrm {erf}(\widetilde{X}_\Gamma )\), the dg category of perfect complexes on \(\widetilde{X}_\Gamma \).


Homological mirror symmetry Ribbon graphs Constructible sheaves 

Mathematics Subject Classification (2010)

32S60 53D37 



We would like to thank Kevin Costello, David Nadler and Dima Tamarkin for discussions around this project. We are greatly indebted to Bohan Fang and Chiu-Chu Melissa Liu for sharing their thoughts, and (for the second- and third-named authors) for our several collaborations with them. The work of E.Z. is supported in part by NSF/DMS-0707064.


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Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Max Planck Institute for MathematicsBonnGermany
  2. 2.Department of MathematicsBoston CollegeChestnut HillUSA
  3. 3.Department of MathematicsNorthwestern UniversityEvanstonUSA

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