Selecta Mathematica

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

Ribbon graphs and mirror symmetry

Article

Abstract

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 \(\widetilde{X}_\Gamma \).

Keywords

Homological mirror symmetry Ribbon graphs Constructible sheaves 

Mathematics Subject Classification (2010)

32S60 53D37 

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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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