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 \(\widetilde{X}_\Gamma \).
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Notes
Locality in the skeleton appeared first in Kontsevich’s [12], but is also a part of a circle of ideas concerning the local nature of the Fukaya category of exact symplectic manifolds, prevalent the work of Abouzaid [1] and Seidel [23, 24] (see also [3]), and in the relation of the Fukaya category to sheaf theory by Nadler [17] and the last author [20].
That is, in the ordinary category of small \(\mathbb {C}\)-linear categories.
For a general treatment of functors induced by contact transformations between microlocal categories of sheaves, see [11, Chapter 7].
The availability of the quotient construction for sheaves of dg categories depends on the existence of Verdier quotients in the dg setting, see [7].
There is a maximal open dense subset \(\Lambda ^{\mathrm{{reg}}} \subset \Lambda \), such that \(\Lambda ^{\mathrm{{reg}}}\) is a possibly non connected smooth submanifold of \(T^*M\): by definition, the set of singular points of \(\Lambda \) is given by \(\Lambda - \Lambda ^{\mathrm{{reg}}}\).
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Acknowledgments
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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Sibilla, N., Treumann, D. & Zaslow, E. Ribbon graphs and mirror symmetry. Sel. Math. New Ser. 20, 979–1002 (2014). https://doi.org/10.1007/s00029-014-0149-7
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DOI: https://doi.org/10.1007/s00029-014-0149-7