Ribbon Graphs and Mirror Symmetry I

Abstract

Given a ribbon graph $\Gamma$ with some extra structure, we define, using constructible sheaves, a dg category $CPM(\Gamma )$ 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 ${\tilde{X}}_{\Gamma }$ with toric components. We prove that our model is equivalent to $Perf({\tilde{X}}_{\Gamma })$, the dg category of perfect complexes on ${\tilde{X}}_{\Gamma }$.

28 pages

References in corpus (9)

• Higher Topos Theory
• Homogeneous coordinate rings and mirror symmetry for toric varieties
• Homological mirror symmetry for the quartic surface
• Derived Algebraic Geometry III: Commutative Algebra
• The Coherent-Constructible Correspondence and Homological Mirror Symmetry for Toric Varieties
• String Topology: Background and Present State
• The Coherent-Constructible Correspondence for Toric Deligne-Mumford Stacks
• Springer theory via the Hitchin fibration
• Derived Algebraic Geometry VI: E_k Algebras

Cited by in corpus (7)

• Fukaya Categories as Categorical Morse Homology
• Skeleta of Affine Hypersurfaces
• Lagrangian Skeleta of Hypersurfaces in $({\mathbb{C}}^{\ast }{)}^n$
• Cyclic symmetries of A_n-quiver representations
• Polytopes and Skeleta
• Mirror Symmetry in dimension one and Fourier-Mukai equivalences
• Line fields on punctured surfaces and twisted derived categories