A Lagrangian filling for every cluster seed

Abstract

We show that each cluster seed in the augmentation variety is inhabited by an embedded exact Lagrangian filling. This resolves the matter of surjectivity of the map from Lagrangian fillings to cluster seeds. The main new technique to produce these Lagrangian fillings is the construction and study of a quiver with potential associated to curve configurations. We prove that its deformation space is trivial and show how to use it to manipulate Lagrangian fillings with $\mathbb{L}$-compressing systems via Lagrangian disk surgeries.

arXiv comment45 pages, 40 figures

References in corpus (18)

• Cluster algebras II: Finite type classification
• Legendrian knots and constructible sheaves
• Cluster varieties from Legendrian knots
• Duality between Lagrangian and Legendrian invariants
• Sheaf quantization of Hamiltonian isotopies and applications to non displaceability problems
• Legendrian Fronts for Affine Varieties
• Knot contact homology
• Legendrian Weaves: N-graph Calculus, Flag Moduli and Applications
• Morsifications and mutations
• Infinitely many monotone Lagrangian tori in ${\mathbb{R}}^6$
• Skeleta of Affine Hypersurfaces
• Mutation of frozen Jacobian algebras
• Domain wall formation and magnon localization in twisted chromium trihalides
• Weave Realizability for D-type
• Computing the Geometric Intersection Number of Curves
• Checkerboard graph monodromies
• Lagrangian Fillings in A-type and their Kalman Loop Orbits
• The valuation pairing on an upper cluster algebra