On the combinatorics of exact Lagrangian surfaces

Abstract

We study Weinstein 4-manifolds which admit Lagrangian skeleta given by attaching disks to a surface along a collection of simple closed curves. In terms of the curves describing one such skeleton, we describe surgeries that preserve the ambient Weinstein manifold, but change the skeleton. The surgeries can be iterated to produce more such skeleta --- in many cases, infinitely many more. Each skeleton is built around a Lagrangian surface. Passing to the Fukaya category, the skeletal surgeries induce cluster transformations on the spaces of rank one local systems on these surfaces, and noncommutative analogues of cluster transformations on the spaces of higher rank local systems. In particular, the problem of producing and distinguishing such Lagrangians maps to a combination of combinatorial-geometric questions about curve configurations on surfaces and algebraic questions about exchange graphs of cluster algebras. Conversely, this expands the dictionary relating the cluster theory of character varieties, positroid strata, and related spaces to the symplectic geometry of Lagrangian fillings of Legendrian knots, by incorporating cluster charts more general than those associated to bicolored surface graphs.

arXiv comment56 pages

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Cited by in corpus (14)

• Cluster varieties from Legendrian knots
• Microlocal Morse theory of wrapped Fukaya categories
• The nonequivariant coherent-constructible correspondence for toric stacks
• The conormal torus is a complete knot invariant
• Homological mirror symmetry for log Calabi-Yau surfaces
• Maximal contact and symplectic structures
• Calabi-Yau structures on topological Fukaya categories
• Immersed two-spheres and SYZ with Application to Grassmannians
• String topology with gravitational descendants, and periods of Landau-Ginzburg potentials
• Sheaf quantization in Weinstein symplectic manifolds
• Infinitely many Lagrangian fillings
• Mirror Symmetry for Truncated Cluster Varieties
• Twisted Polytope Sheaves and Coherent-Constructible Correspondence for Toric Varieties
• Asymptotic behavior of Vianna's exotic Lagrangian tori $T_{a,b,c}$ in ${\mathbb{CP}}^2$ as $a+b+c\to \infty$