Cluster Algebras and Exact Lagrangian Surfaces

We explain a general relationship between cluster theory and the classification of exact Lagrangian surfaces in Weinstein 4-manifolds. A key point is the introduction of an operation on singular Lagrangian skeleta which geometrizes the notion of quiver mutation. This lets us produce large classes of exact Lagrangians labeled by clusters in an associated cluster algebra. When the manifold in question is a cotangent bundle and the exact Lagrangians fill a suitable Legendrian knot in its contact boundary, the microlocalization theory of Kashiwara-Schapira recovers the cluster structures on positroid strata and moduli spaces of local systems from this symplectic paradigm. This is joint with Vivek Shende and David Treumann, part of which is also joint with Eric Zaslow

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