Microlocal theory of Legendrian links and cluster algebras

We show the existence of quasicluster Ꮽ-structures and cluster Poisson structures on moduli stacks of sheaves with singular support in the alternating strand diagram of grid plabic graphs by studying the microlocal parallel transport of sheaf quantizations of Lagrangian fillings of Legendrian links.The construction is in terms of contact and symplectic topology, showing that there exists an initial seed associated to a canonical relative Lagrangian skeleton.In particular, mutable cluster Ꮽ-variables are intrinsically characterized via the symplectic topology of Lagrangian fillings in terms of dually Lcompressible cycles.New ingredients are introduced throughout, including the initial weave associated to a grid plabic graph, cluster mutation along nonsquare faces of a plabic graph, possibly including lollipops, the concept of sugar-free hull, and the notion of microlocal merodromy.Finally, we prove the existence of the cluster DT transformation for shuffle graphs, constructing a contact-geometric realization and an explicit reddening sequence, and establish cluster duality for the cluster ensembles.13F60, 53D12 1. Introduction 901 2. Grid plabic graphs and Legendrian links 909 3. Diagrammatic weave calculus and initial cycles 925 4. Construction of quasicluster structures on sheaf moduli 951 5. Cluster DT transformations for shuffle graphs 987