I will describe how the moduli space of sheaves microsupported on a Legendrian surface forms a Lagrangian patch in a chart of a Fock-Goncharov cluster variety, and how the structure is compatible with quantization and mutation. The classical geometry conjecturally determines open disk invariants of Lagrangian fillings, while the quantum geometry conjecturally encodes open higher-genus invariants. I will also describe dualities relating quiver invariants and refined Kac polynomials with these numbers. This is joint work with Gus Schrader and Linhui Shen.
