I will describe work in progress with Linhui Shen.

The moduli space of Borel-decorated local systems on a punctured surface is a cluster variety with a quantization, after Fock and Goncharov. It contains a (quantum) Lagrangian subspace, per Dimofte-Gabella-Goncharov.

We visit this construction from the perspective of constructible sheaf theory, a version of the Fukaya category. The Lagrangian subspace is the moduli of branes in complex three-space. Following works of Vafa with Aganagic, Hori, Aganagic and Klemm, and Ooguri, we learn that, semi-classically, the function defining the moduli space is a superpotential encoding open Gromov-Witten invariants.

Quantum-mechanically, all-genus invariants are conjecturally encoded in what can be called the wavefunction of the brane.

The talk will outline the mathematics of this perspective. I will show how this works in some simple examples and outline a broader computational scheme. I will also clarify the role of “framings,” and describe a duality with invariants of symmetric quivers without potential.