I will introduce a family of moduli spaces associated with a quiver and a compact Lie group. They are built from solutions of the Lax equation on the edges of the quiver together with Kirchhoff-type compatibility conditions at interior vertices, modulo gauge transformations that are trivial at the boundaries. These moduli spaces are finite-dimensional, smooth, and naturally symplectic, and they carry a canonical Hamiltonian action of the group associated with the boundary of the quiver. I will describe their explicit realization as symplectic reductions of products of cotangent bundles of the group, explain their behaviour under gluing of quivers, and show that they depend only on the topological surface obtained by thickening the quiver. This leads to a 2-dimensional topological quantum field theory valued in a category of Hamiltonian spaces. This talk is based on a recent preprint of the same title (https://arxiv.org/abs/2510.23567) and is joint work with my PhD student Mohamed Maiza.