We introduce the Lax–Kirchhoff moduli space associated with a finite quiver \(\Gamma \) and a compact connected Lie group G. On each oriented edge we consider the Lax equation \(\dot{A}_1 + [A_0, A_1] = 0\) and impose a Kirchhoff-type matching condition for the fields \(A_1\) at interior vertices. Modulo gauge transformations trivial on the boundary, this yields a moduli space \(\mathcal {M}(\Gamma )\) . We prove that \(\mathcal {M}(\Gamma )\) is a finite-dimensional smooth symplectic manifold carrying a Hamiltonian action of \(G^{\partial \Gamma }\) whose moment map records the boundary values of \(A_1\) . Analytically, we construct slices for the infinite-dimensional gauge action and realize \(\mathcal {M}(\Gamma )\) by Marsden–Weinstein reduction. For the quiver consisting of a single edge, we recover the classical identification \(\mathcal {M} \cong T^*G\) . In general, we identify \(\mathcal {M}(\Gamma )\) with a symplectic reduction of \(T^*G^E\) by \(G^{\Gamma _{\textrm{int}}}\) , where E is the set of edges and \(\Gamma _{\textrm{int}}\) is the set of interior vertices. We further show that \(\mathcal {M}(\Gamma )\) is invariant under quiver homotopies, implying that it depends only on the surface with boundary obtained by thickening \(\Gamma \) . We then assemble these spaces into a two-dimensional topological quantum field theory valued in a category of Hamiltonian spaces.
