Lax-Kirchhoff moduli spaces and Hamiltonian 2D TQFT

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 $G^{\Gamma_{\mathrm{int}}}$, where $E$ is the set of edges and $\Gamma_{\mathrm{int}}$ 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.