Lax-Kirchhoff moduli spaces and Hamiltonian 2D TQFT
Mohamed Moussadek Maiza, Maxence Mayrand
TL;DR
The paper develops the Lax--Kirchhoff moduli spaces $\mathcal{M}(\Gamma)$ for quivers $\Gamma$ and a compact Lie group $G$, showing they are finite-dimensional smooth symplectic manifolds with a Hamiltonian $G^{\partial\Gamma}$-action. It provides a concrete identification $\mathcal{M}(\Gamma)\cong T^*G^E\,/\!/_{G^{\Gamma_{\mathrm{int}}}}$ via explicit maps, and proves invariance under quiver homotopies, so $\mathcal{M}(\Gamma)$ depends only on the thickened cobordism $\Sigma_\Gamma$. The authors construct an infinite-dimensional Marsden--Weinstein reduction in this setting and verify a symplectic structure on $\mathcal{M}(\Gamma)$, with a finite-dimensional reduction and boundary moment map capturing $A_1$ at the boundary. Finally, they assemble these spaces into a 2D TQFT valued in a category of Hamiltonian spaces, $\mathbf{Ham}$, yielding a functor from cobordisms to Hamiltonian spaces and establishing a real analogue of Moore--Tachikawa-type theories for Lax--Kirchhoff moduli.
Abstract
We introduce the Lax-Kirchhoff moduli space associated with a finite quiver $Γ$ 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 $\mathcal{M}(Γ)$. We prove that $\mathcal{M}(Γ)$ is a finite-dimensional smooth symplectic manifold carrying a Hamiltonian action of $G^{\partialΓ}$ whose moment map records the boundary values of $A_1$. Analytically, we construct slices for the infinite-dimensional gauge action and realize $\mathcal{M}(Γ)$ 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 $\mathcal{M}(Γ)$ with a symplectic reduction of $T^*G^E$ by $G^{Γ_{\mathrm{int}}}$, where $E$ is the set of edges and $Γ_{\mathrm{int}}$ is the set of interior vertices. We further show that $\mathcal{M}(Γ)$ is invariant under quiver homotopies, implying that it depends only on the surface with boundary obtained by thickening $Γ$. We then assemble these spaces into a two-dimensional topological quantum field theory valued in a category of Hamiltonian spaces.