Nahm-Kirchhoff Moduli Spaces, Hyperkähler Quotient and 2D Field theories

Abstract

To each oriented quiver with boundary $Γ$ and compact connected Lie group $G$, we associate a Nahm--Kirchhoff moduli space $\mathcal{M}_{\mathbb{H}}(Γ)$ of solutions to Nahm's equations along the edges, subject to Kirchhoff matching at the interior vertices, modulo gauge transformations trivial on the boundary. For connected $Γ$, we prove that $\mathcal{M}_{\mathbb{H}}(Γ)$ is a smooth manifold of dimension $4(|E|-|Γ_{\mathrm{int}}|)\dim G$ carrying a hyperkähler structure obtained as an infinite-dimensional hyperkähler quotient. Using a Kempf--Ness argument on $G_{\mathbb{C}}/G$, we establish a quiver analogue of Donaldson's theorem, identifying $\mathcal{M}_{\mathbb{H}}(Γ)$ with the complex-symplectic quotient $\mathcal{M}_{\mathbb{C}}(Γ)$ of solutions to the complex Nahm equations modulo the complexified gauge group. The space $\mathcal{M}_{\mathbb{C}}(Γ)$ is independent of the edge lengths and isomorphic to a finite-dimensional complex-symplectic quotient of $(T^*G_{\mathbb{C}})^E$, whereas the hyperkähler metric depends on them. Through cutting and gluing laws, $Γ\mapsto \mathcal{M}_{\mathbb{C}}(Γ)$ defines a 2D TQFT valued in complex Hamiltonian manifolds, in the spirit of Moore--Tachikawa, while its hyperkähler refinement fails to be functorial, since subdividing an edge into sub-edges of arbitrary lengths alters the metric. Retaining the edge lengths restores functoriality: the spaces $\mathcal{M}_{\mathbb{H}}(Γ)$ assemble into a metric field theory on metric cobordisms, valued in hyperkähler manifolds composed by hyperkähler reduction, which fibres over the tropical moduli space $M_{g,n}^{\mathrm{trop}}$ with fixed complex-symplectic fibre. We also describe the degenerations of the metric as an internal edge collapses or becomes infinitely long.