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

To each oriented quiver with boundary $\mathrm{\Gamma }$ and compact connected Lie group $G$, we associate a Nahm–Kirchhoff moduli space ${\mathcal{M}}_{\mathbb{H}}(\mathrm{\Gamma })$ 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 $\mathrm{\Gamma }$, we prove that ${\mathcal{M}}_{\mathbb{H}}(\mathrm{\Gamma })$ is a smooth manifold of dimension $4(\mathrm{\mid }E\mathrm{\mid }-\mathrm{\mid }{\mathrm{\Gamma }}_{\mathrm{i}\mathrm{n}\mathrm{t}}\mathrm{\mid })\dim G$ carrying a hyperkähler structure obtained as an infinite-dimensional hyperkähler quotient. Using a Kempf–Ness argument on $G_{\mathbb{C}}\mathrm{/}G$, we establish a quiver analogue of Donaldson’s theorem, identifying ${\mathcal{M}}_{\mathbb{H}}(\mathrm{\Gamma })$ with the complex-symplectic quotient ${\mathcal{M}}_{\mathbb{C}}(\mathrm{\Gamma })$ of solutions to the complex Nahm equations modulo the complexified gauge group. The space ${\mathcal{M}}_{\mathbb{C}}(\mathrm{\Gamma })$ is independent of the edge lengths and isomorphic to a finite-dimensional complex-symplectic quotient of $(T^{\ast }{G}_{\mathbb{C}}{)}^E$, whereas the hyperkähler metric depends on them. Through cutting and gluing laws, $\mathrm{\Gamma }\mapsto {\mathcal{M}}_{\mathbb{C}}(\mathrm{\Gamma })$ 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 ${\mathcal{M}}_{\mathbb{H}}(\mathrm{\Gamma })$ assemble into a metric field theory on metric cobordisms, valued in hyperkähler manifolds composed by hyperkähler reduction, which fibres over the tropical moduli space $M_{g,n}^{\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{p}}$ with fixed complex-symplectic fibre. We also describe the degenerations of the metric as an internal edge collapses or becomes infinitely long.