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

Abstract

To an oriented quiver with boundary $\Gamma$ and a compact connected Lie group $G$ we associate a Nahm-Kirchhoff moduli space ${\mathcal{M}}_{\mathbb{H}}(\Gamma )$: solutions of Nahm's equations on the edges with Kirchhoff matching at interior vertices, modulo boundary-trivial gauge transformations. For connected $\Gamma$, ${\mathcal{M}}_{\mathbb{H}}(\Gamma )$ is a smooth manifold of dimension $4(|E|-|{\Gamma }_{\mathrm{int}}|)\dim G$ with a hyperkähler structure arising as an infinite-dimensional hyperkähler quotient. Via a Kempf-Ness argument on the Cartan-Hadamard manifold $G_{\mathbb{C}}/G$, we prove a quiver analogue of Donaldson's theorem identifying it with the complex-symplectic quotient ${\mathcal{M}}_{\mathbb{C}}(\Gamma )$ of the complex Nahm equations by the complexified gauge group. This quotient is independent of edge lengths and isomorphic to a finite-dimensional complex-symplectic quotient of $(T^{\ast }{G}_{\mathbb{C}}{)}^E$, while the hyperkähler metric depends on them: uniform rescaling acts by a homothety, and the transported metrics genuinely vary. By cutting and gluing, $\Gamma \mapsto {\mathcal{M}}_{\mathbb{C}}(\Gamma )$ is a 2D TQFT valued in complex Hamiltonian manifolds, à la Moore-Tachikawa, whereas the hyperkähler refinement is not functorial on ${\mathbf{Cob}}_2$: spines of a cylinder with different lengths give non-isomorphic spaces. Retaining lengths, these spaces form a metric field theory, a functor of non-unital monoidal categories from metric cobordisms to hyperkähler manifolds composed by hyperkähler reduction; over each cone of $M_{g,n}^{\mathrm{trop}}$ they give a family with fixed complex-symplectic fibre and varying metric. Finally, harmonic gauge representatives converge as an internal edge collapses, and on a long edge with regular semisimple Higgs field the Kempf-Ness minimizer is exponentially confined near its centralizer; the limiting metrics are left open.

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