Nahm-Kirchhoff Moduli Spaces, Hyperkähler Quotient and 2D Field theories
Abstract
To an oriented quiver with boundary Γ and a compact connected Lie group G we associate a Nahm-Kirchhoff moduli space ℳℍ(Γ): solutions of Nahm's equations on the edges with Kirchhoff matching at interior vertices, modulo boundary-trivial gauge transformations. For connected Γ, ℳℍ(Γ) is a smooth manifold of dimension 4(|E|−|Γint|)dim G with a hyperkähler structure arising as an infinite-dimensional hyperkähler quotient. Via a Kempf-Ness argument on the Cartan-Hadamard manifold Gℂ/G, we prove a quiver analogue of Donaldson's theorem identifying it with the complex-symplectic quotient ℳℂ(Γ) 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*Gℂ)E, while the hyperkähler metric depends on them: uniform rescaling acts by a homothety, and the transported metrics genuinely vary. By cutting and gluing, Γ↦ℳℂ(Γ) is a 2D TQFT valued in complex Hamiltonian manifolds, à la Moore-Tachikawa, whereas the hyperkähler refinement is not functorial on Cob2: 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 Mg,ntrop 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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BibTeX
@misc{maiza2026nahmkirchhoff,
title = {{Nahm-Kirchhoff Moduli Spaces, Hyperkähler Quotient and 2D Field theories}},
author = {Mohamed Moussadek Maiza},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03384},
eprint = {2610.03384},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03384},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Nahm-Kirchhoff Moduli Spaces, Hyperkähler Quotient and 2D Field theories
AU - Mohamed Moussadek Maiza
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.03384
UR - https://arxiv.org/abs/2610.03384
AB - To an oriented quiver with boundary Γ and a compact connected Lie group G we associate a Nahm-Kirchhoff moduli space ℳ_ℍ(Γ): solutions of Nahm's equations on the edges with Kirchhoff matching at interior vertices, modulo boundary-trivial gauge transformations. For connected Γ, ℳ_ℍ(Γ) is a smooth manifold of dimension 4(|E|−|Γ_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_ℂ/G, we prove a quiver analogue of Donaldson's theorem identifying it with the complex-symplectic quotient ℳ_ℂ(Γ) 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^*G_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, Γ↦ℳ_ℂ(Γ) is a 2D TQFT valued in complex Hamiltonian manifolds, à la Moore-Tachikawa, whereas the hyperkähler refinement is not functorial on 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^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.
ER -