发表机构
Université de Sherbrooke（舍布鲁克大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为带边界箭图构造 Nahm-Kirchhoff 模空间，证明其超凯勒结构并等同于复辛商，进而建立二维场论与度量场论，揭示超凯勒细化对边长的依赖及退化行为。
AI 中文摘要
对于每个带边界的定向箭图 $\Gamma$ 以及紧致连通李群 $G$，我们关联一个 Nahm-Kirchhoff 模空间 $\mathcal{M}_{\mathbb{H}}(\Gamma)$，它由沿边满足 Nahm 方程、在内顶点满足 Kirchhoff 匹配条件、且在边界上平凡化的规范变换下取商的解构成。对于连通的 $\Gamma$，我们证明 $\mathcal{M}_{\mathbb{H}}(\Gamma)$ 是维数为 $4(|E|-|\Gamma_{\mathrm{int}}|)\dim G$ 的光滑流形，其上带有通过无穷维超凯勒商得到的超凯勒结构。利用 $G_{\mathbb{C}}/G$ 上的 Kempf-Ness 论证，我们建立了 Donaldson 定理的箭图类比，将 $\mathcal{M}_{\mathbb{H}}(\Gamma)$ 等同于复 Nahm 方程的解在复化规范群作用下的复辛商 $\mathcal{M}_{\mathbb{C}}(\Gamma)$。空间 $\mathcal{M}_{\mathbb{C}}(\Gamma)$ 不依赖于边长，且同构于 $(T^*G_{\mathbb{C}})^E$ 的有限维复辛商，而超凯勒度量则依赖于边长。通过切割与粘合律，$\Gamma \mapsto \mathcal{M}_{\mathbb{C}}(\Gamma)$ 定义了一个取值于复 Hamiltonian 流形的二维拓扑量子场论（2D TQFT），其精神与 Moore-Tachikawa 的工作一致；而其超凯勒细化则不具有函子性，因为将一条边细分为任意长度的子边会改变度量。保留边长可恢复函子性：空间 $\mathcal{M}_{\mathbb{H}}(\Gamma)$ 组装成一个定义在度量配边上的度量场论，取值于通过超凯勒约化复合的超凯勒流形，该场论以固定复辛纤维纤维化于热带模空间 $M_{g,n}^{\mathrm{trop}}$ 之上。我们还描述了当内部边坍缩或变为无限长时度量的退化行为。
英文摘要
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.
Comments40 pages. All comments are welcome
