AI 中文总结
本文研究8维环面带Spin(7)结构的广义Nahm变换，构造两类Dirac核瞬子丛证明常规变换非良定义，定义渐近和乐并给出其非Spin(7)的实例。
AI 中文摘要
四维平坦超凯勒环面的Nahm变换是环面$T^4$上反自对偶（ASD）瞬子模空间与对偶环面$\tilde{T^4}$上反自对偶瞬子模空间之间的等距映射，该对偶环面参数化$T^4$上的平坦线丛。本文研究带有Spin(7)结构的8维环面上的广义Nahm变换，我分别构造具有正、负手征Dirac核的瞬子丛，证明通常的Nahm变换良定义性不成立；随后定义由瞬子线丛的高次幂$k\neq1$扭曲的瞬子的渐近和乐概念，证明该渐近和乐在$k$的二阶下约化为Spin(7)；最后给出渐近和乐为$\boldsymbol{\frak{u}(1)^4}$（非Spin(7)）的例子。
英文摘要
The Nahm transform for 4-dimensional flat hyperkahler tori is an isometry between the moduli space of anti-self-dual (ASD) instantons on a torus $T^4$ and the moduli space of ASD instantons on the dual torus $\hat{T^4}$ parametrising flat line bundles on $T^4$. This paper studies a generalised Nahm transform on an 8-dimensional torus with a Spin(7) structure. I construct instanton bundles with Dirac kernels respectively in positive and negative chiralities, demonstrating that the usual Nahm transform is not well-defined. I then define a notion of asymptotic holonomy for instantons twisted by a high power $k \gg 1$ of an instanton line bundle, and I show that this asymptotic holonomy reduces to Spin(7) to second order in $k$. Finally, I provide examples for which the asymptotic holonomy is $\mathfrak{u}(1)^4$, and thus not Spin(7).
Comments20 pages
