瞬子数

本书暂定概念：尚未进入正式概念库。下面是本书给出的定义原文。

定义 · 9.4 主丛

其中 $R$ 是正常数，$\epsilon$ 是无穷小正数。细窄的交集 $U_{\mathrm{N}}\cap U_{\mathrm{S}}$ 实质上就是 $S^3$。设 $t_{\mathrm{NS}}(p)$ 为在 $p\in U_{\mathrm{N}}\cap U_{\mathrm{S}}$ 处定义的过渡函数。由于 $t_{\mathrm{NS}}$ 把 $S^3$ 映到 $\mathrm{SU}(2)$，故它由 ${\pi }_3(\mathrm{SU}(2))=\mathbb{Z}$ 分类。刻画该丛的整数称为瞬子数(instanton number)。若取 $t_{\mathrm{NS}}(p)$ 为单位元 $e\in \mathrm{SU}(2)$，则得到平凡丛 $P_0=S^3\times \mathrm{SU}(2)$，它对应于同伦类 0。非平凡丛可如下得到。首先注意到 $\mathrm{SU}(2)\cong S^3$（例 4.12）。元素 $A\in \mathrm{SU}(2)$ 可写成

$$A=\left(\begin{array}{cc}u& v\\ -\bar{v}& \bar{u}\end{array}\right)$$

英文原文

where $R$ is a positive constant and $\epsilon$ is an infinitesimal positive number. The thin intersection $U_{\mathrm{N}}\cap U_{\mathrm{S}}$ is essentially $S^3$ . Let $t_{\mathrm{NS}}(p)$ be the transition function defined at $p\in U_{\mathrm{N}}\cap U_{\mathrm{S}}$ . Since $t_{\mathrm{NS}}$ maps $S^3$ to $\mathrm{SU}(2)$ , it is classified by ${\pi }_3(\mathrm{SU}(2))=\mathbb{Z}$ . The integer characterizing the bundle is called the instanton number. If $t_{\mathrm{NS}}(p)$ is taken to be the unit element $e\in \mathrm{SU}(2)$ , we have a trivial bundle $P_0=S^3\times \mathrm{SU}(2)$ , which corresponds to the homotopy class 0. Non-trivial bundles are obtained as follows. We first note that $\mathrm{SU}(2)\cong S^3$ (example 4.12). An element $A\in \mathrm{SU}(2)$ is written as

$$A=\left(\begin{array}{cc}u& v\\ -\bar{v}& \bar{u}\end{array}\right)$$

出处：9.4 主丛 · node a-883e44046340045d0861c1db · 源节点 SHA-256 581be27a3828836f…