局部规范势

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 · 11.1 不变多项式与陈-韦伊同态

上一章中，我们在主丛上引入了局部规范势(local gauge potential) $\mathcal{A}={\mathcal{A}}_{\mu }\mathrm{d}{x}^{\mu }$ 与场强 $\mathcal{F}=\frac{1}{2}{\mathcal{F}}_{\mu \nu }\mathrm{d}{x}^{\mu }\wedge \mathrm{d}{x}^{\nu }$。我们已经证明，这些几何对象同样描述相伴向量丛。由于联络的集合 $\{{\mathcal{A}}_i\}$ 刻画了纤维丛的扭曲，主丛的非平凡性也为其相伴丛所共有。事实上，若把 (10.57) 用作向量丛中局部联络的定义，那么即使不借助最初与之相伴的主丛，也能定义它。后面我们会遇到必须使用向量丛的情形（惠特尼和丛、分裂原理等）。

英文原文

In the previous chapter, we introduced the local gauge potential $\mathcal{A}={\mathcal{A}}_{\mu }\mathrm{d}{x}^{\mu }$ and the field strength $\mathcal{F}=\frac{1}{2}{\mathcal{F}}_{\mu \nu }\mathrm{d}{x}^{\mu }\wedge \mathrm{d}{x}^{\nu }$ on a principal bundle. We have shown that these geometrical objects describe the associated vector bundles as well. Since the set of connections $\{{\mathcal{A}}_i\}$ describes the twisting of a fibre bundle, the non-triviality of a principal bundle is equally shared by its associated bundle. In fact, if (10.57) is employed as a definition of the local connection in a vector bundle, it can be defined even without reference to the principal bundle with which it is originally associated. Later, we encounter situations in which use of vector bundles is essential (the Whitney sum bundle, the splitting principle and so on).

出处：11.1 不变多项式与陈-韦伊同态 · node a-d7b9870c1431b2239bbb0325 · 源节点 SHA-256 be2623c71a2ec623…