共形Killing向量

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 · 14.2 玻色弦的量子理论

对于这样的 $\delta v$，我们得到 ${\delta }_{\mathrm{D}}{\gamma }_{\alpha \beta }=({\nabla }_{\gamma }\delta v^{\gamma }){\gamma }_{\alpha \beta }$。满足 $\delta v\in \ker P_1$ 的向量称为共形Killing向量(conformal Killing vector)（CKV），见7.7 节。需要注意的是，若取 $\delta \varphi$ 为 ${\nabla }_{\gamma }\delta v^{\gamma }$，则 ${\delta }_{\mathrm{D}}$ 与 ${\delta }_{\mathrm{W}}$ 给出相同的度规形变。因此，CKV 的集合与 ${\mathrm{Diff}}_0({\Sigma }_g)$ 和 Weyl(${\Sigma }_g$) 的交叠等同。设 ${\Sigma }_g$ 上有 $k$ 个独立的 CKV，记作 ${\Phi }_s^{\alpha }(1\le s\le k)$。由黎曼面理论可知

$$k=\{\begin{array}{ll}6& g=0\\ 2& g=1\\ 0& g\ge 2.\end{array}\text{\hspace{1em}(14.58)}$$

英文原文

We find, for such $\delta v$ , that ${\delta }_{\mathrm{D}}{\gamma }_{\alpha \beta }=({\nabla }_{\gamma }\delta v^{\gamma }){\gamma }_{\alpha \beta }$ . A vector $\delta v\in \ker P_1$ is identified with the conformal Killing vector (CKV), see section 7.7. It is important to note that ${\delta }_{\mathrm{D}}$ and ${\delta }_{\mathrm{W}}$ yield the same metric deformations if $\delta \varphi$ is taken to be ${\nabla }_{\gamma }\delta v^{\gamma }$ . Thus, the set of the CKVs is identified with the overlap between ${\mathrm{Diff}}_0({\Sigma }_g)$ and Weyl( ${\Sigma }_g$ ). Let there be $k$ independent CKVs on ${\Sigma }_g$ and denote these by ${\Phi }_s^{\alpha }(1\le s\le k)$ . It is known from the theory of Riemann surfaces that

$$k=\{\begin{array}{ll}6& g=0\\ 2& g=1\\ 0& g\ge 2.\end{array}\text{\hspace{1em}(14.58)}$$

出处：14.2 玻色弦的量子理论 · node a-d76c489ca3456e03db5785dd · 源节点 SHA-256 f07592c2daebd279…