二次微分

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

${\nabla }_{(-2)}^z$ 的显式形式表明 ${\partial }_{\bar{z}}{\varphi }_{izz}=0$，即 $\ker {\nabla }_{(-2)}^z$ 是螺旋度为 $-2$ 的全纯张量集合。张量 ${\varphi }_i={\varphi }_{i\bar{z}z}\mathrm{d}z\otimes \mathrm{d}z$ 称为二次微分(quadratic differential)，而 ${\mu }_i={\mu }_{i\bar{z}z}\mathrm{d}z\otimes \mathrm{d}z$ 称为贝尔特拉米微分(Beltrami differential)，见图 14.7。在实际计算中，用贝尔特拉米微分来指定规范切片往往很方便，这一点后面会看到。现在我们已经确立了

$$\left\{\ker P_1^{†}\right\}=\left\{\text{二次微分}\right\}=\left\{\ker {\nabla }_{(-2)}^z\right\}.\text{\hspace{1em}(14.112)}$$

英文原文

The explicit form of ${\nabla }_{(-2)}^z$ shows that ${\partial }_{\bar{z}}{\varphi }_{izz}=0$ , that is $\ker {\nabla }_{(-2)}^z$ is the set of holomorphic tensors of helicity $-2$ . The tensor ${\varphi }_i={\varphi }_{i\bar{z}z}\mathrm{d}z\otimes \mathrm{d}z$ is called the quadratic differential while ${\mu }_i={\mu }_{i\bar{z}z}\mathrm{d}z\otimes \mathrm{d}z$ is the Beltrami differential, see figure 14.7. In practical computations, it is often convenient to specify the gauge slice by the Beltrami differential, see later. Now we have established that

$$\left\{\ker P_1^{†}\right\}=\left\{\text{Q u a d r a t i c d i f f e r e n t i a l}\right\}=\left\{\ker {\nabla }_{(-2)}^z\right\}.\text{\hspace{1em}(14.112)}$$

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