模变换

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 · 8.1 复流形

变换 $\tau \to {\tau }^{\prime }$ 称为模变换(modular transformation)，它由 $\tau \to \tau +1$ 与 $\tau \to -1/\tau$ 生成。变换 $\tau \to \tau +1$ 沿经线 $m$ 生成一个德恩扭转，其做法如下（图 8.4(a)）：(i) 先沿 $m$ 把环面剪开；(ii) 然后取剪口的一侧，保持另一侧不动，将其旋转 $2\pi$；(iii) 再把两侧重新粘合。另一个变换 $\tau \to -1/\tau$ 则对应于交换纬线 $l$ 与经线 $m$ 的角色（图 8.4(b)）。

英文原文

The change $\tau \to {\tau }^{\prime }$ is called the modular transformation and is generated by $\tau \to \tau +1$ and $\tau \to -1/\tau$ . The transformation $\tau \to \tau +1$ generates a Dehn twist along the meridian $m$ as follows (figure 8.4(a)). (i) First, cut a torus along $m$ . (ii) Then take one of the lips of the cut and rotate it by $2\pi$ with the other lip kept fixed. (iii) Then glue the lips together again. The other transformation $\tau \to -1/\tau$ corresponds to changing the roles of the longitude $l$ and the meridian $m$ (figure 8.4(b)).

出处：8.1 复流形 · node a-2950159a745bbdf6cce5a823 · 源节点 SHA-256 e216d6b32552704f…