Fundamental polygon

A First Course in String Theory

2.5 Constructing simple orbifolds

(b) Consider a torus $\displaystyle{T^2}$ , presented as the $\displaystyle{(x,y)}$ plane with the identifications $\displaystyle{x \sim x + 2}$ and $\displaystyle{y \sim y+2}$ . Choose $\displaystyle{-1 < x, y, \le 1}$ as the fundamental domain. The orbifold $\displaystyle{T^2/\mathbb{Z}_2}$ is defined by imposing the $\displaystyle{\mathbb{Z}_2}$ identification $\displaystyle{(x,y) \sim (-x,-y)}$ .

Prove that there are four points on the torus that are left fixed by the $\displaystyle{\mathbb{Z}_2}$ transformation. Show that the orbifold $\displaystyle{T^2/\mathbb{Z}_2}$ is topologically a two-dimensional sphere, naturally presented as a square pillowcase with seamed edges.

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To find the fixed points, we consider the cases when $\displaystyle{-x = x + 2m}$ and $\displaystyle{-y = y + 2n}$ , where $\displaystyle{m,n \in \mathbb{Z}}$ . Since the length of the interval is only 2, we can consider only the cases when $\displaystyle{m,n = 0, 1}$ . Then the only solutions are