# Problem 2.5a

A First Course in String Theory

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2.5 Constructing simple orbifolds

(a) Consider a circle $\displaystyle{S^1}$, presented as the real line with the identification $\displaystyle{x \sim x + 2}$. Choose $\displaystyle{-1 < x \le 1}$ as the fundamental domain. The circle is the space $\displaystyle{-1 < x \le 1}$ with points $\displaystyle{x = \pm 1}$ identified. The orbifold $\displaystyle{S^1/\mathbb{Z}_2}$ is defined by imposing the (so-called) $\displaystyle{\mathbb{Z}_2}$ identification $\displaystyle{x \sim -x}$. Describe the action of this identification on the circle. Show that there are two points on the circle that are left fixed by the $\displaystyle{\mathbb{Z}_2}$ action. Find a fundamental domain for the two identifications. Describe the orbifold $\displaystyle{S^1/\mathbb{Z}_2}$ in simple terms.

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Put point $\displaystyle{x=0}$ and point $\displaystyle{x=1}$ on the positions that they can form a horizontal diameter.

Then the action is a reflection of the lower semi-circle through the horizontal diameter to the upper semi-circle.

Point $\displaystyle{x=0}$ and point $\displaystyle{x=1}$ are the two fixed points.

A possible fundamental domain is $\displaystyle{0 \le x \le 1}$.

If a variable point $\displaystyle{x}$ moves from 0 to 1 and then keeps going, that point will actually go back and forth between 0 and 1.

— Me@2020-12-31 04:43:07 PM

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