A First Course in String Theory

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2.5 Constructing simple orbifolds

(b) Consider a torus , presented as the plane with the identifications and . Choose as the fundamental domain. The orbifold is defined by imposing the identification .

Prove that there are four points on the torus that are left fixed by the transformation. Show that the orbifold 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 and , where . Since the length of the interval is only 2, we can consider only the cases when . Then the only solutions are

— Me@2021-01-17 04:14:44 PM

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— Wikipedia on *Surface (topology)*

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The formula for this topology is , which is a sphere.

— Me@2021-01-29 06:10:46 PM

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