A First Course in String Theory

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2.5 Constructing orbifold

(c) Determine the three fixed points of the action on the torus. Show that the orbifold is topologically a two-dimensional sphere, naturally presented as a triangular pillowcase with seamed edges and corners at the fixed points.

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[guess]

To find the fixed points, we consider the cases when

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where .

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When ,

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When ,

When ,

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When ,

When ,

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In the fundamental domain, the 3 fixed points are:

when ;

when ;

when .

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Duplicate the fundamental triangle to create a fundamental parallelogram.

If we label the some edges as instead of , the fundamental parallelogram will have a sphere topology .

However, it is not exactly the same as a sphere topology, because a sphere topology would not have the identification.

[guess]

— Me@2021-02-23 03:44:57 PM

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2021.02.24 Wednesday (c) All rights reserved by ACHK

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