A First Course in String Theory

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2.5 Constructing simple orbifolds

(a) Consider a circle , presented as the real line with the identification . Choose as the fundamental domain. The circle is the space with points identified. The orbifold is defined by imposing the (so-called) identification . Describe the action of this identification on the circle. Show that there are two points on the circle that are left fixed by the action. Find a fundamental domain for the two identifications. Describe the orbifold in simple terms.

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Put point and point 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 and point are the two fixed points.

A possible fundamental domain is .

If a variable point 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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