Problem 2.5a

A First Course in String Theory


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.


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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