Fundamental polygon

A First Course in String Theory

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2.5 Constructing simple orbifolds

(b) Consider a torus \displaystyle{T^2}, presented as the \displaystyle{(x,y)} plane with the identifications \displaystyle{x \sim x + 2} and \displaystyle{y \sim y+2}. Choose \displaystyle{-1 < x, y, \le 1} as the fundamental domain. The orbifold \displaystyle{T^2/\mathbb{Z}_2} is defined by imposing the \displaystyle{\mathbb{Z}_2} identification \displaystyle{(x,y) \sim (-x,-y)}.

Prove that there are four points on the torus that are left fixed by the \displaystyle{\mathbb{Z}_2} transformation. Show that the orbifold \displaystyle{T^2/\mathbb{Z}_2} 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 \displaystyle{-x = x + 2m} and \displaystyle{-y = y + 2n}, where \displaystyle{m,n \in \mathbb{Z}}. Since the length of the interval is only 2, we can consider only the cases when \displaystyle{m,n = 0, 1}. Then the only solutions are

\displaystyle{(0,0)}
\displaystyle{(0,1)}
\displaystyle{(1,0)}
\displaystyle{(1,1)}

— 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 \displaystyle{ABB^{-1}A^{-1}ABB^{-1}A^{-1} = ABB^{-1}A^{-1} }, which is a sphere.

— Me@2021-01-29 06:10:46 PM

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