Solving Simple Orbifolds from Zwiebach's String Theory Book

[rbwang1225]

Homework Statement

This problem comes from the string theory book of Zwiebach, prob. 2.5.
I am constructing the orbifolds $S^1/\mathbb Z_2$ and $T^2/\mathbb Z_2$.

Homework Equations

$S^1$ comes from the identification $x\sim x+2$ and choosing the fundamental domain as $-1<x\leq 1$.
$T^2$ are made by $x\sim x+2$ and $y\sim y+2$ and choosing the fundamental domains as $-1<x, y\leq 1$.
$S^1/\mathbb Z_2$ and $T^2/\mathbb Z_2$ are defined by imposing the identification $x\sim-x$ and $(x,y)\sim(-x,-y)$, respectively.

The Attempt at a Solution

By recognizing the identifications, I can know the fixed points of $S^1/\mathbb Z_2$ and $T^2/\mathbb Z_2$.
But my problem is that I can't imagine the resulting pictures of the orbifolds.
Is there any convenient way to figure them out?

Regards.

 

[mark726]

Thank you for sharing your problem with us. Orbifolds can be a bit tricky to visualize, but there are some tools and techniques that can help.

One approach is to use a program like Mathematica or Wolfram Alpha to plot the orbifold. You can input the equations for $S^1/\mathbb Z_2$ and $T^2/\mathbb Z_2$ and see the resulting picture. This can give you a better understanding of the shape and structure of the orbifold.

Another approach is to use a geometric representation of the orbifold. For example, for $S^1/\mathbb Z_2$, you can imagine a circle with a point at the top and bottom, representing the fixed points. Then, when you apply the identification, the top and bottom points will be identified, resulting in a half-circle with two points at the ends. This can help you visualize the shape of the orbifold.

For $T^2/\mathbb Z_2$, you can imagine a square with sides of length 2, representing the fundamental domain. Then, when you apply the identification, the opposite sides of the square will be identified, resulting in a cylinder with two points at the ends. This can also help you visualize the structure of the orbifold.

I hope this helps. Good luck with your orbifold constructions!