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As the title asked, what is the essential difference between the orbifold $S_1/Z_2$ and an interval say, $[0,1]$?

I mean $S_1$ can be [-1,1] with -1 and +1 identified. Now $S_1/Z_2$ then is just [0,1]. But in many models, say the large extra dimensions, people construct the branes on an orbifold $S_1/Z_2$ (as the large extra dimension) and do not say it is simply $[0,1]$, why?

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  • $\begingroup$ You might want to give your definition of an orbifold. All definitions I'm aware of (e.g. via atlases or via stacks/groupoids) encode data about singularities AND keeps track of the isotropy groups at the singular points. The quotient topological spaces in your examples are indeed identical. They're different as orbifolds as there are no non-trivial automorphism groups in the orbifold $[0, 1]/{Id}$ $\endgroup$
    – zzz
    Dec 8, 2017 at 20:08
  • $\begingroup$ @bianchira Thanks for your comment. But I am not aware of the the rigorous theory of orbifolds. I learned the concept of the orbifolds from physics context which are roughly defined as the manifolds obtained by identifications with fixed points. $\endgroup$
    – Wein Eld
    Dec 9, 2017 at 14:07

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In a nutshell, the $\mathbb{Z}_2$ group action. First of all, it should be said that a physical orbifold model is by definition more than just the geometric orbifold $\mathbb{S}^1/\mathbb{Z}_2$ itself. E.g. the $\mathbb{Z}_2$ group also acts on the dynamical variables/fields of the theory.

In contrast, for a general theory on an interval, its (Dirichlet, Neumann, Robin, etc) boundary conditions may be incompatible with a $\mathbb{Z}_2$ group action, and therefore not a $\mathbb{S}^1/\mathbb{Z}_2$ orbifold.

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