I have come across the question whether Stokes' theorem holds also on orbifolds.
Let us take the simple case of $T^2/Z_2$ with a one-form $A$, then the question becomes:
- For a region $\Gamma$ with boundary $\partial\Gamma$ is $\int_\Gamma \mathrm dA = \int_{\partial \Gamma} A$, even if some of the fix poitns of the orbifold lie within $\Gamma$?
- As a special case: Is $\int_\mathcal M \mathrm dA = 0$ if $\mathcal M$ is the entire orbifold?
- What changes if $A$ is a non-trivial $U(1)$ bundle? How do I properly incorporate this?
As you might guess, I'm a physicists and the application I am interested in is a $U(1)$ gauge theory on the $T^2/Z_2$ orbifold.