Understanding the statement "orbifold theories are QFTs with finite gauge group"

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I'd like to understand the equivalence of orbifold theories in string theory and (2D worldsheet) QFTs with finite gauge group, using the path integral.

Suppose my action is $$S= \frac{1}{2\pi \alpha'} \int d^2\sigma (\partial_i X^\mu \partial^i X_\mu)$$ and the $$X^\mu(\sigma)$$ are invariant under some finite group action $$\Gamma$$. To construct the orbifold theory on a Riemann surface $$\Sigma$$, I want to take the path integral over the untwisted and twisted sectors, i.e. I want to average over all boundary conditions on $$X^\mu$$ in which $$X^\mu$$ is periodic up to $$\Gamma$$-action.

Now, on the other hand, if I want to compute the path integral of a QFT with a finite gauge group, I would "gauge-fix" and then compute the path integral.

How can I see that the two approaches are the same?

This post imported from StackExchange Physics at 2019-09-16 20:25 (UTC), posted by SE-user Dwagg
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