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I know that if one gauges the supersymmetry group, you get supergravity. You can then further gauge the $R$-symmetry and these are the so-called gauged supergravities. But I don't think I've seen anyone gauge the $R$-symmetry in a theory with global supersymmetry (ie, just a supersymmetric field theory but not supergravity). So my question is: Can this be done? If not, what is the problem with only gauging the $R$-symmetry?

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Late answer, but the $U(1)$ R-symmetry cannot be gauged in global SUSY. This can be seen from the fact that it rotates the fermionic coordinates $\theta$, which are independent of spacetime coordinates in global SUSY. If you gauge the R-symmetry, the transformation of $\theta$ becomes local, and $\theta$ becomes spacetime dependent -- this means you have curved superspace, i.e. supergravity.

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