It is folklore that quantum gravity cannot have any exact global symmetry (see Global symmetries in quantum gravity). This follows for example from thought experiments involving black holes (no-hair). Yet electrically charged "hair" is allowed. Gauge symmetries seem to be excepted (due to long range forces). But gauge symmetries imply global symmetries. (Invariance under $\phi \to e^{i \theta(x)} \phi$ implies invariance under $\phi \to e^{i \theta} \phi$.) Therefore it seems like global symmetries are allowed, namely the global parts of the local symmetries. That would mean a $U(1)$ global symmetry corresponding to electric charge is allowed for example. What is wrong with this reasoning?
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