# Gauge symmetry implies global symmetry and quantum gravity

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?

• The example given in the question assumes a classical spacetime background, and in that context, the argument is fine. Is the question asking what goes wrong with it when gravity is quantized? That's a hard question. The point of the no-global-symmetries folklore is that (1) something must go wrong with the reasoning and that (2) we don't yet know what goes wrong. The paper Symmetries in Quantum Field Theory and Quantum Gravity spends many pages just to define what global symmetry should mean in quantum gravity before "proving" that AdS/CFT can't have any. Jun 3 '20 at 3:24
• @Chiral, thanks, I will read it. But as a quick reply: Semi classical black hole arguments are often at the basis of the no-go- theorems for global symmetries in quantum gravity. Indeed the very claim is that it holds for any realization of quantum gravity not just in the specific cases we understand a bit better (String Theory/ AdS-CFT). So even if you say there might be other arguments, would you say that those semi-classical thought experiments about black holes say nothing to forbid the global parts of local symmetries (while they do say something already about other global symmetries). Jun 3 '20 at 9:07
• @Chiral, a quick glance seems to suggest that also in this case Harlow means that if there is a global U(1) there has to also be a local U(1) (in a AdS bulk corresponding to a CFT). So indeed this would mean that what is meant is that global symmetries ARE ALLOWED as long as they are the local part of a global symmetry. Is this reading wrong? Jun 3 '20 at 10:51
• If by semi-classical you mean that the spacetime background is fixed, then a no-global-symmetry argument can't be purely semi-classical. For example, one argument involves black hole evaporation: we can use a semi-classical model to derive Hawking radiation, but black holes don't evaporate in such a model. The assumption that the black hole would evaporate in a quantum gravity theory is an essential input to the argument, otherwise the no-go result would not follow. Jun 3 '20 at 12:55
• Not sure which part of Harlow and Ooguri you're referring to (it's a long paper), but the if-then statement you wrote doesn't say that global symmetries must be allowed in the bulk, which is what the no-go folklore is about. In fact the abstract says "We first show that any global symmetry, discrete or continuous, in a bulk quantum gravity theory with a CFT dual would lead to an inconsistency in that CFT, and thus that there are no bulk global symmetries in AdS/CFT." The CFT can have global symmetries, but the no-go folklore doesn't claim to exclude that. Jun 3 '20 at 12:56