As far as I understand, gauging a global symmetry means taking a model with a global symmetry and transforming it into a model such that the previous symmetry group is now the gauge symmetry of your new model.
Given a model with a gauge (local) symmetry, is there any way to transform it into a model with a global symmetry?
As an example, let's say we have a model with a $SU(N)$ gauge symmetry. Is there any way to obtain a model with a $SU(N)$ global symmetry from it?
This is an interesting question.
For this answer, I will not go with the "global symmetries are dea and gauge symmetries are redundant way" because I feel this is not very pedagogical. Yes, having black holes "spawn" kills the idea of exact global symmetries, and local symmetries are more of a "redundancy" for the most part, with some subtleties that I would recommend you look into (Large Gauge Transformations for example")
So first of all, the interesting side is the other way: Given a global symmetry, can you always gauge it?
Why is this more interesting?
Well, a local(gauge) symmetry, gauge symmetry, changes something locally, as the name suggests. So by applying that symmetry at every point in your field, you get an exact global symmetry.
This means that given some infinitesimal generator, you can get a corresponding global transformation from the exponential map. Applying the gauge transformation everywhere seems rather intuitive, but the real question now is whether this is two-way.
I could give you a basic counter-example and call it a day, but the answer to this question is truly fascinating and rather deep.
First of all, let's give an example:
Chiral "Symmetry" The QCD theory is insensitive to independent rotation of right and left-handed spinors:
$\psi_L \rightarrow e^{i\theta_L} \psi_L$ and $\psi_R \rightarrow e^{i\theta_R} \psi_R$
This "vector symmetry" is broken by the addition of a mass term, as we get a $m\bar{\psi}\psi$ term (Why?), and what is very interesting is that this symmetry does not conserve the current from Noether's theorem, and as a result it does not have a conserved quantity. This is very interesting because this makes us question what this "symmetry" is! What is a symmetry?
This is called an anomaly, (look into ABJ anomaly), and this is a class of global symmetries that in general cannot be gauged, after all this is not really the sort of symmetry you are used to seeing, there is no conserved current.
Now let's look at this more mathematically and intuitively. A gauge theory has some information on a more local level. It has a "richer" structure than a global symmetry, as after you have an exact map from local to global symmetries. Intuitively you should not be able to reconstruct the small picture from the bigger picture unless there is a niceness that is obviously present, i.e. having perhaps some sort of periodicity.
Now let's consider a more mathematical picture. The idea of gauging comes from Lie theory. Specifically, we want to take a weird global nonlinear action from some symmetry group G and convert it to relations of the infinitesimal generators of the group. This is where the power of Lie theory lies. Specifically, there is a celebrated theorem that says that for symmetry of the system to be eligible for an expression in terms of the infinitesimal generators, one requires that the system have a sort of maximal rank condition everywhere. If this is not met, even if the generators obey the relations that in general would constitute a symmetry, we don't get a symmetry. The maximal rank condition implies that we have a local diffeomorphism that allows us locally to look at exact maps.
So in general we see that when a transformation is not connected to the identity of our symmetry group, be it the gauge group, or sometimes even not homotopic, like winding numbers and berry phase, we cannot look at the generators of the group to think of the symmetries in the global scene.
I hope this gives some sort of intuition. The last part might feel a bit less motivated or obvious, so in that case, I would like to redirect you to Olver's fantastic book.
