The relation between gauge symmetry and global internal symmetry I'm a little confused about the relation between the gauge symmetry and global internal symmetry of a field theory. I'd appreciate any clarification on this. My question can be phrased as the following:
Is it correct to say that for a global internal symmetry (abelian or non-abelian) like $SU(2)$ or in general a symmetry group $G$, the gauge transformations and the transformations from group $G$ are not related to each others but for a local  internal symmetry the gauge transformations become related to the action of $G$ (by enforcing the invariance of the theory as demonstrated in the usual procedure in textbooks) and the gauge fields become the connections on the manifold associated to $G$?
 A: I believe you may be slightly confused about terminology used. It is really pretty simple.
An internal symmetry is indeed a symmetry of the theory that does not impact the spacetime coordinates. It can be a global symmetry or a local symmetry.
Examples of a global symmetry are $U(1)$ for the EM charge, $SU(3)$ flavour or $SU(2)$ isospin.
When the internal symmetry is local it is a gauge theory. The gauge fields are then a connections of the symmetry.  Examples are $U(1)$ for the EM force, or $SU(2)\times U(1)$ for the electroweak force or $SU(3)$-color for QCD. Note that gravity has a local symmetry as well: diffeomorphism invariance. It is a gauge theory, but not an internal symmetry.
In general the confusing about this comes from:

*

*traditionally gauge theories are taught in the following way: one starts from a global internal symmetry, usually $U(1)$, and then asks oneself the question: what happens if we make this local. One then see that one has to introduce a connection etc.  There is a prior no reason to do it that way; it is  just educational convenience/laziness


*a local internal symmetry is, strictly speaking, not a symmetry. It is a redundancy in the description of the system that allows to simplify it. It introduces d.o.f. that then need to be gauged away by fixing the gauge.


*the example of gravity that is a gauge theory but not an internal symmetry


*the fact that some people use gauge theory and Yang-Mills theory interchangeably. They are not the same. YM are just special examples of gauge theories.
