Conformal coordinate change: gauge symmetry or global symmetry

Question

While reading the fourth chapter "Introducing Conformal Field Theory" of D. Tong's string theory notes, I read that

A transformation of the form $\sigma^\alpha\to\tilde{\sigma}^\alpha(\sigma)$ (i.e. a conformal transformation) has a different interpretation depending on whether we are considering a fixed background metric $g_{\alpha\beta}$, or a dynamical background metric. When the metric is dynamical, the transformation is a diffeomorphism; this is a gauge symmetry. When the background is fixed, the transformation should be thought of as an honest, physical symmetry, taking the point $\sigma^\alpha$ to point $\tilde{\sigma}^\alpha(\sigma)$. This is now a global symmetry with the corresponding conserved currents.

What is the real mathematical difference between the two? Since they are very different concepts, how can one say that they are equivalent?

I stumbled upon

Conformal transformation vs diffeomorphisms

Conformal transformation/ Weyl scaling are they two different things? Confused!

Simple conceptual question conformal field theory

but I don't feel that they answer the same question.

Answer 1

This is actually more general than just diffeomorphisms, but it is a but more intuitive there since we are so used to coordinate transformations.

For any quantity, one can simply decide to measure it in different units, different scale, or shift the scale. This doesn't change physical properties in any way, and in this sense corresponds to a gauge transformation. You are changing the gauge with which you measure quantities (this is actually the sense that Weyl introduced the term "eichinvarianz"). Note that such a gauge transformation can be global (the same for any point in space) or local (space-dependent). Although all physical quantities remain unchanged, it is not a "physical" symmetry since there are no conservation laws etc. associated with it; everything physical is invariant under gauge transformations by definition, so there is no use in trying to assign deeper meaning to this invariance.

On the other hand, global symmetries must always be defined to some external reference. If you want to decide if some object has rotation symmetry, you must first provide a reference measuring device/object/background space. This seems a bit silly for rotations, but if you think of broken $U(1)$-symmetry in superfluid (or superconductors), you see that you need a reference superfluid to be able to determine it. These global symmetries (that can be broken in principle) are associated with conserved quantities.

Here is another way of looking at it. Say I have some symmetry group $G$, and two objects $A$ and $B$ (one of these could also refer to the background space). If I do a gauge transformation, this can act on each object separately, so the total transformation group is $G_A \times G_B$. However, if I decide to measure $A$ with respect to $B$, I cannot transform $A$ separately anymore. One can still do gauge/coordinate transformations on the whole system, so the gauge transformation group is now the diagonal subgroup $G_{A+B}$.

So coming back to conformal transformation: if you have a fixed background metric, i.e. a reference, then such a transformation is a global symmetry. It leads to conservation laws, and can be broken. If the metric is dynamical, then such a transformation is equivalent to a local coordinate change, a diffeomorphism, which is a gauge transformation in the above sense, and cannot lead to anything physical.

I have a reference where we go into this in some detail, but since this is my own work, I'll only give it when asked explicitly.