I'm just brushing up on a bit of CFT, and I'm trying to understand whether conformal symmetry is local or global in the physics sense.
Obviously when the metric is viewed as dynamical then the symmetry is local, because essentially then we're dealing with a change of variables under which the metric transforms with a local scale factor $\Omega = \Omega(x)$.
Usually, however, we think of the metric as fixed. David Tong's excellent notes suggest that in this case the symmetry should be thought of as global. But I'm not sure I agree.
Say we work in 2D and have general conformal transformation given by holomorphic function $f(z)$. Under a conformal transformation $z\to w$ say, viewed actively, a general field $\Phi$ will transform to
$$(\frac{dw}{dz})^h \Phi$$
where the prefactor is clearly dependent on the spacetime point. This would suggest that the transformation is local in the physics sense.
Perhaps the distinction he is trying to make is between physical and gauge transformations. But then again I might be wrong because I thought that only global transformations had nonzero conserved quantities, and there's definitely a conserved current for conformal symmetry.
Could anyone help to clarify this for me?
Best Answer
Whether the conformal symmetry is local or global depends on the theory! More precisely, the symmetry that may be local is not really conformal symmetry but ${\rm diff}\times {\rm Weyl}$.
For example, in all the CFTs we use in the AdS/CFT correspondence, for example the famous ${\mathcal N}=4$ gauge theory in $d=4$, the conformal symmetry is global – and, correspondingly, it is a physical symmetry with nonzero values of generators. This is related to the fact that the CFT side of the holographic duality is a non-gravitational theory so it avoids all local symmetries related to spacetime geometry.
The previous paragraph holds even if the dimension of the CFT world volume is $d=2$. In $d=2$, it may happen that the global conformal symmetry is extended to the infinite-dimensional local symmetry where $\Omega(x)$ depends on the location. However, such an enhancement looks "automatic" only classically. Quantum mechanically, a nonzero central charge $c\neq 0$ prevents one from defining the general local conformal transformations. In all the CFTs from AdS/CFT, we have $c\geq 0$. Such a nonzero $c$ leads to the "conformal anomaly" (proportional to the world sheet Ricci scalar and $c$).
On the contrary, the world sheet $d=2$ CFT theories used to describe perturbative string theory always have a local diffeomorphism and local Weyl symmetry. This is needed to decouple all the unphysical components of the world sheet metric tensor; and a necessary condition is the incorporation of the conformal (and other) ghosts so that in the critical dimension, we have the necessary $c=0$. We say that the world sheet CFT is "coupled to gravity" as we add the world sheet metric tensor, the diff symmetry, and the Weyl symmetry. The Weyl symmetry is the symmetry under a general scaling of the world sheet metric by $\Omega(x)$ that depends on the location on the world sheet. One may gauge-fix this local Weyl symmetry along with the 2-dimensional diffeomorphism symmetry, e.g. by demanding the $\delta_{ij}$ form of the metric tensor. This gauge-fixing still preserves some residual symmetry, a subgroup of the originally infinite-dimensional "diff times Weyl" symmetry. This residual symmetry is nothing else than the infinite-dimensional conformal symmetry generated by $L_n$ and $\tilde L_n$. Because its being infinite-dimensional, we may call it a local conformal symmetry but it's really just a residual symmetry from "diff times Weyl". The global $SL(2,C)\sim SO(3,1)$ global subgroup is the Mobius group generated by $L_{0,\pm 1}$ and those with tildes, too.
As far as I know, this local conformal symmetry is a special case of some $d=2$ theories. In higher dimensions, the Weyl and diff aren't enough to kill all the components of the metric tensor and the "partially killed" theories with a dynamical metric are still inconsistent as the usual naively quantized versions of general relativity.
In all the cases above and others, it is true that the local symmetries – where the parameter $\Omega(x)$ is allowed to depend on time and space coordinates (if the latter exist) – are gauge symmetries (in the sense that the generators are obliged to annihilate physical states) while the global symmetries are always "physical" in your sense of the charge's being nonzero. These equivalences follow from some easy logical argument. When you have infinitely many generators of the (space)time-dependent symmetry transformations, it follows that all the quanta associated with these generators exactly decouple – have vanishing interactions – with the gauge-invariant degrees of freedom. So we always study the physical part of the theory only, and it's the theory composed of the gauge symmetry's singlets.
Greetings to David.