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.

TL;DR. The main point is that Tong only needs to identify the leading singularity in order to determine the Weyl anomaly

$$ \langle T^{\alpha}{}_{\alpha}\rangle~=~-\frac{c}{12} R^{(2)}. \tag{4.34} $$

Concerning OP's question it is indeed unclear how to properly account for subleading terms in Tong's approach. They are presumably either contact terms or vanish on-shell.

Let us introduce a regulator $\varepsilon>0$ in the $XX$ OPE

$$\begin{align} {\cal R} X(z,\bar{z})X(w,\bar{w})~=~&-\frac{\alpha^{\prime}}{2} \ln(|z-w|^2+\varepsilon)\cr
&+~: X(z,\bar{z})X(w,\bar{w}): \end{align}\tag{4.22}$$

to better identify the singular structure. The $\partial X\partial X$ OPE becomes:

$$\begin{align}
{\cal R} \partial_zX(z,\bar{z})& \partial_wX(w,\bar{w}) \cr
~\stackrel{(4.22)}{=}&~-\frac{\alpha^{\prime}}{2}\frac{(\bar{z}-\bar{w})^2}{(|z-w|^2+\varepsilon)^2}\cr
&~+~:\partial_zX(z,\bar{z}) \partial_wX(w,\bar{w}): ~.\end{align} \tag{4.23}$$

The stress-energy-momentum tensor is

$$ T_{zz}~~=~ -\frac{1}{\alpha^{\prime}} :\partial_zX\partial_zX:~.\tag{4.25} $$

The $TT$ OPE becomes

$$ \begin{align}
{\cal R}T_{zz}(z,\bar{z}) &T_{ww}(w,\bar{w})\cr
~\stackrel{(4.23)+(4.25)}{=}&~\frac{c}{2}\frac{(\bar{z}-\bar{w})^4}{(|z-w|^2+\varepsilon)^4} \cr
&-\frac{2}{\alpha^{\prime}} \frac{(\bar{z}-\bar{w})^2}{(|z-w|^2+\varepsilon)^2}:\partial_zX(z,\bar{z}) \partial_wX(w,\bar{w}):\cr
&+~\ldots. \end{align}\tag{4.28}$$

We next use the energy conservation

$$ \partial_z T_{\bar{z}z} + \partial_{\bar{z}} T_{zz}~\approx~0, \tag{4.35z}$$

which holds on-shell up to contact terms.
We calculate$^1$

$$ \begin{align} {\cal R}\partial_z T_{z\bar{z}}(z,\bar{z})
&\partial_wT_{w\bar{w}}(w,\bar{w}) \cr
~\stackrel{(4.35z)}{\approx}&~{\cal R}\partial_{\bar{z}} T_{zz}(z,\bar{z})\partial_{\bar{w}}T_{ww}(w,\bar{w})\cr
~\stackrel{(4.28)}{=}&~
\partial_{\bar{z}}\partial_{\bar{w}} \left[ \frac{c}{2} \frac{(\bar{z}-\bar{w})^4}{(|z-w|^2+\varepsilon)^4} +\ldots \right] \cr
~=~&\partial_{\bar{w}} \left[ 2c \frac{\varepsilon(\bar{z}-\bar{w})^3}{(|z-w|^2+\varepsilon)^5} +\ldots \right] \cr
~=~&\frac{c}{12}\partial_{\bar{w}}\partial_z\partial_w\partial_z\frac{\varepsilon}{(|z-w|^2+\varepsilon)^2}+\ldots,
\end{align}\tag{4.36}$$

which leads to the sought-for OPE

$$\begin{align}
{\cal R}T_{z\bar{z}}(z,\bar{z}) &T_{w\bar{w}}(w,\bar{w}) \cr
~\stackrel{(4.36)}{=}&~\frac{c}{12}\partial_z\partial_{\bar{w}}\frac{\varepsilon}{(|z-w|^2+\varepsilon)^2}+\ldots \cr
~\stackrel{(4.2d)}{=}&~\frac{c\pi}{12}\partial_z\partial_{\bar{w}}\delta^2(z\!-\!w,\bar{z}\!-\!\bar{w}) +\ldots .
\end{align} \tag{4.38}$$

Here we use the following representation of the 2D Dirac delta distribution$^2$

$$\begin{align} \delta^2(z\!-\!w,\bar{z}\!-\!\bar{w})~:=~&
\delta({\rm Re} (z\!-\!w))~\delta({\rm Im} (z\!-\!w))\cr
~=~&\lim_{\varepsilon\searrow 0^+} \frac{1}{\pi}\frac{\varepsilon}{(|z-w|^2+\varepsilon)^2}. \end{align} \tag{4.2d}$$

Now proceed as in Tong's notes. $\Box$

References:

- D. Tong,
*Lectures on String Theory*; Subsection 4.4.2.

--

$^1$ Tong's trick (4.36) suggests another route: Let us instead consider the $\partial X \bar{\partial}X$ OPE

$$\begin{align}
\left. \begin{array}{c}
{\cal R} \partial_zX(z,\bar{z})
\partial_{\bar{w}}X(w,\bar{w})\cr\cr
{\cal R} \partial_{\bar{z}}X(z,\bar{z}) \partial_wX(w,\bar{w})\end{array}\right\}
~=~&\frac{\alpha^{\prime}}{2}\frac{\varepsilon}{(|z-w|^2+\varepsilon)^2}+\ldots
\cr
~\stackrel{(4.2d)}{=}&~\frac{\alpha^{\prime}\pi}{2}\delta^2(z\!-\!w,\bar{z}\!-\!\bar{w}) +\ldots. \end{align}$$

It is comforting that the regularization $\varepsilon>0$ correctly predicts that the leading singularity is a 2D Dirac delta distribution.
Then the $T\bar{T}$ OPE becomes

$$ \begin{align} {\cal R}T_{zz}(z,\bar{z})&T_{\bar{w}\bar{w}}(w,\bar{w})\cr
~=&~\frac{c}{2}\frac{\varepsilon^2}{(|z-w|^2+\varepsilon)^4}\cr
&+\frac{2}{\alpha^{\prime}} \frac{\varepsilon}{(|z-w|^2+\varepsilon)^2}:\partial_zX(z,\bar{z})\partial_{\bar{w}}X(w,\bar{w}):\cr
&+~\ldots. \end{align}$$

The leading singularity is given by double contractions, which are proportional to the square of the 2D Dirac delta distribution. This is ill-defined, cf. e.g. this Phys.SE post.

Nevertheless, let us now formally apply Tong's trick: Using the energy conservation (4.35z) leads to

$$ {\cal R}\partial_z T_{z\bar{z}}(z,\bar{z}) \partial_{\bar{w}}T_{w\bar{w}}(w,\bar{w})
~\stackrel{(4.35z)}{\approx}~{\cal R}\partial_{\bar{z}} T_{zz}(z,\bar{z})\partial_w T_{\bar{w}\bar{w}}(w,\bar{w}), $$

so that the sought-for OPE leads to the square of the 2D Dirac delta distribution as well

$$ \begin{align} {\cal R}T_{z\bar{z}}(z,\bar{z}) &T_{w\bar{w}}(w,\bar{w})\cr
~=~&\frac{c}{2}\frac{\varepsilon^2}{(|z-w|^2+\varepsilon)^4}+\ldots\cr
~\stackrel{(4.2d)}{=}&~\frac{c}{2}\pi^2\delta^2(z\!-\!w,\bar{z}\!-\!\bar{w})^2+\ldots. \end{align}$$

There might be a way to resolve the square of the 2D Dirac delta distribution, and argue that the leading singularity is instead given by the second derivative of the 2D Dirac delta distribution,

$$\begin{align} \frac{c}{12}\partial_z\partial_{\bar{w}}\underbrace{\frac{\varepsilon}{(|z-w|^2+\varepsilon)^2}}_{=\pi\delta^2(z-w,\bar{z}-\bar{w})}
~=~&\frac{c}{2}\underbrace{\frac{\varepsilon^2}{(|z-w|^2+\varepsilon)^4}}_{=\pi^2\delta^2(z-w,\bar{z}-\bar{w})^2}\cr
~-~&\frac{c}{3}\underbrace{\frac{\varepsilon}{(|z-w|^2+\varepsilon)^3}}_{=4\pi^2\delta^2(z-w,\bar{z}-\bar{w})^2},\end{align}$$

as in eq. (4.38), although we shall not pursue the matter here. $\Box$

$^2$ Note that there is a factor of 2 in Tong's definition of the 2D Dirac delta distribution, cf. the end of Section 4.0.1.

## Best Answer

The number of $r_{ij}$'s is $N(N-1)/2$. A general monomial $\prod_{1\leq i<j\leq N} r_{ij}^{\mu_{ij}}$ is conformally invariant if and only if $d_i = \sum_{j=1}^{i-1} \mu_{ji} + \sum_{j=i+1}^N \mu_{ij} =0$ for all $i=1,\ldots, N$. This are $N$ equations. So we get $N(N-1)/2-N=N(N-3)/2$ cross-ratios.

But note also if your dimension is $D$ then the dimension of the conformal group is $(D+2)(D+1)/2$ and because the cross-ratios itself just depend on $D\cdot N$ variables and there are $(D+2)(D+1)/2$ constraints the number of "algebraically independent" variables can actually be just $D\cdot N-(D+2)(D+1)/2$, i.e. if the number of cross ratios is bigger than this there are have to be (complicated) relations between the cross ratios. This you can see in the example for $D=1$, see comment above.