In many introductory textbooks on conformal field theories in two dimensions, the flat Euclidean manifold $\mathbb{R}^2$ is considered. Later, when the global conformal transformation is derived, $\mathbb{R}^2$ is compactified to a Riemann sphere $S^2$ so that the infinity can be included by the compactification so that the global conformal transformation is PSL$(2,\mathbb{C})$ with elements: \begin{eqnarray} z\rightarrow\frac{az+b}{cz+d}. \end{eqnarray} My concern is whether such compactification will induce some problems as follows. We know that the compactification $S^2$ has nontrivial Euler characteristics, so it cannot be globally flat. Thus, although we can still force the metric to be Euclideanly flat at any finite point, the metric must be curved (singular) at the infinity $\infty$ so that $\infty$ contributes to all the Euler density. In this sense, we are inevitably staying on a curved geometry after compactification. What I am worried about is that the global transformation, e.g. $z\rightarrow z\exp(\Delta\xi)$ with $\Delta\xi\in\mathbb{R}$ potentially has a Weyl anomaly in the integral measure. Then I am wondering whether our usage of these global conformal transformations to argue the general properties of correlation functions, e.g. $\langle\Phi(z,\bar{z})\Phi(w,\bar{w})\rangle=1/(z-w)^{2h}\cdot 1/(\bar{z}-\bar{w})^{2\bar{h}}$, would be problematic due to the inevitably curved nature of Riemann sphere. Let us take the (infinitesmal) dilation transformation above $\tilde{z}\equiv\exp(\Delta\xi)z;\tilde{w}\equiv\exp(\Delta\xi)w$ on the Riemann sphere $S^2$ with $g_{ij}=\delta_{ij}\exp(2\xi)$: \begin{eqnarray} &&G(\tilde{z}-\tilde{w})\\ &\equiv&\langle\Phi(\tilde{z})\Phi(\tilde{w})\rangle_{S^2;g_{\mu\nu}}\\&=&\int\mathcal{D}\Phi^{g}\Phi(\tilde{z})\Phi(\tilde{w})\exp[-S(g,\Phi)]\nonumber\\ &=&\int\mathcal{D}\tilde{\Phi}^{g}\tilde{\Phi}(\tilde{z})\tilde{\Phi}(\tilde{w})\exp[-S(g,\tilde{\Phi})]\,\,\,\,\text{Renaming of the dummy variable $\Phi$ to $\tilde{\Phi}$}\nonumber\\ &=&\int\mathcal{D}\tilde{\Phi}^{g}\tilde{\Phi}(\tilde{z})\tilde{\Phi}(\tilde{w})\exp[-S(g,{\Phi})]\,\,\,\,\text{apply the invariance on the action}, \end{eqnarray} where (the antiholomorphic coordinates are suppressed above just for clearness) \begin{eqnarray} \tilde{\Phi}\left(\tilde{z},\bar{\tilde{z}}\right)=\exp(-\Delta\xi h-\Delta\xi\bar{h})\Phi(z,\bar{z}), \end{eqnarray} and $\mathcal{D}\Phi^{g}$ denotes the measure of $\Phi$ defined under the metric $g_{\mu\nu}$.
Then let us focus on the measure: \begin{eqnarray} \int\mathcal{D}\tilde{\Phi}^{g}&=&\int\mathcal{D}\Phi^{g\exp(-2\Delta\xi)}\,\,\,\,\text{Diffeomorphism invariance of the measure}\\ &=&\int\mathcal{D}\Phi^{g}\exp\left(\frac{c}{24\pi}\int\partial^2\xi\Delta\xi\right)\,\,\,\text{Weyl anomaly of the measure}\\ &=&\int\mathcal{D}\Phi^{g}\exp\left(\frac{c}{24\pi}8\pi\Delta\xi\right)\,\,\,\text{By Gauss-Bonnet $\int R=8\pi$; $R=\partial^2\xi$; $\Delta\xi$ a constant}. \end{eqnarray} Therefore, combined the equations above, \begin{eqnarray} G(\tilde{z}-\tilde{w})=\exp\left[\left(2\frac{2c}{12}-2h-2\bar{h}\right)\Delta\xi\right]G(z-w), \end{eqnarray} which gives $G(z-w,\bar{z}-\bar{w})\propto1/(z-w)^{2(h-c/12)}\cdot 1/(\bar{z}-\bar{w})^{2\left(\bar{h}-{c}/12\right)}$.
In my understanding, the correlator calculated above is not the conventional correlation function which is defined as $G(z-w)/Z$ where $Z$ is the partition function. Due to Weyl anomaly, $Z$ as the denominator will also obtain a phase ambiguity to cancel that from the $G(z-w)$. Then $G(z-w)/Z=1/(z-w)^{2h}\cdot 1/(\bar{z}-\bar{w})^{2\bar{h}}$. However, it is interesting to observe that the phase ambiguity from $Z$ is $z$- or $w$-independent while it cancels the $z$- or $w$-dependence of the numerator. It is within our expectation actually, because $G(z-w)/Z$ is potentially a ''zero-over-zero'' game since $Z$ vanishes due to Weyl anomaly. Thus, if we know $G(z-w)/Z$ is well-defined, $G(z-w)$ is actually zero as a direct consequence of Weyl anomaly.
