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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.

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  • $\begingroup$ In my understanding, the correlator in my post corresponds to the full correlator in contrast to the conventional correlator defined as the ratio of the full correlator and the partition function. Textbooks are using the conventional one which does not have an anomaly factor due to the cancellation from the partition function in the denominator. $\endgroup$
    – Yuan Yao
    Jul 10, 2020 at 13:49
  • $\begingroup$ You can either store curvature in local or global information, it’s up to you. In the former case a non-trivial metric and curvature etc will appear explicitly in the formalism (but they may drop out at the end of a calculation depending on what one is calculating). In the latter case you can store curvature implicitly in the transition functions on patch overlaps, which allows you to work with locally flat surfaces in any given local chart on your surface. These comments hold for all Riemann and super-Riemann surfaces. In the usual CFT texts the transition function approach is implied. $\endgroup$ Jul 10, 2020 at 19:29
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    $\begingroup$ Having said that (and in the spirit of the Socratic method), I’ll ask you a question: you quote the Gauss-Bonnet theorem in your last sentence; the integrand you display is only locally defined, so does your expression make sense? What should you have written instead, and how do you calculate it (using BOTH methods mentioned above)? I’m asking because this easier calculation automatically leads to the answer to your question. $\endgroup$ Jul 10, 2020 at 19:48
  • $\begingroup$ @Wakabaloola I have edited the post and I hope it would be clear to you. Well, I avoided using the coordinate patches by playing with Ricci curvature $\partial^2\xi$ whose value is globally defined. $\endgroup$
    – Yuan Yao
    Jul 11, 2020 at 1:59
  • $\begingroup$ @Wakabaloola By the way, I just found a relevant reference ias.ac.in/article/fulltext/pram/035/03/0205-0286 Please check Eqs.(3.2, 3.3), which implies my calculation seems correct. $\endgroup$
    – Yuan Yao
    Jul 11, 2020 at 2:03

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