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In 2D we can always choose coordinates in a coordinate patch so that the metric is conformally flat $$g_{\mu\nu}(x)=\kappa(x)\delta_{\mu\nu}$$ A simple example is the sphere $S^2$ in stereographic coordinates.

If we put a massless scalar field on this manifold in these coordinates, the action just reduces to that of a scalar field on a flat manifold $$\int d^2 x \sqrt{g} \,\,g^{\mu\nu}\partial_\mu \phi\partial_\nu \phi=\int d^2 x \,\,\delta^{\mu\nu}\partial_\mu \phi\partial_\nu \phi$$ If I'm understanding correctly, if the manifold has the topology of a sphere, or a sphere with punctured points, then there are also no topological obstructions to treating this as a global integral over $R^2$.

Since the action is the same, were it not for the conformal anomaly it seems that we could just take the correlators for a massless field in flat space and directly apply it to any curved manifold (at least with compatible topology) in conformally flat coordinates.

But this can't be right, since say for $S^2$ we would expect the correlation function to have symmetry appropriate to the sphere. If $\mathbf{x}_1,\mathbf{x}_2$ are stereographic coordinates on the sphere, the preceding argument would give (please forgive constant factors) $$\langle\phi(\mathbf{x}_1)\phi(\mathbf{x}_2)\rangle_{S^2}=-\log |\mathbf{x}_1-\mathbf{x}_2|^2$$ But this correlation function should depend on the geodesic distance between $x_1,x_2$ as points on $S^2$, not the Euclidean distance between this specific choice of coordinates $x_1,x_2$ themselves.

Now of course the metric still appears in the definition of the measure of the path integral. But are the correlation functions really so sensitive to this? Is it possible to use the flat space correlators for the curved space theory, and if so, how do we resolve the symmetry issue I brought up?

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  • $\begingroup$ Here's a closely related question (that apparently I even upvoted at some point and forgot) physics.stackexchange.com/questions/245089/… $\endgroup$
    – octonion
    Oct 12 '19 at 7:28
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    $\begingroup$ you can work in conformally flat coordinates, globally as well as locally. that there is nontrivial euler number can be packaged into the transition functions (and corresponding cocycle relations) on patch overlaps. $\endgroup$ Oct 12 '19 at 18:29
  • $\begingroup$ @octonion Eqs.(6.2.7-6.2.17) in Polchinski Vol. 1 can be helpful to you. $\endgroup$
    – Yuan Yao
    Feb 23 '21 at 11:47

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