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In AdS/CFT, the bulk-to-bulk propagator can be obtained as the limit of the bulk-to-bulk propagator with one point approaching the boundary. For example in the scalar case \begin{equation} K_{\Delta}(t,x,z|t',x')=\lim_{z'\rightarrow 0}z'^{-\Delta}\langle \varphi(t,x,z)\varphi(t',x',z')\rangle, \end{equation} which turns out to be a function of a "renormalized" distance $\tilde{\sigma}=\lim_{z'\rightarrow 0}z'\sigma$, where $\sigma$ is the AdS invariant distance \begin{equation} \sigma(t,x,z|t',x',z')=\frac{z^2+z'^2-(t-t')^2+|x-x'|^2}{2zz'}. \end{equation} I'm sure that the bulk-to-bulk propagator is a scalar under diffeos, but I'm not convinced on how the bulk-to-boundary should transform under change of coordinates. Is one allowed to consider only transformations that leave the boundary unchanged?

Maybe I should add that this question arised reading this article, where the spinor counterpart is obtained, for example in equation (60) of the reference, where the found expression doesn't strike me as a scalar (mostly because of the $\Gamma^Ax_A$ term).

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