What is the expression of Bulk-to-bulk propagator in global coordinates? I mean, I know that there is the standard expression in terms of hypergemotric funciton depending on the invariant chordal distance. For instance, in $AdS_3$

$$ G(z,x|z',x')\sim \xi^{\Delta} F_{21}[\Delta/2,\Delta/2+1/2,\Delta,\xi^2] $$ with $\xi$ the geodesic distance. The point is that everyone refers to this object written in Poincarè coordinates so that this distance becomes $$ \xi=\frac{2zz'}{z^2+z'^2+(x-x')^2} $$

My question is: in global coordinates, does the expression of the propagator remain the same as the one written above, with the chordal distance $\xi$ written in global coordinates?


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