Can the geodesic propagators in the Euclidean BTZ black hole can be written in terms of meromorphic functions on its conformal boundary? I'm interested in knowing if ,in the context of $AdS_{3}/CFT_{2}$, we can (and how to) express the geodesic propagators on the bulk space of the Euclidean $AdS_{3}$ black holes, in terms of  meromorphic functions on its conformal boundary $CFT_{2}$.
 A: This is an interesting question, which does have a bearing on the problem of qubits and black hole categories, or BPS charges.  I can only lay down some possible ideas for how to think about this.  The anti-de Sitter spacetime is the quotient space
$$
AdS_n~=~\frac{SO(n-1,2)}{SO(n-1,1)}
$$
We denote by $g~=~so(n-1,2)$ and $h~=~so(n-1,1)$ the Lie algebras and by   the projection
$g~\rightarrow~g/h$. An involutive automorphism $\mu:g~\rightarrow~g$ which fixes elements of $h$, and we call $f$ the eigenspace of eigenvalue $−1$ of $\mu$. Thus $g$ decomposes as an algebra as $g~=~h\oplus f$. The compact part of $SO(n-1,2)$ is the decomposition into $K~=~SO(2)\times SO(n − 1)$.  So this defines a set of elements which are in involution with this automorphism.  This defines a certain orbit or path.
For points which are nilpotent under the group $AdS_2~\simeq~SL(2,R)$ and $AdS_4~\sim~SL(2,R)^2$ we can define the orbits of the AdS by nilpotent conditions on ${\cal G}~=~SL(2,R)^n$.  A set $S$ which defines a horizon $t^2~-~x^2~=~0$ define the BTZ as $BTZ~\sim~AdS_n\setminus S$, and where the BTZ is given by the fixed or nilpotent points of the group.  So we might then consider looking at these dynamics on the moduli space, and where the Killing vectors are timelike, with no spacelike isometries due to spatial stationary conditions.  
SO if I were to set this up I might try the following. Let the $AdS_4$ spacetime with the IR decomposition $AdS_2\times {\cal M}$. The gauge action in $AdS_4$ spacetime is in the STU model is similar to 
$$
S~=~\int d^4x (R~+~G_{ab}\partial{\bar z}^a\partial z^b~-~m_{IJ}F^IF^J~-~n_{IJ}F^I(*F)^J
$$
The equations of motion may be solved with the metric for a charged black hole, or equivalently a $D$-brane geometry
$$
ds^2~=~r^2 F(r)(-dt^2~+~dx^idx_i)~+~{1\over{r^2F(r)}}dr^2,~F(r)~=~1~+~{m\over r}
$$
For the near horizon condition we have the metric reduce to $AdS_2$ on the $t,~r$ coordinates may be removed from the $x^i$ coordinates of the $2$-sphere.  A substitution of variables $R(r)~=~e^{-2U}/r^2$ gives the $AdS_2$ metric $ds^2~=~-e^{-2U}dt^2~-~e^{2U}dr^2$.  The dynamics is then on $D2$-branes.  The portion of the STU action which gives the hyperbolic dynamics on the space ${\cal L}~=~\sqrt{-h}\partial g_{ab}{\bar z}^a z^b$ gives the geodesic equation with $\tau~=~1/r$ where the scalars and gravity decouple:
$$
{{d^2\phi^i}\over{d\tau^2}}~+~{\Gamma^i}_{jk}{{d\phi^j}\over{d\tau}}{{d\phi^j}\over{d\tau}}
$$
This geodesic equation the defines the orbits, where the moduli are determined by the nilpotent orbits of the group.  This would then be used to address the question   The condition for the BTZ horizon is then mapped to the boundary of the $AdS_n$ or $CFT_{n-1}$.  This then should be a copy of the conformal group of quantum mechanics $SL(2,R)$.
