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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 $CFT_{2}$.

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What are geodesic propagators? You mean propagators of (say) scalar field in the geodesic approximation? In any event, BTZ is an orbifold of AdS3, you can just use the method of images to construct any object in BTZ if you know the “parent” object in AdS3. –  user566 Mar 7 '11 at 1:27
thanks @Moshe. Yes, I meant for instance the propagator in BTZ of an scalar field. I know that the BTZ background is an orbifold of pure AdS3 and that this allows to construct any object in BTZ from AdS3 by orbifolding but I wanted to know how, if possible, can we writte these objects in terms of meromorphic functions on the conformal boundary, say, products of Jacobi theta functions. –  xavimol Mar 7 '11 at 8:58
So, you can get the propagator in BTZ (with or without the geodesic approximation) as sum over images - this is done in many places in the literature, and it’s probably easiest to just do it yourself. If I understand you correctly, you want to know if the resulting sum has a closed form in terms of known functions? I am not sure if that’s the case, but what is the significance of this? Or, maybe I am misunderstanding and you are asking for the analytic structure of that propagator? –  user566 Mar 7 '11 at 16:58
Well, my interest in writing the sum over images (in the geodesic approximation) in terms of some known functions comes because I'm trying to match some result derived with CFT techniques on the torus by means of the holographic approach in the geodesic limit. Problably it's naive but, as the BTZ background has, as a conformal boundary, the torus T2, I was asking myself if there is some method to write these geodesic propagators in terms of "canonical functions" of T2 (i.e Jacobi theta functions). –  xavimol Mar 7 '11 at 22:03
Oh, I see. Unfortunately I don't know the answer to your question off hand. But, for what its worth, it sounds right to me: there should be an expression in terms of the building blocks for a modular invariant partition function. –  user566 Mar 8 '11 at 0:19

1 Answer 1

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 $SL(2,R)$.

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Thanks for the answer and the interest. –  xavimol Mar 7 '11 at 8:59

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