# Construction of the supergravity side explicitly in gauge/gravity dualities

Although the motivation of this question comes from the AdS/CFT correspondence, it actually is related to a more general principle of gauge/gravity duality. We know from Maldacena's conjecture that a $\mathcal{N}=4$ SYM theory is dual to a Type IIB String Theory on an $AdS_5 \times S^5$ background. Mathematically, the identification is $$\left\langle \int \phi_0 \mathcal{O}\right\rangle_{SYM} = Z[\phi(\mathbf{x},0)=\phi_0]$$ where $\phi_0$ are boundary fields and $\phi$ s are bulk fields while $\mathcal{O}$ are single trace operators of the boundary theory.\ Now, this comes mainly by studying the symmetries of the two theories. (Both of them have the symmetry groups $SO(2,4)$ and $SO(6)$). Also, in the low energy limit, i.e when $\alpha^{\prime} = l_s^2 \rightarrow 0$, we see that the two theories decouples from the system. In the bulk we have a theory of 10D free supergravity. Now, say I start with some gauge theory (not necessarily $\mathcal{N}=4$ SYM theory) on the boundary which is Minkowski like. Starting with that theory, how do we construct the correct gravity theory in the bulk that will be dual to the gauge theory on the boundary. What i understand is that there are some things about the bulk metric that we can say from the holographic principle. Since, we started with a gauge theory in Minkowski background, the metric in the bulk should be such that its conformal boundary must be Minkowski. My questions is:\ Is there an explicit prescription which will allow me to find out what is the theory in the bulk if I know what my boundary theory is?\ Also, there is this issue of the weak-strong coupling duality. the strong coupling regime of the gauge theory is dual to the weak coupling regime of the gravity side and vice versa. Anyways, we cannot evaluate the LHS in the strong coupling regime since, perturbation theory will just break down. So, we have to take the weak coupling limit on the LHS. But, that would imply working in the strong coupling regime on the gravity side. Then, we basically extend the result of AdS/CFT and we can say that we at least know the partition function of the gravity side when it is strongly coupled.

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What you are describing is a general problem in AdS/CFT. It is easy to "put a CFT on the boundary" but there is no golden recipe to construct the bulk theory. Indeed you realize this yourself, in the end of your post: if you take CFT with small gauge group on the bd. you will get not some nice SUGRA model but a spaghetti theory in the bulk. –  Vibert May 11 '13 at 9:13
Actually I should have mentioned in the post itself. My question was also motivated by the paper (arxiv.org/pdf/1011.2986.pdf) of Gopakumar & Gaberdiel. They have shown the duality of two theories again. This time, the theories are a 2d $\mathcal{W}_{N}$ minimal model with large $N$ ('t Hooft limit) and a higher spin theory in $AdS_3$. What I'm asking is this. If I would have started with the 2d $\mathcal{W}_{N}$ minimal model then, can the fact that it is dual to a higher spin theory in $AdS_3$ automatically emerge out from my calculations? –  Debangshu May 11 '13 at 9:24
Also, although I have not mentioned it explicitly, I guess I am also assuming a very big result though I am not sure if it is true: Every CFT will have an AdS dual theory. If this is false, then obviously my question itself becomes irrelevant. –  Debangshu May 11 '13 at 9:29