Duality between gravitation and $O(N)$ model Does there exist any gravity dual theory for theory with $N$-component scalar field with $(\phi^2)^2$ interaction?
 A: The 4d AdS dual of the conformally-symmetric case of the 3d $(\phi\cdot\phi)^2$ model with $\phi$ in the fundamental representation of O($N$) is studied in


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*Klebanov and Polyakov, "AdS Dual of the Critical O(N) Vector Model," https://arxiv.org/abs/hep-th/0210114
Specifically, they consider the 4d AdS dual of the 3d O($N$) model in the large $N$ limit. This is reviewed in


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*Giombi, "TASI Lectures on the Higher Spin - CFT duality," https://arxiv.org/abs/1607.02967
and is also mentioned in the more general review


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*Penedones, "TASI lectures on AdS/CFT," https://arxiv.org/abs/1608.04948
around equation (128). The corresponding 4d AdS theory has one massless field for each even spin $s$, remarkably including $s\geq 4$. This is a gravitational dual because it includes a massless $s=2$ field. This dual 4d AdS theory gives correlation functions of the singlet currents (such as $J=\phi\cdot\phi$) in the 3d O($N$) theory.
For other examples of recent work with the O($N$) CFTs, see https://arxiv.org/abs/0912.3462 and https://arxiv.org/abs/1307.6856.
