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Does there exist any gravity dual theory for theory with $N$-component scalar field with $(\phi^2)^2$ interaction?

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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

Specifically, they consider the 4d AdS dual of the 3d O($N$) model in the large $N$ limit. This is reviewed in

and is also mentioned in the more general review

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.

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