# 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?

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
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.