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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 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 ...


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Sponteneous symmetry breaking occurs when the ground state has a different (reduced) symmetry from the Hamiltonian/Lagrangian. So $\mathcal{L}$ retains the $U(1)$ or shift symmetry, but you have to look at whether or not the ground state also does. The "gauged $XY$" model is also called the Abelian-Higgs model. The first is the usual $U(1)$ gauge symmetry ...


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