Singleton deformation of higher-spin theory and the phase structure of the three-dimensional O(N) vector model

We consider a singleton deformation of the AdS₄ higher-spin theory dual to the three-dimensional O(N) vector model. The singleton couples to the higher-spin multiplet only through a marginal boundary interaction. We argue that the effect of such a deformation is to shift N→N+1 in both sides of the holographic correspondence and we show how the gap equations of the three-dimensional O(N) vector model arise from the higher-spin theory. The singleton deformation breaks higher-spin symmetry and gives rise to the well-known 1/N anomalous dimensions of the boundary theory.