We study the O(N) and Gross-Neveu models at large N on AdSd+1 background. Thanks to the isometries of AdS, the observables in these theories are constrained by the SO(d, 2) conformal group even in the presence of mass deformations, as was discussed by Callan and Wilczek [1], and provide an interesting two-parameter family of quantities which interpolate between the S-matrices in flat space and the correlators in CFT with a boundary. For the actual computation, we judiciously use the spectral representation to resum loop diagrams in the bulk. After the resummation, the AdS 4-particle scattering amplitude is given in terms of a single unknown function of the spectral parameter. We then “bootstrap” the unknown function by requiring the absence of double-trace operators in the boundary OPE. Our results are at leading nontrivial order in 1N, and include the full dependence on the quartic coupling, the mass parameters, and the AdS radius. In the bosonic O(N) model we study both the massive phase and the symmetry-breaking phase, which exists even in AdS2 evading Coleman’s theorem, and identify the AdS analogue of a resonance in flat space. We then propose that symmetry breaking in AdS implies the existence of a conformal manifold in the boundary conformal theory. We also provide evidence for the existence of a critical point with bulk conformal symmetry, matching existing results and finding new ones for the conformal boundary conditions of the critical theories. For the Gross-Neveu model we find a bound state, which interpolates between the familiar bound state in flat space and the displacement operator at the critical point.
