Holography in 4D (Super) Higher Spin Theories and a Test via Cubic Scalar Couplings

TL;DR

The paper develops a supersymmetric extension of the Klebanov–Polyakov HS/CFT program in AdS$_4$, introducing an ${ m N}=1$ HS theory with a parity-preserving structure and two bosonic truncations (Type A and Type B). It derives the scalar field equation to quadratic order, proving the vanishing of all non-derivative and quadratic scalar self-couplings, and analyzes the holographic duals: Type A aligns with the strongly coupled IR fixed point of the ${ m O}(N)$ vector model, while Type B is conjectured to dual the 3d Gross–Neveu model, with boundary-condition–driven Legendre transformations linking free and strongly coupled fixed points. The work further tests holography by showing that certain cubic scalar amplitudes vanish in the appropriate boundary conditions, consistent with CFT expectations. Overall, it extends HS/CFT dualities to an ${ m N}=1$ setting, clarifies parity constraints, and provides concrete 3-point/covariant tests of the dualities with implications for boundary deformations and RG flows in three-dimensional SUSY vector models.

Abstract

The correspondences proposed previously between higher spin gauge theories and free singleton field theories were recently extended into a more complete picture by Klebanov and Polyakov in the case of the minimal bosonic theory in D=4 to include the strongly coupled fixed point of the 3d O(N) vector model. Here we propose an N=1 supersymmetric version of this picture. We also elaborate on the role of parity in constraining the bulk interactions, and in distinguishing two minimal bosonic models obtained as two different consistent truncations of the minimal N=1 model that retain the scalar or the pseudo-scalar field. We refer to these models as the Type A and Type B models, respectively, and conjecture that the latter is holographically dual to the 3d Gross-Neveu model. In the case of the Type A model, we show the vanishing of the three-scalar amplitude with regular boundary conditions. This agrees with the O(N) vector model computation of Petkou, thereby providing a non-trivial test of the Klebanov-Polyakov conjecture.