Conformal Coupling of Higher Spin Gauge Fields to a Scalar Field in $AdS_{4}$ and Generalized Weyl Invariance

TL;DR

This work analyzes how higher-spin gauge fields in $AdS_{4}$ couple to a conformally coupled scalar, addressing two scalar boundary conditions ($\beta=1$ and $\beta=2$) that correspond to the $O(N)$ vector model's free and interacting fixed points in $CFT_{3}$. Using Noether's procedure and generalized Weyl invariance, the authors construct linearized interactions for spin-2 and spin-4 sectors, clarifying that $\beta=1$ couples to conformal, traceless currents while $\beta=2$ requires double-trace currents within the Fronsdal framework, with Weyl invariants connecting the two pictures. They derive explicit spin-2 and spin-4 couplings, including curvature-like invariants $r^{(2)}$ and $r^{(4)}$ and the corresponding transformation laws, yielding a linearized, gauge- and Weyl-invariant action. The results illuminate the $AdS_{4}/CFT_{3}$ holographic structure of the critical $O(N)$ model and provide concrete starting points for extending higher-spin interactions to higher spins in a controlled, symmetric framework.

Abstract

The higher spin interaction currents for the conformally coupled scalar in $AdS_{4}$ space for both regular and irregular boundary condition corresponding to the free and interacting critical point of the boundary O(N) sigma model are constructed. The explicit form of the linearized interaction of the scalar and spin two and four gauge fields in the $AdS_{D}$ space using Noether's procedure for the corresponding spin two and four linearized gauge and generalized Weyl transformations are obtained.