S=1 in O(N)/HS duality

TL;DR

The paper investigates how the Coleman-Mandula theorem manifests in the $O(N)$/Higher-Spin duality by formulating an $S$-matrix for scattering of collective bi-local dipoles in the $O(N)$ vector model. It shows that at the free UV fixed point, the collective $S$-matrix is trivial ($S=1$) because nonlinear $1/N$ interactions can be removed by a nonlinear field transformation, while boundary conditions or external potentials can render the $S$-matrix nontrivial. A detailed LSZ-like reduction for bi-local correlators demonstrates the vanishing of three- and four-point amplitudes (and generically higher points), consistent with a free-field structure despite the underlying nonlinear bulk theory. The authors explicitly construct the nonlinear bi-local transformation to a quadratic (free) theory, illustrating the Coleman-Mandula mechanism within the AdS$_4$/CFT$_3$ higher-spin context and clarifying the distinction between boundary and collective $S$-matrices. This work reinforces the view that large-$N$ vector models with higher-spin duals admit a solvable bulk description via bi-local collective fields and clarifies the role of boundary conditions in determining scattering data.

Abstract

Following the work of Maldacena and Zhiboedov, we study the implementation of the Coleman-Mandula theorem in the free O(N)/Higher Spin correspondence. In the bi-local framework we first define an S-matrix for scattering of collective dipoles. Its evaluation in the case of free UV fixed point theory leads to the result S=1 stated in the title. We also present an appropriate field transformation that is seen to transform away all the non-linear 1/N interactions of this theory. A change of boundary conditions and/or external potentials results in a nontrivial S-matrix.

Figures (4)

• Figure 1: Illustration of the four-point collective field diagrams.
• Figure 2: Scattering of three dipoles.
• Figure 3: The $s$-channel diagram of four-dipole scattering.
• Figure 4: The cross-shaped diagram of four-dipole scattering.