Higher-Spin Algebras, Holography and Flat Space

TL;DR

The work demonstrates that the cubic couplings of the type-A minimal bosonic higher-spin theory on $AdS_{d+1}$, reconstructed holographically from the free $O(N)$ model, realize the unique higher-spin algebra in generic dimensions, providing a nontrivial test of HS holography beyond $AdS_4$. It further shows that in four-dimensional flat space, Metsaev’s cubic vertices imply a compatible higher-spin structure that matches the AdS$_4$ Lorentz sector under a contraction, hinting at an underlying HS symmetry for flat space. The analysis combines ambient-space formalism, Noether deformations, and spinor-helicity techniques, and culminates in explicit matches of HS structure constants and bilinear forms with known Eastwood–Vasiliev algebras. These results strengthen the link between holographic duality for HS theories and the algebraic underpinning of their interactions, while highlighting the subtle role of lower-derivative (exotic) couplings in flat space and their interpretation as minimal couplings or indicators of underlying symmetry. The findings have implications for higher-order amplitudes and the realization of HS symmetry in both AdS and flat backgrounds, and motivate further investigation into covariantising nonlocal structures and the possible oscillator realizations of the flat-space HS algebra.

Abstract

In this article we study the algebra generated by the holographically reconstructed cubic couplings for the type A minimal bosonic higher-spin theory on AdS$_{d+1}$, which were recently extracted from the free scalar $O(N)$ model. We demonstrate that it is equal to the unique higher-spin algebra for bosonic totally symmetric higher-spin fields in generic dimensions. This provides an explicit check of the holographic reconstruction and of the duality between higher-spin theories and the free $O(N)$ model in general dimensions, extending the result of Giombi and Yin in AdS$_4$. For completeness, we also address the same problem in the flat space for the cubic couplings derived by Metsaev in 1991, which are recovered in the flat limit of the AdS type-A cubic couplings. We observe that both flat and AdS$_4$ higher-spin Lorentz subalgebras coincide, hinting towards the existence of a full higher-spin symmetry behind the flat-space cubic couplings of Metsaev.

Figures (1)

• Figure 1: Holographic reconstruction of the bulk $s_1$-$s_2$-$s_3$ cubic vertex from the three-point function $\langle {\cal J}_{s_1}{\cal J}_{s_2}{\cal J}_{s_3} \rangle$ of single-trace operators of spins $s_i$ in the free scalar $O\left(N\right)$ model.