Higher-Spin Algebras, Holography and Flat Space

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.