Higher-spin Interactions from CFT: The Complete Cubic Couplings

In this letter we provide a complete holographic reconstruction of the cubic couplings in the minimal bosonic higher-spin theory in AdS$_{d+1}$. For this purpose we also determine the OPE coefficients of all single-trace conserved currents in the $d$-dimensional free scalar $O\left(N\right)$ vector model, and compute the tree-level three-point Witten diagram amplitudes for a generic cubic interaction of higher-spin gauge fields in the metric-like formulation.