Higher spin interactions in four-dimensions: Vasiliev versus Fronsdal

We consider four-dimensional higher-spin (HS) theory at the first nontrivial order corresponding to the cubic action. All HS interaction vertices are explicitly obtained from Vasiliev’s equations. In particular, we obtain the vertices that are not determined solely by the HS algebra structure constants. The dictionary between the Fronsdal fields and HS connections is found and the corrections to the Fronsdal equations are derived. These corrections turn out to involve derivatives of arbitrary order. We observe that the vertices not determined by the HS algebra produce naked infinities, when decomposed into the minimal derivative vertices and improvements. Therefore, standard methods can only be used to check a rather limited number of correlation functions within the HS AdS/CFT duality. A possible resolution of the puzzle is discussed.