Limiting shifted homotopy in higher-spin theory and spin-locality

Abstract

Higher-spin vertices containing up to quintic interactions at the LagTangian level are explicitly calculated in the one-form sector of the non-linear unfolded higher-spin equations using a \U0001d6fd →-∞-shifted contracting homotopy introduced in the paper. The problem is solved in a background independent way and for any value of the complex parameter \U0001d702 in the higher-spin equations. All obtained vertices are shown to be spin-local containing a finite number of derivatives in the spinor space for any given set of spins. The vertices proportional to \U0001d7022 and {overline{eta}}^2 are in addition ultra-local, i.e., zero-forms that enter into the vertex in question are free from the dependence on at least one of the spinor variables y or overline{y} . Also the \U0001d7022 and {overline{eta}}^2 vertices are shown to vanish on any purely gravitational background hence not contributing to the higher-spin current interactions on AdS4. This implies in particular that the gravitational constant in front of the stress tensor is positive being proportional to eta overline{eta} . It is shown that the \U0001d6fd-shifted homotopy technique developed in this paper can be reinterpreted as the conventional one but in the \U0001d6fd-dependent deformed star product.