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

Abstract

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 𝛽 →-∞-shifted contracting homotopy introduced in the paper. The problem is solved in a background independent way and for any value of the complex parameter 𝜂 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 𝜂2 and $$ {\overline{\eta}}^2 $$ η ¯ 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 $$ \overline{y} $$ y ¯ . Also the 𝜂2 and $$ {\overline{\eta}}^2 $$ η ¯ 2 vertices are shown to vanish on any purely gravitational background hence not contributing to the higher-spin current interactions on AdS 4 . This implies in particular that the gravitational constant in front of the stress tensor is positive being proportional to $$ \eta \overline{\eta} $$ η η ¯ . It is shown that the 𝛽-shifted homotopy technique developed in this paper can be reinterpreted as the conventional one but in the 𝛽-dependent deformed star product.