Limiting Shifted Homotopy in Higher-Spin Theory and Spin-Locality

Higher-spin vertices containing up to quintic interactions at the Lagrangian level are explicitly calculated in the one-form sector of the non-linear unfolded higher-spin equations using a $\beta \to -\mathrm{\infty }$–shifted contracting homotopy introduced in the paper. The problem is solved in a background independent way and for any value of the complex parameter $\eta$ in the HS 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 ${\eta }^2$ and ${\bar{\eta }}^2$ 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 $\bar{y}$. Also the ${\eta }^2$ and ${\bar{\eta }}^2$ vertices are shown to vanish on any purely gravitational background hence not contributing to the higher-spin current interactions on $Ad{S}_4$. This implies in particular that the gravitational constant in front of the stress tensor is positive being proportional to $\eta \bar{\eta }$. It is shown that the $\beta$-shifted homotopy technique developed in this paper can be reinterpreted as the conventional one but in the $\beta$-dependent deformed star product.