Moderately non-local $\eta \bar{\eta }$ vertices in the $Ad{S}_4$ higher-spin gauge theory

A new concept of moderate non-locality in higher-spin gauge theory is introduced. Based on the recently proposed differential homotopy approach, a moderately non-local scheme, that issofter than those resulting from the shifted homotopy approach available in the literature so far, is worked out in the mixed $\eta \bar{\eta }$ sector of the Vasiliev higher-spin theory. To calculate moderately non-local vertices ${\mathrm{{\rm Y}}}^{\eta \bar{\eta }}(\omega ,C,C,C)$ for all ordering of the fields $\omega$ and $C$ we apply an interpolating homotopy, that respects the moderate non-locality in the perturbative analysis of the higher-spin equations.