Moderately non-local $\eta \bar\eta$ vertices in the $AdS_4$ 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 $\Upsilon^{\eta\bar \eta}(\omega, C,C,C)$ 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.