Shifted Homotopy Analysis of the Linearized Higher-Spin Equations in Arbitrary Higher-Spin Background

Analysis of the first-order corrections to higher-spin equations is extended to homotopy operators involving shift parameters with respect to the spinor $Y$ variables, the argument of the higher-spin connection $\omega(Y)$ and the argument of the higher-spin zero-form $C(Y)$. It is shown that a relaxed uniform $(y+p)$-shift and a shift by the argument of $\omega(Y)$ respect the proper form of the free higher-spin equations and constitute a one-parametric class of vertices that contains those resulting from the conventional (no shift) homotopy. A pure shift by the argument of $\omega(Y)$ is shown not to affect the one-form higher-spin field $W$ in the first order and, hence, the form of the respective vertices.