Shifted homotopy analysis of the linearized higher-spin equations in arbitrary higher-spin background

Abstract

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 ω(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 ω(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 ω(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.