Minimal models of field theories: Chiral higher spin gravity

There exists a unique class of local higher spin gravities with propagating massless fields in $4d$—chiral higher spin gravity. Originally, it was formulated in the light-cone gauge. We construct a covariant form of this theory as a free differential algebra up to the next-to-leading order, i.e., at the level of equations of motion. It also contains the recently discovered covariant forms of the higher spin extensions of self-dual Yang-Mills and self-dual gravity, as well as self-dual Yang-Mills and self-dual gravity themselves. From the mathematical viewpoint the result is equivalent to taking the minimal model (in the sense of $L_{\infty }$-algebras) of the jet-space extension of the BV-BRST formulation of chiral higher spin gravity, thereby, containing also information about (presymplectic AKSZ) action, counterterms, anomalies, etc.