where V ( ω ω , . . . , C ) are the interaction vertices that are of higher and higher order in the C -field. The product in the higher-spin algebra is denoted by ⋆ ⋆ . The explicit form of the vertices can be extracted from the Vasiliev equations. The vertices that are bilinear in the fields are determined by the higher-spin algebra. Automorphism π π is induced by the automorphism of the anti-de Sitter algebra that flips the sign of translations, see below. If we truncate away higher orders in the C -expansion, the equations are just the zero-curvature condition for a connection ω ω of the higher-spin algebra and the covariant constancy equation for a zero-form C that takes values in the twisted-adjoint representation (twist is by the automorphism π π ).