Group manifold approach to higher spin theory

We consider the group manifold approach to higher spin theory. The deformed local higher spin transformation is realized as the diffeomorphism transformation in the group manifold $\mathbf{\text{M}}$. With the suitable rheonomy condition and the torsion constraint imposed, the unfolded equation can be obtained from the Bianchi identity, by solving which, fields in $\mathbf{\text{M}}$ are determined by the multiplet at one point, or equivalently, by $(W_{\mu }^{[a(s-1),b(0)]},H)$ in $Ad{S}_4\subset \mathbf{\text{M}}$. Although the space is extended to $\mathbf{\text{M}}$ to get the geometrical formulation, the dynamical degrees of freedom are still in $Ad{S}_4$. The $4d$ equations of motion for $(W_{\mu }^{[a(s-1),b(0)]},H)$ are obtained by plugging the rheonomy condition into the Bianchi identity. The proper rheonomy condition allowing for the maximum on-shell degrees of freedom is given by Vasiliev equation. We also discuss the theory with the global higher spin symmetry, which is in parallel with the WZ model in supersymmetry.