Invariant Functionals in Higher-Spin Theory

A new construction for gauge invariant functionals in the nonlinear higher-spin theory is proposed. Being supported by differential forms closed by virtue of the higher-spin equations, invariant functionals are associated with central elements of the higher-spin algebra. In the on-shell $AdS_4$ higher-spin theory we identify a four-form conjectured to represent the generating functional for $3d$ boundary correlators and a two-form argued to support charges for black hole solutions. Two actions for $3d$ boundary conformal higher-spin theory are associated with the two parity-invariant higher-spin models in $AdS_4$. The peculiarity of the spinorial formulation of the on-shell $AdS_3$ higher-spin theory, where the invariant functional is supported by a two-form, is conjectured to be related to the holomorphic factorization at the boundary. The nonlinear part of the star-product function $F_*(B(x))$ in the higher-spin equations is argued to lead to divergencies in the boundary limit representing singularities at coinciding boundary space-time points of the factors of $B(x)$, which can be regularized by the point splitting. An interpretation of the RG flow in terms of proposed construction is briefly discussed.