Invariant Functionals in Higher-Spin Theory

Abstract

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 $Ad{S}_4$ 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 $Ad{S}_4$. The peculiarity of the spinorial formulation of the on-shell $Ad{S}_3$ 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 $F_{\ast }(B(x))$ 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.

39 pages; V2 40 pages, typos corrected clarifications and references added. V3: The to be published version. Definition of the invariant functionals is slightly changed to make it globally defined, interpretation of the boundary singularities associated with nonlinear terms in higher-spin equations is modified, typos corrected, acknowledgement added, references updated

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