Cubic Vertices for Symmetric Higher-Spin Gauge Fields in $(A)d{S}_d$

Abstract

Cubic vertices for symmetric higher-spin gauge fields of integer spins in $(A)d{S}_d$ are analyzed. $(A)d{S}_d$ generalization of the previously known action in $Ad{S}_4$, that describes cubic interactions of symmetric massless fields of all integer spins $s\ge 2$, is found. A new cohomological formalism for the analysis of vertices of higher-spin fields of any symmetry and/or order of nonlinearity is proposed within the frame-like approach. Using examples of spins two and three it is demonstrated how nontrivial vertices in $(A)d{S}_d$, including Einstein cubic vertex, can result from the $AdS$ deformation of trivial Minkowski vertices. A set of higher-derivative cubic vertices for any three bosonic fields of spins $s\ge 2$ is proposed, which is conjectured to describe all vertices in $Ad{S}_d$ that can be constructed in terms of connection one-forms and curvature two-forms of symmetric higher-spin fields. A problem of reconstruction of a full nonlinear action starting from known unfolded equations is discussed. It is shown that the normalization of free higher-spin gauge fields compatible with the flat limit relates the noncommutativity parameter $\hslash$ of the higher-spin algebra to the $(A)dS$ radius.

78 pages. V2: typos corrected, coefficients in sections 5.1 and 6.3 corrected and undetermined coefficients of the supertrace of higher-spin algebra in section 6.1 and of the relation between tensor and generating function formalism in section 7.3 are computed. Clarifications, graphs, references and acknowledgments added. V3: typos corrected, reference added, the version to appear in Nucl.Phys.B

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