Cubic Vertices for Symmetric Higher-Spin Gauge Fields in $(A)dS_d$

TL;DR

This paper develops a new formalism to analyze cubic interaction vertices for symmetric higher-spin gauge fields in (A)dS_d, generalizing previous results and proposing a comprehensive set of higher-derivative vertices.

Contribution

It introduces a cohomological formalism within the frame-like approach and constructs a set of cubic vertices for higher-spin fields in (A)dS_d, extending known flat-space results.

Findings

Derived (A)dS generalization of known $AdS_4$ cubic action.

Demonstrated how (A)dS vertices emerge from Minkowski vertices.

Proposed a set of higher-derivative cubic vertices for all spins $s extgreater=2$.

Abstract

Cubic vertices for symmetric higher-spin gauge fields of integer spins in $(A)dS_d$ are analyzed. $(A)dS_d$ generalization of the previously known action in $AdS_4$, that describes cubic interactions of symmetric massless fields of all integer spins $s\geq 2$, 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 $(A)dS_d$, 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 $s\geq 2$ is proposed, which is conjectured to describe all vertices in $AdS_d$ 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 $\hbar$ of the higher-spin algebra to the $(A)dS$ radius.