Using Noether's procedure we present a complete solution for the trilinear interactions of arbitrary spins s 1 , s 2 , s 3 in a flat background, and discuss the possibility to enlarge this construction to higher order interactions in the gauge field. Some classification theorems of the cubic (self)interaction with different numbers of derivatives and depending on relations between the spins are presented. Finally the expansion of a general spin s gauge transformation into powers of the field and the related closure of the gauge algebra in the general case are discussed.

The higher-spin (HS) algebras relevant to Vasiliev's equations in various dimensions can be interpreted as the symmetries of the minimal representation of the isometry algebra. After discussing this connection briefly, we generalize this concept to any classical Lie algebra and consider the corresponding HS algebras. For sp 2N and so N , the minimal representations are unique so we get unique HS algebras. For sl N , the minimal representation has one-parameter family, so does the corresponding HS algebra. The so N HS algebra is what underlies the Vasiliev theory while the sl 2 one coincides with the 3D HS algebra hs [λ]. Finally, we derive the explicit expression of the structure constant of these algebras -more precisely, their bilinear and trilinear forms. Several consistency checks are carried out for our results.

Abstract:We make an explicit link between the cubic interactions of off-shell fields and the on-shell three-point amplitudes in four dimensions. Both the cubic interactions and the on-shell three-point amplitudes had been independently classified in the literature, but their relation has not been made explicit. The aim of this note is to provide such a relation and discuss similarities and differences of their constructions. For the completeness of our analysis, we also derive the covariant form of all parity-odd massless vertices.

We propose a theory of three-dimensional (anti) de Sitter gravity carrying Chan-Paton color charges. We define the theory by Chern-Simons formulation with the gauge algebra (gl 2 ⊕ gl 2 ) ⊗ u(N ), obtaining a color-decorated version of interacting spinone and spin-two fields. We also describe the theory in metric formulation and show that, among N 2 massless spin-two fields, only the singlet one plays the role of metric graviton whereas the rest behave as colored spinning matter that strongly interacts at large N . Remarkably, these colored spinning matter acts as Higgs field and generates a non-trivial potential of staircase shape. At each extremum labelled by k = 0, . . . , [ N −1 2 ], the u(N ) color gauge symmetry is spontaneously broken down to u(N − k) ⊕ u(k) and provides different (A)dS backgrounds with the cosmological constants N N −2k 2 Λ. When this symmetry breaking takes place, the spin-two Goldstone modes combine with (or are eaten by) the spin-one gauge fields to become partially-massless spin-two fields. We discuss various aspects of this theory and highlight physical implications.

Abstract:The global symmetry algebras of partially-massless (PM) higher-spin (HS) fields in (A)dS d+1 are studied. The algebras involving PM generators up to depth 2 (ℓ − 1) are defined as the maximal symmetries of free conformal scalar field with 2 ℓ order wave equation in d dimensions. We review the construction of these algebras by quotienting certain ideals in the universal enveloping algebra of (A)dS d+1 isometries. We discuss another description in terms of Howe duality and derive the formula for computing trace in these algebras. This enables us to explicitly calculate the bilinear form for this one-parameter family of algebras. In particular, the bilinear form shows the appearance of additional ideal for any non-negative integer values of ℓ − d/2 , which coincides with the annihilator of the one-row ℓ-box Young diagram representation of so d+2 . Hence, the corresponding finite-dimensional coset algebra spanned by massless and PM generators is equivalent to the symmetries of this representation.

We present an off-shell generating function for all cubic interactions of Higher Spin gauge fields constructed in [1]. It is a generalization of the on-shell generating function proposed in [2], is written in a very compact way, and turns out to have a remarkable structure.

Abstract:We study the color-decoration of higher-spin (anti)-de Sitter gravity in three dimensions. We show that the rainbow vacua, which we found recently for the colored gravity theory, also pertain in the colored higher-spin theory. The color singlet spin-two plays the role of first fundamental form (metric). The difference is that when spontaneous breaking of color symmetry takes place, the Goldstone modes of massless spin-two combine with all other spins and become the maximal-depth partially massless fields of the highest spin in the theory, forming a Regge trajectory.

We study the cubic vertices for Maxwell-like higher-spins in flat and (A)dS background spaces of any dimension. Reducibility of their free spectra implies that a single cubic vertex involving any three fields subsumes a number of couplings among different particles of various spins. The resulting vertices do not involve traces of the fields and in this sense are simpler than their Fronsdal counterparts. We propose an extension of both the free theory and of its cubic deformation to a more general class of partially reducible systems, that one can obtain from the original theory upon imposing trace constraints of various orders. The key to our results is a version of the Noether procedure allowing to systematically account for the deformations of the transversality conditions to be imposed on the gauge parameters at the free level.