# Non-abelian cubic vertices for higher-spin fields in AdS(d)

TL;DR: In this article, the Fradkin-Vasiliev procedure was used to construct the full set of non-Abelian cubic vertices for totally symmetric higher spin gauge fields in AdS d in flat space.

Abstract: We use the Fradkin-Vasiliev procedure to construct the full set of non-Abelian cubic vertices for totally symmetric higher spin gauge fields in AdS
d
space. The number of such vertices is given by a certain tensor-product multiplicity. We discuss the one-to-one relation between our result and the list of non-Abelian gauge deformations in flat space obtained elsewhere via the cohomological approach. We comment about the uniqueness of Vasiliev’s simplest higher-spin algebra in relation with the (non)associativity properties of the gauge algebras that we classified. The gravitational interactions for (partially)-massless (mixed)-symmetry fields are also discussed. We also argue that those mixed-symmetry and/or partially-massless fields that are described by one-form connections within the frame-like approach can have non-Abelian interactions among themselves and again the number of non-Abelian vertices should be given by tensor product multiplicities.

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TL;DR: In this paper, the quartic vertex is obtained from the field theory four-point function of the operator dual to the bulk scalar, by making use of previous results for the Witten diagrams of higher-spin exchanges.

Abstract: Clarifying the locality properties of higher-spin gravity is a pressing task, but notoriously difficult due to the absence of a weakly-coupled flat regime. The simplest non-trivial case where this question can be addressed is the quartic self-interaction of the AdS scalar field present in the higher-spin multiplet. We investigate this issue in the context of the holographic duality between the minimal bosonic higher-spin theory on AdS4 and the free O(N) vector model in three dimensions. In particular, we determine the exact explicit form of the derivative expansion of the bulk scalar quartic vertex. The quartic vertex is obtained from the field theory four-point function of the operator dual to the bulk scalar, by making use of our previous results for the Witten diagrams of higher-spin exchanges. This is facilitated by establishing the conformal block expansions of both the boundary four-point function and the dual bulk Witten diagram amplitudes. We show that the vertex we find satisfies a generalised notion of locality.

220 citations

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TL;DR: In this paper, the authors studied the uniqueness of higher-spin algebras and showed that the Eastwood-Vasiliev algebra is the unique solution for d = 4 and d > 7.

Abstract: We study the uniqueness of higher-spin algebras which are at the core of higher-spin theories in AdS and of CFTs with exact higher-spin symmetry, i.e. conserved tensors of rank greater than two. The Jacobi identity for the gauge algebra is the simplest consistency test that appears at the quartic order for a gauge theory. Similarly, the algebra of charges in a CFT must also obey the Jacobi identity. These algebras are essentially the same. Solving the Jacobi identity under some simplifying assumptions spelled out, we obtain that the Eastwood-Vasiliev algebra is the unique solution for d = 4 and d > 7. In 5d there is a one-parameter family of algebras that was known before. In particular, we show that the introduction of a single higher-spin gauge field/current automatically requires the infinite tower of higher-spin gauge fields/currents. The result implies that from all the admissible non-Abelian cubic vertices in AdSd, that have been recently classified for totally symmetric higher-spin gauge fields, only one vertex can pass the Jacobi consistency test. This cubic vertex is associated with a gauge deformation that is the germ of the Eastwood-Vasiliev’s higher-spin algebra.

168 citations

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TL;DR: In this paper, a fully-gauge and o(d, 2 ) -covariant approach to boundary values of AdS d + 1 gauge fields is presented.

161 citations

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TL;DR: In this article, the authors compute the complete bulk-to-bulk propagators for massless bosonic higher-spin fields in the metric-like formulation, in any dimension and in various gauges.

Abstract: Within holography, we calculate the contribution of an arbitrary spin-s gauge boson exchange in AdS
d+1 to the four-point function with scalar operators on the boundary. As an important ingredient, we first compute the complete bulk-to-bulk propagators for massless bosonic higher-spin fields in the metric-like formulation, in any dimension and in various gauges. The split representation of the bulk-to-bulk propagators in terms of bulk-to-boundary propagators allows to present the higher-spin exchange diagram in the form of a conformal partial wave expansion. Our results provide a step towards the larger goal of the holographic reconstruction of bulk interactions, and of clarifying bulk locality.

143 citations

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TL;DR: In this paper, the authors make an explicit link between the cubic interactions of off-shell fields and the on-shell three-point amplitudes in four dimensions, and derive the covariant form of all parity-odd massless vertices.

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.

111 citations

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TL;DR: In this article, nonlinear field equations for totally symmetric bosonic massless fields of all spins in any dimension are presented, and a nonlinear nonlinear model of the field equations is proposed.

756 citations

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TL;DR: In this paper, consistent equations of motion of interacting gauge fields of all spins in 3+1 dimensions are formulated in a closed form, which are explicitly general coordinate invariant, possess all necessary higher spin gauge symmetries and reduce to the usual equations of free massless fields at the linearized level.

754 citations

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Yale University

^{1}TL;DR: In this article, a unified geometric formulation of gravitation and supergravity is presented, which is constructed out of the components of the curvature tensor for bundle spaces with four-dimensional Lorentz base manifold and structure groups Sp(4) for gravity and OSp(1, 4) for supergravity.

Abstract: A unified geometric formulation of gravitation and supergravity is presented. The action for these theories is constructed out of the components of the curvature tensor for bundle spaces with four-dimensional Lorentz base manifold and structure groups Sp(4) for gravity and OSp(1,4) for supergravity. The requirement of invariance under reflections, local Lorentz transformations, and general coordinate transformations uniquely determines the action and ensures the existence of local supersymmetry in supergravity.

744 citations

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TL;DR: In this article, the cohomology groups of the differential introduced by Becchi, Rouet, Stora and Tyutin are computed in a self-contained manner, with the sources of the BRST variations of the fields included in the problem.

611 citations

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TL;DR: In this paper, nonlinear field equations for totally symmetric bosonic massless fields of all spins in any dimension are presented, and a nonlinear nonlinear model of the field equations is proposed.

Abstract: Nonlinear field equations for totally symmetric bosonic massless fields of all spins in any dimension are presented.

587 citations