We address the uniqueness of the minimal couplings between higher-spin fields and gravity. These couplings are cubic vertices built from gauge non-invariant connections that induce non-abelian deformations of the gauge algebra. We show that Fradkin-Vasiliev's cubic 2 − s − s vertex, which contains up to 2s − 2 derivatives dressed by a cosmological constant Λ, has a limit where: (i) Λ → 0; (ii) the spin-2 Weyl tensor scales non-uniformly with s; and (iii) all lower-derivative couplings are scaled away. For s = 3 the limit yields the unique non-abelian spin 2 − 3 − 3 vertex found recently by two of the authors, thereby proving the uniqueness of the corresponding FV vertex. We extend the analysis to s = 4 and a class of spin 1 − s − s vertices. The non-universality of the flat limit high-lightens not only the problematic aspects of higher-spin interactions with Λ = 0 but also the strongly coupled nature of the derivative expansion of the fully nonlinear higher-spin field equations with Λ = 0, wherein the standard minimal couplings mediated via the Lorentz connection are subleading at energy scales |Λ| < < E < < M p . Finally, combining our results with those obtained by Metsaev, we give the complete list of all the manifestly covariant cubic couplings of the form 1 − s − s and 2 − s − s , in Minkowski background.

We solve the problem of constructing consistent first-order cross-interactions between spin-2 and spin-3 massless fields in flat spacetime of arbitrary dimension n > 3 and in such a way that the deformed gauge algebra is non-Abelian. No assumptions are made on the number of derivatives involved in the Lagrangian, except that it should be finite. Together with locality, we also impose manifest Poincaré invariance, parity invariance and analyticity of the deformations in the coupling constants.1 Chargé de Recherches FNRS (Belgium); nicolas.boulanger@umh.ac.be 2 serge.leclercq@umh.ac.be plying γ to Equation (4.16), dγb 1 = 0 is obtained. Thanks to the Poincaré lemma and Proposition 3, we see that b 1 can be taken inLet us now introduce the differential D defined in Section 3.4, we obtainThis is because the left-hand side is strictly non γ-exact. Let us label the ghosts more preciselyThen, as D raises by 1 the D-degree, the only non zero components of the matrix A are A J i J i+1 and the last equation decomposes into:The first equation means thatthanks to Proposition 5 and Proposition 6. The λ's are constants, because of the Poincaré invariance. We obtainThus, β J 0 A J 0 J 1 depends only on the underivated antifields, which cannot be δ-exact modulo d unless they vanish, because a δ-exact term depends on the derivatives of the antifields or on the equations of motion, and because of the Poincaré invariance. Thus, δα J 1 + dβ J 1 = 0 and β J 0 A J 0 J 1 = 0 independently. By applying the same reasoning recursively, the same decomposition appears to occur at every D-degree, so we finally obtain:(4.18)

Based on a talk given by X.B. at the RTN Workshop "Constituents, Fundamental Forces and Symmetries of the Universe" (Corfu, 20-26th September 2005). Contribution to the Proceedings. AbstractThe problem of determining all consistent non-Abelian local interactions is reviewed in flat space-time. The antifield-BRST formulation of the free theory is an efficient tool to address this problem. Firstly, it allows to compute all on-shell local Killing tensor fields, which are important because of their deep relationship with higher-spin algebras. Secondly, under the sole assumptions of locality and Poincaré invariance, all non-trivial consistent deformations of a sum of spin-three quadratic actions deforming the Abelian gauge algebra were determined. They are compared with lower-spin cases.

In the eighties, Berends, Burgers and van Dam (BBvD) found a nonabelian cubic vertex for self-interacting massless fields of spin three in flat spacetime. However, they also found that this deformation is inconsistent at higher order for any multiplet of spinthree fields. For arbitrary symmetric gauge fields, we severely constrain the possible nonabelian deformations of the gauge algebra and, using these results, prove that the BBvD obstruction cannot be cured by any means, even by introducing fields of spin higher (or lower) than three.

The problem of constructing consistent parity-violating interactions for spin-3 gauge fields is considered in Minkowski space. Under the assumptions of locality, Poincaré invariance and parity non-invariance, we classify all the nontrivial perturbative deformations of the Abelian gauge algebra. In space-time dimensions n = 3 and n = 5, deformations of the free theory are obtained which make the gauge algebra non-Abelian and give rise to nontrivial cubic vertices in the Lagrangian, at first order in the deformation parameter g. At second order in g, consistency conditions are obtained which the five-dimensional vertex obeys, but which rule out the n = 3 candidate. Moreover, in the five-dimensional first-order deformation case, the gauge transformations are modified by a new term which involves the second de Wit-Freedman connection in a simple and suggestive way.

We give an explicit proof of the result that non trivial conserved n − 2 forms for a spin 2 field on a background corresponding to a solution to Einstein's equation (with or without cosmological constant) are characterized uniquely by the Killing vector fields of the background.