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**Unformatted text preview: **l. Even more interesting, these currents can be
obtained from some global invariances of the free theory by a
Noether-like procedure, provided the constant parameters
associated with these rigid symmetries be replaced by the
gauge parameters of the spin-s1 ﬁeld (also internal color
indices must be treated appropriately) (Berends, Burgers,
and van Dam, 1986; Deser and Yang, 1990). The simplest
class of cubic interactions below the Bell-Robinson line is
provided by the couplings between scalar ﬁelds (s2 ¼ 0)
and a collection of higher-spin tensor gauge ﬁelds through
the Berends–Burgers–van Dam currents containing s1 derivatives of the scalar ﬁelds (Berends, Burgers, and van Dam,
1986). Recently, they were reexamined by Bekaert (2006),
Fotopoulos et al. (2007), and Bekaert, Joung, and Mourad
(2009) as a toy model for higher-spin interactions. Note that
these cubic interactions induce, at ﬁrst order in the coupling
constant, gauge transformations for the scalar ﬁeld which are
non-Abelian at second order and reproduce the group of
unitary operators acting on free scalars on Minkowski spacetime (Bekaert, 2006; Bekaert, Joung, and Mourad, 2009).
As demonstrated by Boulanger, Leclercq, and Sundell
(2008), in a ﬂat background the non-Abelian 2-s-s vertex is
unique and involves a total number of 2s À 2 derivatives.
From s ¼ 3 on, the non-Abelian 2-s-s vertex in Minkowski
spacetime is thus ‘‘nonminimal’’ and the full Lagrangian (if
any) has no chance of being diffeomorphism invariant, a fact
explicitly shown by Boulanger and Leclercq (2006) and
Boulanger, Leclercq, and Sundell (2008). It was also shown
by Boulanger, Leclercq, and Sundell (2008) that the unique
and non-Abelian 2-s-s vertex in Minkowski spacetime is
nothing but the leading term in the ﬂat limit of the corresponding AdS Fradkin-Vasiliev vertex that, among others,
contains the Lorentz minimal coupling. That the minimal
Lorentz coupling term in the Fradkin-Vasiliev vertex is subleading in the ﬂat limit shows that the Weinberg equivalence
principle is restored for higher spins in AdS spacetime but is
lost in the ﬂat limit. This supports the need to consider higherspin interactions in the AdS background, at least if one wants
to make a contact between higher-spin gauge ﬁelds and lowspin theories including Einstein-Hilbert gravity. 21 Note that one can trivially write down higher-derivative BornInfeld–like consistent cubic interactions involving only gaugeinvariant linearized ﬁeld-strength tensors (Damour and Deser,
1987). However, these interactions deform neither the gauge algebra
nor the gauge transformations at ﬁrst order in some coupling
constant. Nevertheless, they might be needed when pushing the
non-Abelian cubic vertices to the next order in the coupling
constants. 996 Xavier Bekaert, Nicolas Boulanger, and Per A. Sundell: How higher-spin gravity surpasses the spin- . . . Recently, general results on the structure of cubic s-s0 -s00
couplings (s s0 s...

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