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**Unformatted text preview: **00 ) non-Abelian already at this order
were given, showing, in particular, that the maximum number
of derivatives involved in a non-Abelian coupling is 2s0 À 1
or 2s0 À 2, depending on the parity of the sum s þ s0 þ s00
(Bekaert, Boulanger, and Leclercq, 2010). It was also shown
that the cubic vertices saturating the upper derivative bound
have a good chance of being extended to second order in the
deformation parameter, as far as the Jacobi identity for the
gauge algebra is concerned. Later on, the generic nonAbelian vertices were studied and explicitly built by
Manvelyan, Mkrtchyan, and Ruehl (2010a, 2010b). Some
classiﬁcation results were also obtained about the structure
of the Abelian cubic vertices. A posteriori, the approach
(Manvelyan, Mkrtchyan, and Ruehl, 2010a, 2010b) to the
writing of covariant non-Abelian vertices can be seen as the
covariantization of the vertices already obtained in the lightcone approach by Bengtsson, Bengtsson, and Brink (1983a,
1983b) and Metsaev (2006, 2007) where, on top of the cubic
coupling given by the light-cone gauge approach, terms are
added which vanish in the spin-s De Donder gauge.
With the advent of string ﬁeld theory in the second half of
the 1980s, the construction of higher-spin cubic vertices in
ﬂat space was carried out by Koh and Ouvry (1986),
Bengtsson (1988), and Cappiello et al. (1989) in the socalled Becchi-Rouet-Stora-Tyutin (BRST) approach. This
approach was indeed motivated by the BRST ﬁrst quantization of the string and by the tensionless limit of open string
ﬁeld theory. More recently, this analysis was pursued by
Bonelli (2003) and Buchbinder et al. (2006), Fotopoulos
and Tsulaia (2007), Fotopoulos et al. (2007) [a review of the
last three works plus other works can be found in Fotopoulos
and Tsulaia (2009)]. The results obtained in this framework
are encouraging, for instance, in the case of non-Abelian
s-0-0 interactions (Fotopoulos et al., 2007), although the
higher-spin gauge ﬁeld (self- and cross) interactions found in
Fotopoulos and Tsulaia (2007) are Abelian, and therefore can
hardly be related to the non-Abelian higher-spin theory of
Vasiliev.
Before turning to the cubic interactions in AdS background, we continue with our review of positive results for
higher-spin cubic vertices in ﬂat space. Important results have
recently been obtained by analyzing the tree-level amplitudes
of the tensile (super)string. In what could be called a string
and S-matrix approach, Polyakov (2009, 2010), Taronna
(2010), and Sagnotti and Taronna (2011) obtained a plethora
of vertices and recovered the vertices obtained in the previously cited approaches, thereby creating a direct link between
open string theory and higher-spin gauge theory at the
dynamical level. Moreover, in light of the uniqueness results
of Boulanger, Leclercq, and Sundell (2008), one has a precise
relation between the Fradkin-Vasiliev vertices and string
theory.
Generically, the idea is that the non-Abelian ﬂat-space
cubic...

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