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**Unformatted text preview: **cetime as background. Using the
light-cone gauge approach, higher-spin s-s0 -s00 cubic vertices
in four spacetime dimensions were found by Bengtsson,
Bengtsson, and Brink (1983a, 1983b), Bengtsson,
Bengtsson, and Linden (1987), and Fradkin and Metsaev
(1991). These results, in the light-cone gauge approach,
were considerably generalized later by Metsaev, 1993a,
1993c, 2006, 2007) and Fradkin and Metsaev (1995) with a
complete classiﬁcation of cubic (self- and cross)couplings for
arbitrary massive and massless higher-spin ﬁelds, bosonic
and fermionic, in dimensions four, ﬁve, and six. Mixedsymmetry ﬁelds were also considered therein. Moreover,
Metsaev (1993b) obtained a wide class of cubic interactions
for arbitrary ﬁelds in arbitrary dimension.
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As far as manifestly Poincare-invariant vertices in the
Lagrangian approach are concerned, Berends, Burgers, and
van Dam obtained a class of manifestly covariant, nonAbelian cubic couplings (Berends, Burgers, and van Dam,
1984, 1985). They used a systematization of the Noether
procedure for introducing interactions, where the couplings
are not necessarily of the form ‘‘gauge ﬁeld times conserved
current.’’ Berends, Burgers, and van Dam (1985) obtained
consistent and covariant cubic couplings of the kind s1 -s2 -s2 ,
for the values of s1 and s2 indicated in Table I. Of course,
some of the vertices were already known before, such as, for
example, in the cases 1-1-1, 2-2-2, and 2- 3 - 3 corresponding
22
to Yang-Mills, Einstein-Hilbert, and ordinary supergravity
theories, respectively. There is a class of cross interactions
s1 -s2 -s2 for which the cubic vertices could easily been
written. This class corresponds to the ‘‘Bell-Robinson’’ line
s1 ¼ 2s2 and below this line s1 > 2s2 (Berends, Burgers, and
van Dam, 1986); see Deser and Yang (1990) in the s1 ¼ 4 ¼
2s2 case and some more recent considerations by Manvelyan,
Mkrtchyan, and Ruehl (2010c). In the aforementioned region
s1 ! 2s2 , the gauge algebra remains Abelian at ﬁrst order in a
coupling constant although the gauge transformations for the
spin-s2 ﬁeld are deformed. The reason is that the ﬁrst-order
20
As a matter of fact, a nonstandard action principle for Vasiliev’s
equations, which leads to a nontrivial quantization, was proposed by
Boulanger and Sundell (2011). Rev. Mod. Phys., Vol. 84, No. 3, July–September 2012 995 deformation of the free spin-s2 gauge transformations involves the spin-s2 ﬁeld only through its gauge-invariant
Weinberg–de Wit–Freedman ﬁeld strength (Weinberg,
1965; de Wit and Freedman, 1980).21 Although they do not
lead to non-Abelian gauge algebras, it is interesting that the
cubic interactions on and below the Bell-Robinson line (i.e.,
for s1 ! 2s2 ) have the form ‘‘spin-s1 ﬁeld times current’’
where the current is quadratic in (the derivatives of) the
spin-s2 ﬁeld strength (Berends, Burgers, and van Dam,
1986; Deser and Yang, 1990) and is conserved on the
spin-s2 shel...

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