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**Unformatted text preview: **vertices obtained by Bekaert, Boulanger, and
Cnockaert (2006) and Boulanger, Leclercq, and Sundell
(2008) (which were shown to be related to the, appropriately
taken, ﬂat-space limit of the corresponding Fradkin-Vasiliev
vertices) are also the seed for the construction of consistent
massive higher-spin vertices in ﬂat and AdS spacetimes.
Rev. Mod. Phys., Vol. 84, No. 3, July–September 2012 From these non-Abelian ﬂat-space vertices, one can
systematically construct massive and massless vertices in
`
AdS and ﬂat spaces by switching on mass terms a la
¨
Stuckelberg and cosmological constant terms. This approach
was used with success by Zinoviev (2009a, 2009b). See also
the recent work by Zinoviev (2010) where the framelike
formalism for higher-spin gauge ﬁelds is used.
B. Cubic vertices in AdS spacetime As mentioned in the previous section, at cubic level (i.e., at
ﬁrst order in perturbative deformation) Fradkin and Vasiliev
found a solution to the higher-spin (gravitational, self- and
cross) interaction problem by considering metric perturbations around ðAÞdS4 background (Fradkin and Vasiliev,
1987b, 1987c). This was later extended to ﬁve dimensions
(Vasiliev, 2001a), N ¼ 1 supersymmetry (Alkalaev and
Vasiliev, 2003) and arbitrary dimensions (Vasiliev, 2011).
For a recent analysis of the Fradkin-Vasiliev mechanism in
arbitrary dimension D and in the cases 2-s-s and 1-s-s, see
Boulanger, Leclercq, and Sundell (2008).
The Fradkin-Vasiliev construction was the starting point of
dramatic progress leading recently to fully nonlinear ﬁeld
equations for higher-spin gauge ﬁelds in arbitrary dimension
(Vasiliev, 2003). We will not detail their construction here but
we simply comment that the use of a twistor variable and a
Moyal-Weyl star product is central, although historically the
usefulness of the star product was not immediately recognized. In a few words, the main problem with the higher-spin
gravitational interaction was that, introducing the Lorentz
minimal coupling terms in the action and gauge transformations, higher-spin gauge invariance could not be satisﬁed
anymore. The solution provided by Fradkin and Vasiliev
was to introduce a nonvanishing cosmological constant Ã
and expand the metric around an (A)dS background. The
gauge variation of the cubic terms coming from the Lorentz
minimal coupling around (A)dS are now canceled on the free
shell, by the variation of a ﬁnite tail of additional nonminimal
cubic vertices, each of them proportional to the linearized
Riemann tensor around (A)dS and involving more and more
(A)dS-covariant derivatives compensated by appropriate
negative powers of the cosmological constant. In that gauge
variation, the terms proportional to the free equations of
motion are absorbed through appropriate corrections to the
gauge transformations. This solution is the Fradkin-Vasiliev
mechanism, and we call the gravitational cubic coupling they
obtained the quasiminimal coupling, in the sense that th...

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