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**Unformatted text preview: **m ﬁeld theory
(Bern et al., 2007, 2009) as well as higher-spin gravity
(Giombi and Yin, 2009, 2010), one may start raising criticism
against the very assumptions behind the Fronsdal program:
the higher-derivative nature of higher-spin interactions leads
ultimately to a conceptual breakdown of the standard canonical approach to quantum ﬁeld theory based on time slicing in
ordinary spacetime. Although one can refer perturbatively to
the canonical structure of the free ﬁelds (thought of as
ﬂuctuations around the spin-two background), the nonperturbative formulation of higher-spin symmetries leads toward an
extension of spacetime by extra bosonic coordinates on which
higher-spin translations act by linear differentiation. One may
therefore think of a bosonic generalization of the superspace
approach to supergravities, which is precisely what is provided by the unfolded dynamics program initiated by Vasiliev
[for an illustration of the basic ideas in the context of higherspin supergravity, see, for example, Engquist, Sezgin, and
Sundell (2003)].
IV. VARIOUS YES-GO EXAMPLES In this section we give a review of the various positive
results obtained over the years concerning consistent higherspin cubic couplings in ﬂat and AdS backgrounds.
Section IV.A gathers together the results for cubic vertices
in ﬂat space, while Sec. IV.B essentially mentions the results
obtained by Fradkin and Vasiliev in the late 1980s for cubic
vertices in ðAÞdS4 . Section IV.C consists of a summary in the
form of a general picture for non-Abelian higher-spin gauge
theory, which seems to emerge from the known no-go theorems and yes-go examples. Of course, a word of caution
should be added: the existence of consistent cubic couplings
does not imply that a complete theory exists at all. However,
the existence of full interacting equations (Vasiliev, 1990, 17 Including the Coleman-Mandula theorem, since the conserved
charges used in its arguments depend on the asymptotic behavior of
interactions at large distances.
18
See Eq. (B2) of Appendix B or Eq. (26) in Porrati (2008).
Rev. Mod. Phys., Vol. 84, No. 3, July–September 2012 19
This criterion is subtle, however, since for nonvanishing Ã,
generic spins cannot have as many gauge symmetries as for
vanishing Ã. Xavier Bekaert, Nicolas Boulanger, and Per A. Sundell: How higher-spin gravity surpasses the spin- . . . TABLE I. s1 -s2 -s2 covariant vertices obtained by Berends,
Burgers, and van Dam (1985).
#s1 !s2 0 1
2 1 3
2 2 5
2 3 0
1
2
3
n Â
Â
Â
Â
Â Â
Â
Â
Â Â
Â
Â
Â Â
Â
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Â Â 1992, 2003) is a strong indication that a complete interacting
Lagrangian20 may exist, at least in (A)dS background.
Actually, one of the open problems in higher-spin gravity is
whether or not the Fronsdal program can be pursued beyond
the cubic order in a standard fashion.
A. Consistent cubic vertices in Minkowski spacetime In the 1980s, the quest for high-spin interactions successfully started, taking ﬂat spa...

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