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Unformatted text preview: cated understanding of their ﬁrst quantizations (Ã and Ts expansions). Now to its advantage string
theory offers a massless window where its ﬁrst-quantization
is weakly coupled, whereas in dealing with unbroken higherspin gravity one must face the whole packed-up content of its
A striking similarity between open string theory and
higher-spin gravities occurs when one considers
(Konstein, Vasiliev, and Zaikin, 2000) extensions of the
higher-spin algebra by an internal, associative algebra [see
also Vasiliev (2004b, 2006)]. In such cases, there exist
colored, massless spin-two ﬁelds resembling the spin-two
states of open strings. These states can be given Chan-Paton
factors since their interactions are based on an associative
algebra. This similarity was pointed out by Francia and
Sagnotti (2003, 2006) to which we refer the interested
reader for related discussions. We note that the existence
of colored gravitons in extended higher-spin theories does
not enter in contradiction with the results of Boulanger
et al. (2001), since there it was assumed that the ﬁelds
considered could have spin two at most and the background
was taken to be ﬂat.
At the classical level, the possibilities remain of having
consistent truncations of closed-string theory down to higherspin gravity, and of higher-spin gravity down to general
relativity. For example, both of these types of truncations
may turn out to be relevant in the case of the hypothetical
tensionless type-IIB closed-string theory on AdS5 Â S5 that
should be the antiholographic dual of free four-dimensional
maximally supersymmetric Yang-Mills theory in its 1=N
expansion (Sundborg, 2001; Sezgin and Sundell, 2002b).
Here the hypothetical ﬁve-dimensional maximally supersymmetric higher-spin gravity [for the linearized theory, see
Sezgin and Sundell (2001b)] can be identiﬁed as the
Kaluza-Klein reduction of the ‘‘bent’’ ﬁrst Regge trajectory
of the ﬂat-space string theory (Bianchi, Morales, and
Samtleben, 2003; Sezgin and Sundell, 2005). The full tensionless string theory then involves a much larger higher-spin
symmetry algebra bringing in mixed-symmetry ﬁelds with
critical masses such that they ﬁt into multipletons (Bianchi,
Morales, and Samtleben, 2003; Sezgin and Sundell, 2005).
As for consistent truncations of higher-spin gravity down to
possibly matter-coupled (super)gravities, a look at the state of
affairs in gauged supergravities arising from sphere reductions (de Wit and Nicolai, 1987; Nastase, Vaman, and van
Nieuwenhuizen, 1999; Cvetic et al., 2000) suggests that one
Rev. Mod. Phys., Vol. 84, No. 3, July–September 2012 1003 should conjecture their existence in the case of maximal
As far as the type-IIB superstring is concerned, its graviton in ten-dimensional ﬂat spacetime admits a deformation
into a graviton of ﬁve-dimensional anti–de Sitter spacetime.
More generally, a key physical effect of having a negative
cosmological constant i...
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