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**Unformatted text preview: **gravity is
essentially nonlocal in spacetime but admits a quasilocal
formulation in terms of star products on the direct product
of commutative spacetime and noncommutative twistor
space, where one can then proceed building classical observables and geometries for the theory.
This somewhat awkward albeit mathematically completely
well-deﬁned situation raises the issue of whether Vasiliev’s
equations should be viewed as natural representative for
higher-spin gravity or not? Since there are no other known
examples of classes of higher-spin gravities with local degrees of freedom, it is difﬁcult to make any direct comparisons. However, lessons can be drawn by looking at the AdS/
CFT correspondence.
D. AdS/CFT correspondence: Vasiliev’s theory
from free conformal ﬁelds In the previous sections we attempted to deﬁne a relation
between the S matrix and Lagrangian approaches in the case
of vanishing cosmological constant. Switching on the cosmological constant the notion of the S matrix becomes deformed
into that of a holographic conformal ﬁeld theory. Thus, one
way of assessing to what extent a higher-spin gravity is
‘‘natural’’ is to ask oneself to what extent its dual conformal
ﬁeld theory is natural. 1002 Xavier Bekaert, Nicolas Boulanger, and Per A. Sundell: How higher-spin gravity surpasses the spin- . . . Shortly after Maldacena’s version of the AdS/CFT conjecture, which was derived within a stringy context involving
strong- and weak-coupling dual descriptions of branes, the
question came as to what the antiholographic dual of a weakly
coupled CFT could be. Since a free CFT has inﬁnitely many
conserved currents of arbitrary spin, in addition to the stressenergy tensor, it was natural to expect the AdS dual to be a
higher-spin gauge theory containing a graviton. With a
noticeable precursor (Bergshoeff et al., 1988), such ideas
emerged progressively in a series of papers (Haggi-Mani and
Sundborg, 2000; Konstein, Vasiliev, and Zaikin, 2000;
Shaynkman and Vasiliev, 2001; Sundborg, 2001; Sezgin
and Sundell, 2001a, 2002b, 2005; Witten, 2001; Klebanov
and Polyakov, 2002; Mikhailov, 2002): the idea was born in
the context of the type-IIB theory on AdS5 Â S5 (Haggi-Mani
and Sundborg, 2000; Sezgin and Sundell, 2001a; Sundborg,
2001), and then pursued in a more general D-dimensional
context, ﬁrst at the level of kinematics (Konstein, Vasiliev,
and Zaikin, 2000; Shaynkman and Vasiliev, 2001) and later at
a dynamical level leading to the duality conjecture between a
pure bosonic higher-spin gravity in any dimension and a
theory of (a large number of) free conformal scalars in the
vector representation of an internal symmetry group (Witten,
2001; Mikhailov, 2002; Sezgin and Sundell, 2002b), reﬁned
to include the strongly coupled ﬁxed points of the threedimensional OðN Þ model and the Gross-Neveu model, in
Klebanov and Polyakov (2002) and Sezgin and Sundell
(2005), respectively. More precisely, the bilinear operators
formed out o...

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