**Unformatted text preview: **f free ﬁelds couple to higher-spin sources identiﬁed as the boundary data of bulk higher-spin gauge ﬁelds.
One should stress that although the boundary CFT is quadratic, it is nevertheless nontrivial since the bilinear operators
actually couple to background sources; therefore the bulk
dual theory is interacting. The concrete relation with
Vasiliev’s unfolded equations in four and ﬁve dimensions
was elaborated on by Sezgin and Sundell (2001a, 2002b,
2005), and the fully nonlinear bosonic higher-spin gravity
in any dimension was then found in Vasiliev (2003).
The agreement between Vasiliev’s four-dimensional
higher-spin gravity and the sector of bilinear operators
formed out of free conformal scalars and spinors in three
dimensions was veriﬁed at the level of scalar cubic couplings
by Petkou (2003) and Sezgin and Sundell (2005), and, more
recently, at the general cubic level by Giombi and Yin (2009,
2010) under certain prescriptions which still remain to be
spelled out in their entirety. Thus the question of whether
Vasiliev’s higher-spin gravity is natural or not is equivalent to
the question of whether free scalars (and spinors) are natural
building blocks for three-dimensional conformal ﬁeld theories with (unbroken or weakly broken) higher-spin currents.
Or stated differently, thinking about Vasiliev’s higher-spin
gravity is about as natural as thinking about threedimensional conformal ﬁeld theories starting from free ﬁelds.
Intermediate developments are given by Das and Jevicki
(2003), Leigh and Petkou (2003), Leonhardt, Meziane, and
Ruehl (2003), Bonelli (2004), Leonhardt and Ruehl (2004),
Hartnoll and Prem Kumar (2005), Ruehl (2005), Diaz and
Dorn (2006), Elitzur et al. (2007), and Yonge (2007). More
recently, the full checks of the conjecture for AdS4 =CFT3 at
the cubic level (Giombi and Yin, 2009, 2010) prompted a
Rev. Mod. Phys., Vol. 84, No. 3, July–September 2012 revived interest in the correspondence.24 For instance, the
conjecture was generalized in the presence of a ChernSimons gauge ﬁeld on the three-dimensional boundary
(Aharony, Gur-Ari, and Yacoby, 2011; Giombi et al.,
2011). Another duality was proposed relating bosonic
Vasiliev’s theory on de Sitter bulk spacetime dS4 and fermionic scalar ﬁelds Euclidean CFT3 (Anninos, Hartman, and
Strominger, 2011). The thermodynamic behavior of
Vasiliev’s higher-spin gravity was inferred from Conformal
Field Theory computations (Shenker and Yin, 2011). Several
attempts toward a constructive derivation of the bulk dual of a
free CFT in the vector representation have been proposed,
such as the bilocal ﬁeld approach (Das and Jevicki, 2003;
Jevicki, Jin, and Ye, 2011; Koch et al., 2011) and the
renormalization group (Douglas, Mazzucato, and Razamat,
2010).
Here we also stress that AdS/CFT is more to gauge ﬁeld
theory than what standard global-symmetry current algebra is
to quantum ﬁeld theory, essentially since the boundary currents are coupled to bulk gauge ﬁel...

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