The strong form of the AdS/CFT correspondence posits that every CFT is dual to a
theory of quantum gravity in AdS. At the least, a subspace of all CFTs can be mapped via
holography to the space of weakly coupled theories of gravity or string/M-theory in AdS.
Given that string and M-theory are tightly constrained by their symmetries, this suggests
that any consistent CFT possesses a level of substructure over and above the manifest
requirements of conformal symmetry. One might hope to enlist chaos in the quest to “see”
the structure of AdS string or M-theory compactifications from CFT.
At large central charge c and with a sufficiently sparse spectrum of light operators
∆
i
c, a universality emerges: such CFTs appear to be dual to weakly coupled theories
of AdS gravity, that in the simplest cases contain Einstein gravity. These CFTs obey
certain other unobvious constraints: for example, corrections to a − c in four-dimensional
CFTs are controlled by the higher spin spectrum [21]. Identifying the set of sufficient
conditions for the emergence of a local bulk dual is an open problem. There is already
evidence that λ
L
= 2π/β is at least a necessary criterion, but one would like to make a
sharper statement. Explicitly connecting the value of λ
L
with the strong coupling OPE
data would permit a direct derivation of λ
L
= 2π/β from CFT, which is presently lacking
in d > 2 and has been done under certain conditions on the operators V and W in d = 2 [7].
Not all weakly coupled theories of gravity are local: one can, for instance, add higher
spin fields. In AdS
D>3
, there are no-go results: namely, one cannot add a finite number of
either massive or massless higher spin fields, for reasons of causality [21] and — in the case
of massless fields — symmetry [22, 23]. In AdS
3
, the constraints are less strict. For one,
the graviton is non-propagating. Moreover, higher spin algebras, i.e. W-algebras, with a
finite number of currents do exist.
Consider theories which augment the metric with an infinite tower of higher spin gauge
fields. Other than string theory, these include the Vasiliev theories [24–26]; see [27, 28] for
recent reviews. These are famously dual to O(N) vector models in d ≥ 3 CFT dimensions
(and, in d = 3, Chern-Simons deformations thereof [29, 30]). One widely held motivation
for studying the Vasiliev theories in d dimensions is that they morally capture the leading
Regge trajectory of tensionless strings in AdS [31]. For the supersymmetric AdS
3
Vasiliev
theory with so-called shs
2
[λ] symmetry, this is now shown to be literally true [32–34]: CFT
arguments imply that this super-Vasiliev theory forms a closed subsector of type IIB string
theory on AdS
3
× S
3
× T
4
in the tensionless limit, α
0
→ ∞. More generally, it is unclear
whether other, e.g. non-supersymmetric, Vasiliev theories are UV complete, or whether
they can always be viewed as a consistent subsector of a bona fide string theory.
In AdS
3
, there seem to be other consistent theories of higher spin gravity: to every W-
algebra arising as the Drinfeld-Sokolov construction of a Lie algebra G, one can associate
a pure higher spin gravity in AdS
3
cast as a G × G Chern-Simons theory. This builds on
the original observation that general relativity in AdS
3
can be written in this fashion with
G = SL(2, R) [35, 36]. Such pure Chern-Simons theories have been studied in the context
of AdS/CFT, especially for G = SL(N, R) and G = hs[λ]. The former contains a single
higher spin gauge field at every integer spin 2 ≤ s ≤ N which generate an asymptotic
W
N
symmetry [37]. The latter, a one-parameter family labeled by λ, contains one higher
spin gauge field at every integer spin s ≥ 2 which generate an asymptotic W
∞
[λ] symme-
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