Ofer Aharony gave an interesting Duality Seminar about the
- CFT/AdS correspondence, massive gravitons, and connectivity index conjecture.
Ofer views the opposite statement, denoted as AdS/CFT, to be obvious: one simply defines the CFT to be the set of all possible correlators among all possible local operators. This set of functions can be simply extracted from the scattering amplitudes of all physical states created on the boundary and interacting in the bulk of the AdS. Because the AdS space always has gravity on it, you always obtain the stress-energy tensor in the spectrum of the operators. Other consistency rules of CFTs probably hold, too.
The main statement of Ofer Aharony, Andreas Karch, and Adam Clark is that the opposite implication doesn't hold. One can construct CFTs whose dual doesn't deserve to be called "quantum gravity on AdS" despite the fact that you might think that "quantum gravity" without the CFT is so loose and ill-defined that any theory can be called quantum gravity.
The reason why some CFTs are not dual to gravity on AdS according to these three guys is that these CFTs can really be shown to be dual to gravity on the union of several copies of AdS - copies that are connected through the boundary conditions relating their AdS boundaries only.
Their examples involve the CFTs constructed in the following way. Choose a dimension of the CFT spacetime and two copies of a CFT that are originally decoupled. They have two different stress-energy tensors. Now, add a small coupling that relates them. Only the overall stress-energy tensor will be conserved. The anti-diagonal combination that generates the relative shifts of the "two" spacetimes will acquire an anomalous dimension and it will no longer be conserved. It will become a massive spin-two field or, using a different terminology, a massive graviton.
This interpretation of massive spin-two fields is analogous to their interpretation in the language of deconstruction. Every massive spin-two field is morally another metric tensor that acquired mass through a Higgs-like mechanism. Incidentally, they also do the bulk calculation of the mass of the anti-diagonal graviton and this calculation yields a result that exactly agrees with the anomalous dimension of the stress-energy tensor, using the usual AdS/CFT dictionary that translates the dimensions into masses. In the flat space, the limit "m goes to zero" is discontinuous. But in the AdS space, the limit turns out to be smooth.
They can show that even after these previously independent CFTs are coupled, the bulk interpretation of this coupling is only via the AdS boundaries. The particular examples include either
- two two-dimensional CFTs with an added (1,1) tensor coupling that is constructed as a product of a (1,0) current from the first CFT and a (0,1) current from the second CFT
- two four-dimensional theories analogous to the Klebanov-Witten fixed points, with a particular trace used as a perturbation
Their reasoning leads them to conjecture that the set of all CFTs can be split to subsets with non-negative integer labels "n" that tell you how many copies of AdS space the dual gravitational theory has. For "SU(K) x SU(L)", a decoupled pair of two gauge theories, the value of "n" is clearly two. But they propose models where "n" is greater than one even though the theory cannot be split into decoupled subtheories.
They reasonably believe that if the gravitational theory can be interpreted in terms of a disjoint union of AdS spaces, there should be no other description of the same physics that involves one AdS space only. This leads them to argue that it is not true that every CFT has a dual gravitational description that involves one AdS space only.
I asked Ofer about the interpretation of "SU(15)" broken to "SU(10) x SU(5)" by the scalar vevs. You can integrate out all the off-diagonal, bi-fundamental degrees of freedom (W-bosons et al.) and write down the resulting theory as "SU(10)" combined with "SU(5)" gauge theory, with small corrections that couple them. Such a description will have a gravitational dual geometry composed of two independent near-horizon geometries. However, the effective theory with the W-bosons integrated out will break down at energies equal to the W-mass or higher. Still, the full theory is continuously connected to a theory whose dual manifestly lives on one AdS space only - the dual to the "SU(15)" gauge theory.
I asked Ofer whether he didn't think that this behavior was universal. Ofer argued that it was not. In the Klebanov-Witten-like examples, he argues that the deformed theory with the mixed interaction is valid up to arbitrarily high energy scales, unlike the theory with the W-bosons integrated out. The main possible loophole could be that there are some new operators with dimensions that diverge for "lambda=0" and that become finite for non-zero "lambda" (the coupling relating the two CFTs).
This effect would be analogous to the fact that there exist D-branes and other solitons and instantons whose tensions or masses or actions diverge in the "g=0" limit. Aren't there some operators dual to these heavy objects? If this is the case, then one could revive the idea that for every CFT, the physics of all interacting CFTs is qualitatively analogous to the "SU(15)" example and "morally", the number of AdS components is always one as long as you look at the theory with a high enough energy cutoff.
In other words, the different CFTs can be decoupled in the middle of the AdS space where the volumes are large - the infrared regime - but they always become coupled, even in the bulk, in the ultraviolated regime - near the AdS boundary where the volumes are large. Note that Ofer et al. essentially agree that the the coupling between the theories becomes important near the boundary but they always view such a relation to be nothing else than a backreaction reflecting the coupled boundary conditions at the exact boundaries.
Meanwhile, until someone finds the high-dimension operators of the KW-KW system mentioned above, Ofer et al. can legitimately believe that the set of CFTs sharply splits into classes labeled by integers "n" that tell you how many components the dual AdS-like bulk description has. That's a somewhat unusual statement to conjecture that every random CFT has this kind of an "index" called "n" but there is no completely obvious way to disprove this bold and interesting assertion.
Jacques has written about this topic here.