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As I work through AdS/CFT exercises, it struck me that there seemed no one doing the following.

Suppose we have a holographic CFT. By some reeconstruction method, we can write CFT operators in terms of bulk fields and vice versa.

Now we add the d-dimensional Einstein-Hilbert (EH) term and the Gibbons-Hawking-York (GHY) term to the d-dimensional CFT action to gravitize the boundary CFT.

Rewrite the EH and GHY terms in terms of bulk fields, though we utilize the original CFT holographic relation/reconstruction. This re-written EH and GHY terms are directly added to the original bulk theory action.

The question is, the new bulk theory resulting from this procedure seems very interesting in its own, but I have not seen anyone studying such a theory. At least we could apply the GPKW dictionary naively to get the dual boundary theory of the new bulk theory, though I do not think the new boundary theory is a CFT.

So what are the reasons for this theory not being useful, usable or potentially inconsistent? And it also seems that people are not interested in gravitating the boundary, and I sort of get why - bulk theory contains gravity, so why do that - but there seems to be little to almost zero interest, and it feels weird to me. Maybe I am wrong about this assessment though.

(As a precaution: this is not about finding the holographic dual of a CFT with EH and GHY term, it is about the new bulk theory with boundary EH and GHY written in terms of original bulk fields.)

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  • $\begingroup$ AdS/CFT usually refers to a correspondence between a gravitational AdS partition function with a fixed boundary in the bulk and a CFT generating functional on the boundary. The only instance I know of AdS/CFT where the CFT side is gravitational is the 3D Chern-Simons/2D Liouville gravity duality arxiv.org/abs/gr-qc/9506019 . Though it is not what you are searching for, I think my first sentence explains why no such construction is found in the literature. $\endgroup$
    – Jeanbaptiste Roux
    Commented Apr 9 at 8:15
  • $\begingroup$ @JeanbaptisteRoux But in the new bulk theory, all that happens in the path integral is the same background space, and same dynamic gravity if we allow for dynamic gravity, the only different thing is the integrand that contains the new terms that are functions of bulk fields, which changes the partition function. The "boundary" remains fixed as before, which thus does not prevent the use of GPKW relations that map partition functions. $\endgroup$
    – Bulldozer
    Commented Apr 9 at 9:32
  • $\begingroup$ @JeanbaptisteRoux Maybe it's better to forget about dynamic gravity in the bulk and focus on static gravity, just with boundary Einstein-Hilbert terms added in terms of original bulk fields. The boundary remains perfectly fixed as before, what changed was the bulk theory content. We then compute by GPKW or BDHM to obtain the new boundary theory, ignoring that this theory is not actually conformal. $\endgroup$
    – Bulldozer
    Commented Apr 9 at 9:34
  • $\begingroup$ It's hard to know what you're asking. Boundaries don't have a boundary so I don't know what the other GHY term will be. It sounds like you want the other EH term to be a constant, i.e. the Ricci scalar evaluated at some background metric because if you made it dynamical the theory that was formerly a CFT would almost surely end up in the swampland. And does it matter which conformal frame you pick for this fixed metric? $\endgroup$
    – Connor Behan
    Commented Apr 14 at 19:24

2 Answers 2

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This actually is something related to something that people have tried, and is called double holography. Instead of two dual theories, there are three perspectives one could take on this system. See Figure 1 of the link, but for posterity I'll describe it here as well.

One is as a BCFT. Basically, instead of a CFT on a closed manifold, you take it to live on a manifold with boundary. Additionally, you place another, lower dimensional CFT on that boundary, and couple these two theories in a way that preserves the maximal amount of conformal symmetry.

The second is a halfway step, where you instead view the system as a non-gravitational CFT coupled to a gravitational one of the same dimension, joined where the boundary of the CFT used to live. This gravitational portion is essentially the system you were proposing, at least in spirit.

Finally, the bulk dual of this system is the usual AdS spacetime, together with a defect called an end-of-the-world (EOTW) brane. An EOTW brane effectively chops off a part of the spacetime, and is anchored to (you guessed it) to the boundary of perspective one.

For more details, see the link above and references therein. Also, these systems are sometimes called Karch-Randall braneworlds.

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  • $\begingroup$ This is what I've been looking for! Thanks! $\endgroup$
    – Bulldozer
    Commented May 17 at 5:05
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Usually the point of the duality is to study gravity theories in an equivalent theory without gravity. Hopefully working in this theory is going to be easier, even due to the fact that this theory has 1 less dimension. This is also in line with the rough idea of spacetime being emergent from a purely quantum theory.

So adding gravity in the dual CFT seeems to defeat the point of the duality in the first place. It is not clear that all this added complexity is worth with respect to simply solving the original bulk gravity theory.

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