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The number of degrees of freedom of a CFT is given by its central charge $c$. From the bootstrap point of view, any CFT is characterized by the knowledge of its "CFT data", i.e. the scaling dimensions of its operators $\Delta_i$ and its 3-point functions $\lambda_{ijk}$.

My question is, when we say a CFT has a holographic dual, exactly what conditions do we need to impose on the CFT? Does it involve restriction only on the central charge (the repeated meme is large $c$, but large compared to what?), or do we need to fine tune the CFT data also?

Also are these conditions necessary and sufficient for a holographic bulk dual, in the sense that can one always perform a HKLL causal bulk reconstruction starting from such a CFT, or a bulk reconstruction by deriving the Einstein equations via RT formula for these CFTs?

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    $\begingroup$ This question was asked before on this website (and in the literature): physics.stackexchange.com/q/27662 $\endgroup$ – Hans Moleman Nov 30 '19 at 9:50
  • $\begingroup$ @HansMoleman Thanks, but the answer just repeats the standard AdS/CFT meme and doesn't answer my question at all. Yes there is large $c$, but large compared to what? And do we need specific CFT data? And are these conditions necessary and sufficient for a holographic dual, in the sense that can one perform a HKLL bulk reconstruction starting from such a CFT, or a bulk reconstruction by deriving the Einstein equations via RT formula? $\endgroup$ – Bruce Lee Nov 30 '19 at 11:58
  • $\begingroup$ @BruceLee The central charge is dimensionless, so large just means large, no need to compare it to anything. $\endgroup$ – AccidentalFourierTransform Nov 30 '19 at 12:31
  • $\begingroup$ @AccidentalFourierTransform It could be compared to a dimensionless parameter. Anyway I got my answer, have a look! $\endgroup$ – Bruce Lee Nov 30 '19 at 12:39
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The introduction to this answers a part of my question in a concise manner. The following points are necessary in order for the CFT to have a holographic dual.

  1. The first AdS/CFT papers (Maldacena, Witten, GKP) state that the three point functions like $\langle TTT \rangle$ must have specific structures for a holographic dual to exist. (This means that any anomalous terms in these 3-point functions must not appear.)

  2. The dual quantum gravity has an action which is Einstein-Hilbert + higher derivative terms. However higher derivative terms are known to violate causality. CEMZ state that causality dictates that the graviton 3-point coupling must be universal in quantum gravity, by requiring the theory to be causal in shockwave states. This constraint has been derived in a number of papers for $d\geq 3$ under the following assumptions:

    a. The central charge $c$ is large, $c \gg 1$. (Note that for gauge theories this corresponds to the large $N$ limit)

    b. The spectrum is sparse, i.e. the lightest single trace operator with spin $>2$ has dimension $\Delta_{gap} \gg 1$.

CFTs in this class, irrespective of microscopic details admit a gravitational dual description at low energies.

Note 1 : Given such a CFT, HKLL reconstruction can be trivially done.

Note 2 : In this paper, the linearized Einstein equation for any holographic CFT is derived using the RT.

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  • $\begingroup$ Dear Bruce Lee. It is often frown upon to post nearly identical answers to similar posts. (The problem is if everybody start to copy-paste identical answers en mass.) $\endgroup$ – Qmechanic Nov 30 '19 at 14:28

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