# What are the necessary conditions for a CFT to have a holographic dual? [duplicate]

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?

• This question was asked before on this website (and in the literature): physics.stackexchange.com/q/27662 – Hans Moleman Nov 30 '19 at 9:50
• @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? – Bruce Lee Nov 30 '19 at 11:58
• @BruceLee The central charge is dimensionless, so large just means large, no need to compare it to anything. – AccidentalFourierTransform Nov 30 '19 at 12:31
• @AccidentalFourierTransform It could be compared to a dimensionless parameter. Anyway I got my answer, have a look! – Bruce Lee Nov 30 '19 at 12:39

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.

• 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.) – Qmechanic Nov 30 '19 at 14:28