From what I understand the AdS-CFT correspondence states that the bulk dynamics of a $n$-dimensional gravitational theory are encoded in the degrees of freedom of its dual CFT in the $(n-1)$ dimensional boundary.

The question is the following: Suppose we start with a $1D$ CFT theory. This will describe the dynamics of a dual gravitational theory in $2D$. Suppose also that the graviational theory is conformally invariant. Now, knowing the dynamics of the 2D theory we could find the dynamics of the $3D$ and so on. Is this possible? And if not, what am I missing?

Also an extra question: Suppose we have an $AdS_{1}$ gravitational theory. It seems that the correspondace saturates since we can't define a $0$-dimensional dual CFT?

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    $\begingroup$ An n-dimensional spacetime consists of (n-1) dimensions of space and one dimension of time. Therefore, you supposedly cannot set the n-dimensional spacetime as the world for the next conformal field theory. Meaning a gravitational theory cannot be conformally invariant because of the opposite sign of the time dimension. $\endgroup$ May 16, 2022 at 9:05

1 Answer 1


A $d$-dimensional conformal field theory with the right properties is `holographic', meaning that it's dual to a $(d+1)$-dimensional gravitational theory in AdS. But that $(d+1)$-dimensional theory cannot itself be holographic in the same sense, for a few reasons:

  • It has gravity! The original $d$-dimensional theory is a conventional QFT defined on a fixed spacetime, not on a dynamical spacetime as gravity demands.
  • Conformally invariance means (roughly speaking) that your theory has no intrinsic length scale. But the gravitational theory has several: the curvature length of AdS and the Planck length, for example.

You could imagine something slightly different, which is a gravitational theory with conformally invariant, holographic matter. This has been considered recently in this paper, for example. Sometimes it's called a Karch-Randall model.

On the second question: there is not even AdS$_2$/CFT$_1$ in the same sense as higher dimensions, since a one-dimensional theory (quantum mechanics) can't be conformally invariant in the sense you'd need.


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