1
$\begingroup$

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?

$\endgroup$
1
  • 1
    $\begingroup$ 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

0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.