# Simplest the AdS/CFT correspondence is a conjecture

Does anyone have a simple explanation for this?

josh1
At it's simplest the AdS/CFT correspondence is a conjecture about certain solutions in string theory and M-theory that include gravity which says they have an equivalent purely non-gravitational description in terms of a solution in one spacetime dimension less.

The way that it's been explained to me is a matching of degrees of freedom, which came from looking at black holes.

I run the danger of making an ass out of myself in front of experts, but that's never stopped me before. Plus, if they see my mangled description, it may encourage them to step in and correct me, thereby teaching me something :)

You can think of entropy'' as information'' in some sense---this was pioneering work done by Shannon long ago. Either way, black holes tell you that information'' scales as the surface area of some volume, not as the actual volume itself. This is because the entropy of a black hole is proportional to its surface area. Anyway, this gives you some motivation to think about why all of the information about what's inside the black hole should be stored at it's surface, as opposed to the singularity in the middle where (naively) it SHOULD be.

AdS space has a boundary, so you might think you should be looking on that boundary for some information about what's going on inside the AdS space. If you would have had this idea in 1997 you could have a pemanent position at IAS right now. It turns out that if you do some gravity calculations inside the AdS space, you get the exact same result as if you had done the calculations using a very special field theory living on the boundary of that space. In some cases, the gravity calculation is much easier than the field theory calculation, and in some cases it's the other way around.

Either way, this should be very exciting to you, because you have now a tool that actually LINKS gravity (which is a non-renormalizable field theory) with Super(symmetric) Yang Mills theory, which you can treat perturbatively in some cases. (Sometimes you can't treat it perturbatively, too.)

(Patiently awaiting criticism!)

nrqed
Homework Helper
Gold Member
The way that it's been explained to me is a matching of degrees of freedom, which came from looking at black holes.

I run the danger of making an ass out of myself in front of experts, but that's never stopped me before. Plus, if they see my mangled description, it may encourage them to step in and correct me, thereby teaching me something :)

You can think of entropy'' as information'' in some sense---this was pioneering work done by Shannon long ago. Either way, black holes tell you that information'' scales as the surface area of some volume, not as the actual volume itself. This is because the entropy of a black hole is proportional to its surface area. Anyway, this gives you some motivation to think about why all of the information about what's inside the black hole should be stored at it's surface, as opposed to the singularity in the middle where (naively) it SHOULD be.

AdS space has a boundary, so you might think you should be looking on that boundary for some information about what's going on inside the AdS space. If you would have had this idea in 1997 you could have a pemanent position at IAS right now. It turns out that if you do some gravity calculations inside the AdS space, you get the exact same result as if you had done the calculations using a very special field theory living on the boundary of that space. In some cases, the gravity calculation is much easier than the field theory calculation, and in some cases it's the other way around.

Either way, this should be very exciting to you, because you have now a tool that actually LINKS gravity (which is a non-renormalizable field theory) with Super(symmetric) Yang Mills theory, which you can treat perturbatively in some cases. (Sometimes you can't treat it perturbatively, too.)

(Patiently awaiting criticism!)

Sounds right to me.

One thing I never got around quite understanding is the boundary of an AdS space. Can someone shed some light on this? How can a spacetime have a boundary?

Haelfix
Its really a conformal boundary, and you should really think about the boundary in terms of the topology. I found this helpful PDF via Wiki, which shows how its defined in a rigorous mathematical sense.

As for the AdS/CfT correspondance, amusingly I just finished listening to Maldacena's lectures (Via NEQNET). I found them really well explained, and I learned a few new things that I didn't know... So highly recommended:

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Demystifier
Gold Member
It turns out that if you do some gravity calculations inside the AdS space, you get the exact same result as if you had done the calculations using a very special field theory living on the boundary of that space.
The crucial expression is "It turns out". I think that nobody has been able to provide a SIMPLE explanation of WHY it turns out? And I think THIS is the thing that kurt.physics is looking for.

I have seen AdS/CFT compared to Stokes' theorem, and this might be a good analogy, including the limitations. Recall that Stokes' theorem says two things:

1. Every d-form on a closed, d-dimensional boundary is equivalent to some (d+1)-form in the buld.

2. Most (d+1)-forms in the bulk are *not* equivalent to some d-form on the boundary - it only works if the (d+1)-form is closed, which in general it is not.

The analogy with AdS/CFT would then be:

1. Every gauge theory on the boundary is equivalent to some string theory in the bulk. Whether or not this is useful depends on whether the bulk theory is tractable, which seems to be the case if there is enough (too much) supersymmetry around.

2. Whether a gravity theory in the bulk must be equivalent to a CFT on the boundary is much less clear, especially since it involves the assumption of a negative cosmological constant (AdS), in disagreement with observation.

The crucial expression is "It turns out". I think that nobody has been able to provide a SIMPLE explanation of WHY it turns out? And I think THIS is the thing that kurt.physics is looking for.

Agreed. It DOES, we have good evidence for that---every quantum gravity theory that I have seen (by accident or by design) reproduces this behavior.

Demystifier
Gold Member
... every quantum gravity theory that I have seen (by accident or by design) reproduces this behavior.
Every?!?
How about quantum gravity without supersymmetry? More specifically, how about canonical quantum gravity based on the Wheeler-DeWitt equation, or on the loop representation? How about perturbative quantum gravity based on quantization of the Einstein-Hilbert action?

Every?!?
How about quantum gravity without supersymmetry?

Like Loop Quantum Gravity? :) They DO seem to engineer the constant 1/4, but at least they get a reltionship between entropy and surface area.

More specifically, how about canonical quantum gravity based on the Wheeler-DeWitt equation, or on the loop representation? How about perturbative quantum gravity based on quantization of the Einstein-Hilbert action?

Well...I DID qualify the statement :) Every QG theory that I have seen...

It seems at least reasonable that this should be the case, though, as the calculation of Hawking is semi-classical. That is, unless the quantum gravity effects come in at a very large scale (which seems unlikely), then any approach to reconcile gravity and QFT SHOULD give the A/4 relation because the space-time across an horizon is nice and flat, and this is where all of the fun is happening (as I understand it).

But I am far from an expert in this field. Either way, check a paper by Steve Carlip: http://arxiv.org/abs/0705.3024. In this paper, he addresses WHY it seemes to be that (many) of the QG theories that we have get A/4 for black hole enropy.

Haelfix
Certainly the latter, but only as a crude limiting approximation (since Supergravity trivially contains EH gravity in a certain decoupling regime). So it should hold for canonical gravity as well, though that is riddled with conceptual problems, leading some to believe they are inequivalent quantizations (See Hartle).

What is true is that quantum gravity without supersymmetry in the AdS/CFT picture is very touchy, you probably can't (as currently formulated) get away with anything goes (so some sort of restriction on the classes of Gauge theories involved). That hasn't been made entirely precise, at least as I understand it.

Demystifier
Gold Member
Like Loop Quantum Gravity? :) They DO seem to engineer the constant 1/4, but at least they get a reltionship between entropy and surface area.
More precisely, they get a relationship between entropy OF THE AREA and surface area, which is a hardly surprising result. The nontrivial achievement of loop quantum gravity is that this entropy is finite (UV divergences are automatically removed without an ad hoc UV cutoff) and that the constant of proportionality is universal (if it is 1/4 for one kind of black hole, then it is also so for any kind of black hole). But it cannot explain why the entropy of the bulk is equal to the entropy of the surface. Instead, it merely uses a classical (not quantum) argument that simply states that the degrees of freedom inside the black hole are not visible to an outside observer.

Instead, it merely uses a classical (not quantum) argument that simply states that the degrees of freedom inside the black hole are not visible to an outside observer.

So then it's exactly Hawking's argument?

Demystifier
Gold Member
So then it's exactly Hawking's argument?
Essentially, yes.

Does anyone have a simple explanation for this?

I am not sure if this is qualifies as simple but AdS/CFT explains gravity as an effect produced by projecting particles and fields from a lower dimensional reality to a higher dimensional one. The screen is the boundary.

The link below is to an article in Scientific America by Juan Maldacena. You may find some insight by looking into the holographic principle as well.

http://homepage.mac.com/photomorphose/documents/qpdf.pdf" [Broken]

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AdS/CFT explains gravity as an effect produced by projecting particles and fields from a lower dimensional reality to a higher dimensional one.
Ah but you are missing half of the point : as Ben mentionned in #3 already, calculations are sometimes simpler in the bulk/gravity and sometimes simpler on the boundary/CFT.

Ah but you are missing half of the point : as Ben mentionned in #3 already, calculations are sometimes simpler in the bulk/gravity and sometimes simpler on the boundary/CFT.

Agreed. And I think the half you and Ben are referring to is more evident in our universe than the form in which Juan Maldacena envisioned. So is the symmetry there or is something broken? I certainly don't have the answer. But either way, it's interesting speculation.

So is the symmetry there or is something broken?

Can you clarify this?

Can you clarify this?

Essentially I was referring to what Thomas Larsson wrote in post #7, at least as I understand it.

[QUTOE=Thomas Larsson]1. Every d-form on a closed, d-dimensional boundary is equivalent to some (d+1)-form in the bulk.

2. Most (d+1)-forms in the bulk are *not* equivalent to some d-form on the boundary - it only works if the (d+1)-form is closed, which in general it is not.[/QUOTE]

So our universe permits black holes, but it does not appear to be anti-de Sitter (closed). Though, pre-inflation, it does (it was closed). Both de Sitter and anti-de Sitter spaces are symmetric but the transition between the two does not appear to be. That seems odd to me.