Approximations and limits in the AdS/CFT correspondence

[physicus]

Hi, sorry for the quite long text. Thanks in advance for any help!

I am a little confused about the different limits in which the AdS/CFT correspondence is conjectured to hold in its stong, intermediate, weak form.

I am trying to understand the correspondence motivated by Maldacena's original decoupling argument. This is what I understand:

We look at the situation of $N$ D3-branes from two points of view:
1. The branes are objects on which open strings can end. So the spectrum of states on the brane worldvolume is given by the open string excitations. The massless excitations form a vector multiplet for each brane. When the branes are coincident we get a $U(N)$ gauge theory by the Chan-Paton mechanism in the low energy limit (only massless excitations survive).
Away from the branes we only have closed strings, i.e. simply Type IIB closed strings ins flat space.
However, there are interactions between the two sectors since, e.g., two open strings on the brane can join to form a closed string and propagate away from the brane.
In the low energy limit ($\alpha' \to 0$) the two sectors decouple (is there an easy argument for the interactions to vanish without looking at the action?). On the brane we get $\mathcal{N}=4$ SYM. In the bulk the low energy limit of type IIB string theory is type IIB supergravity.
2. The geometry interpreted as D-branes ("extended balck holes") is a solution of supergravity. So we can treat the branes as deformation of the background and study string theory in this background. The background is $AdS_5\times S^5$ close to the branes and flat ten dimensional space far away from the branes. In the low energy limit, far away from the branes we get type IIB supergravity as before (only massless excitations of the closed string remain). Because of redshift the energy of excitations close to the brane is proportional to $r$, the distance from the brane. So even higher energy excitations seem very low energy from far away. What remains here in the low energy limit? Is it also only supergravity in the deformed background? Or are there also massive excitations?
Again the two sectors decouple in the low energy limit. The gravitational potential confines excitations that are close to the brane. Is that correct or are there better explanations?

Using this we come to identify $\mathcal{N}=4$ SYM with type IIB supergravity (or string theory?) in $AdS_5 \times S^5$.

So, it is very clear to me that the duality holds in the low energy limit. Often one reads, that in addition we must have $N\to\infty, g_s\to 0, \lambda=g_s N=const.$ Where does this come from?

In its weakest formulation the duality is claimed to hold only if $N\to\infty$ and $\lambda$ very large. Apparently this reduces the supergravity to its classical limit while the gauge theory becomes strongly coupled and has a large number of colours. A (modestly) strong form still requires $N\to\infty$, but $\lambda$ can have any value. What changes on both sides? The gauge theory is not neccessarily strongly coupled any more. What happens on the string theory side? Do we get stringy effects, or do we still have supergravity? In the strongest form both $N$ and $g_s$ are arbitrary.

How is it motivated that the duality might only hold in the weak form?

I would be very glad about any clarifiation and answer to my questions.

Cheers, physicus

 

[Physics Monkey]

Hey Physicus,

If I understand your question correctly, the basic intuition you're looking for is encoded in the following four possibilities:

Large N/large $\lambda$: classical supergravity
Large N/small $\lambda$: classical string theory
Small N/large $\lambda$: quantum supergravity
Small N/small $\lambda$: quantum string theory

Regarding weak/strong versions of duality etc., I'm not crazy about this terminology, but I would say it is logically possible that the duality could only apply to certain sectors of the theory. For example, it might hold only at infinite N, e.g. there is a phase transition at infinite N. Or the duality might only work for supergravity excitations. Such a situation is believed to apply to something like Kerr/CFT where the near horizon captures only some subset of the CFT states.

One thing is clear, whatever quantum string theory is, it doesn't look that much like what we normally think of as gravity. Related to this point, we mostly have evidence for the duality at large N and strong coupling, but in some cases one can check that 1/N and stringy corrections match field theory expectations.

I certainly believe that the duality is completely general and applies in some sense to all quantum field theories.

 

[physicus]

Many thanks for your answer. I have some questions remaining:

1. You say it is "logically possible" that the duality only holds in certain limits. Do you mean that, because there is no proper proof, we simply cannot exclude the possibility that the duality does not hold generally?

2. In Maldacena's decoupling argument: If we only take the low energy limit in the supergravity solution point of view, do we get supergravity or full string theory close to the branes? Because of the redshift that makes all excitations close to the brane look low energy, it seems reasonable to me, that we can have massive excitations close to the branes.

3. When trying to find a motivation for claiming that the duality was only valid in the large N limit, I came across the following:
"We can find supergravity solutions carrying the same mass and charges [as the D-branes in the open strng point of view]. Naively the supergravity solution describes only the long range fields of the D-branes, since we do not expect supergravity to be valid at short distances. General covariance, however, tells us that we can trust the supergravity solution as long as curvatures are locally small compared to the string scale (or the Planck scale). A more careful analysis shows that for a system with a large number of branes, large N, the curvatures are small and we can trust the supergravity solutions even at the substringy distances involved in the decoupling limit described above." [arXiv:hep-th/9802042, page 2]
One point of view that we take is to consider closed strings moving in the background that is deformed by the branes. This background is a supergravtiy solution and I understand from the above extract, that it is only valid for large distances. How does it make sense to study string theory in such a background, when it is not even valid at the string length scale?

Thank you for any answers!

Edit:
4. In Maldacena's original paper that conjectured AdS/CFT from 1997 [arXiv:hep-th/9711200] he writes on page 4:
"We can trust the supergravity solution when $gN \gg 1$."
I get the derivation of this starting from the fact that SUGRA is only valid if the curvature radius is smaller than the string length. But then he says in a footnote:
"In writing $[gN\gg 1]$ we assumed that $g ≤ 1$, if $g > 1$ then the condition is $N/g \gg 1$. In other words we need large N, not large g."
Does anyone know where this comes from?

 

[Physics Monkey]

I believe the duality, say between n=4 sym and IIB strings in asymptotically ads, is true for all N and all coupling. Not that we know what one means by IIB string theory in general. What I meant by "logically possible" is that in the larger context of related and generalized dualities, it is perfectly legitimate for a duality to only relate subsectors of the two sides e.g. Kerr/CFT. Hence things look very modular: a piece of this theory may match a piece of that theory, but you can patch the pieces together in different ways to potentially produce two different objects that are not globally identical.

 

[trimok]

"In writing [gN≫1] we assumed that g≤1, if g>1 then the condition is N/g≫1. In other words we need large N, not large g."
Does anyone know where this comes from?

.

I think it is because S-Duality maps string theories with coupling g, with string theories with coupling 1/g. So, beginning with a string theory with g>1, we can find another string theory with g' = 1/g <1, and we may apply the condition g'N>>1, that is N/g >>1

 

[physicus]

This is still not very clear to me:

Large N/large $\lambda$: classical supergravity
Large N/small $\lambda$: classical string theory
Small N/large $\lambda$: quantum supergravity
Small N/small $\lambda$: quantum string theory

The string theory parameters $\alpha'=l_s^2$ (where $l_s$ is the string length) and $g_s$ are related to $N,\lambda$ via
$\frac{\lambda}{N}=4\pi g_s,$
$\lambda=\frac{L^4}{\alpha'^2},$
where $L$ is the $AdS$-radius.
$\frac{1}{\sqrt{\lambda}}$ is also the string worldsheet theory coupling constant since the Polyakov action in an $AdS_5\times S^5$ background has the prefactor of $\frac{L^2}{4\pi \alpha'}=\frac{\sqrt{\lambda}}{4\pi}$.

Therefore, stringy effects vanish in the low energy limit if $l_s \ll L \;\Leftrightarrow\; \lambda \gg 1$. This is equivalent to $\alpha'$ being very small, i.e. only massless supergravity excitations survive. So I agree that $\lambda$ controls if we are in the SUGRA limit or not.

What do we mean by "classical" string theory? Does this mean that the parameter for quantum corrections in the string worldsheet theory (which is $\lambda$) is small? That can't be true if your if Physics Monkey's statement is correct. Large $N$ rather leads to a small string coupling $g_s$. Isn't the terminology a little strange there? Why are string loops labeled quantum effects while the QFT loops in the worldsheet theory are not?

Thanks for any answers.

 

[Physics Monkey]

One can mean different things by classical string theory. Here what I meant was that although the theory was still at large N, so that the path integral could be approximated by saddle point, the saddle point involves string-like excitations in addition to the supergravity modes.

Does this make sense?

PS What is and isn't regarded as quantum is a vexing notion in string theory. For example, dualities can really muddle things up. Switching between strongly fluctuating and weakly fluctuating variables or between momentum and winding modes both involve some non-trivial mixing of "classical" and "quantum" effects.