Hi, sorry for the quite long text. Thanks in advance for any help!(adsbygoogle = window.adsbygoogle || []).push({});

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 [itex]N[/itex] 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 [itex]U(N)[/itex] 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 ([itex]\alpha' \to 0[/itex]) the two sectors decouple (is there an easy argument for the interactions to vanish without looking at the action?). On the brane we get [itex]\mathcal{N}=4[/itex] 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 [itex]AdS_5\times S^5[/itex] 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 [itex]r[/itex], 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 [itex]\mathcal{N}=4[/itex] SYM with type IIB supergravity (or string theory?) in [itex]AdS_5 \times S^5[/itex].

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 [itex]N\to\infty, g_s\to 0, \lambda=g_s N=const.[/itex] Where does this come from?

In its weakest formulation the duality is claimed to hold only if [itex]N\to\infty[/itex] and [itex]\lambda[/itex] 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 [itex]N\to\infty[/itex], but [itex]\lambda[/itex] 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 [itex]N[/itex] and [itex]g_s[/itex] 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

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# Approximations and limits in the AdS/CFT correspondence

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