Very basic question on AdS/CFT

Question

I was going through the introductory material by Horatiu in Ads-CFT. It says that $N+1$ D-branes are split into $N$ D-Branes and a probe D-Brane. The Wilson loop is located on the probe D-brane, which is at the Minkowski boundary of the AdS space.

The AdS space is given by $f^{-1/2}dx_{||}^2 + f^{1/2}(du^2 + d\Omega^2)$, where $f$ is the harmonic function = $\frac{R^4}{u^4}$.

My question is, what is causing this AdS metric (what is the source of the AdS space)? Is it the N D3 branes? or something else? If there is no source, then the space time would be flat.

Is there an assumption that the probe D3-brane is not modifying the metric of the AdS space at all?

Appreciate any clarification on this.

Answer 1

Here is an answer to the question of why the $AdS_5\times S^5$ metric is appearing. This is taken almost directly from the TASI lectures I cite at the end.

If you consider N coincident Dp-branes, the background solution has a metric and dilaton which we can write as $$ds^2 = H^{-1/2}(r)\left[-f(r)dt^2 +\sum_{i=1}^p(dx^i)^2\right]+H^{1/2}(r)\left[f^{-1}(r) dr^2+r^2 d\Omega_{8-p}^2\right]$$ $$e^{\Phi}=H^{(3-p)/4}(r)$$ with the warp-factors $$H(r)=1+\frac{L^{7-p}}{r^{7-p}}, \quad f(r)=1-\frac{r_0^{7-p}}{r^{7-p}}$$ If you take $p=3$, such that you are considering now a stack of D3-branes and additionally take the so-called extremal limit ($r_0\rightarrow 0$), then this metric becomes identical to the one you are asking about. This isn't quite $AdS_5\times S^5$ yet. All you need to do now is to take the limit $\frac{r}{L}\rightarrow 0$ and you will be left with none other than $$ds^2=\frac{L^2}{z^2}(-dt^2+d\vec{x}^2+dz^2)+L^2 d\Omega_5^2$$ which is the usual metric for $AdS_5\times S^5$.

References: "TASI Lectures: Introduction to the AdS/CFT Correspondence", https://arxiv.org/abs/hep-th/0009139

Answer 2

First, by $S$-duality of Type IIB, it is always possible to keep $g_s<1$. We are going to talk about a stack of $N$ D3-branes in two situations:

1. $Ng_s$ small: the stack will not deform the background, and only perturbs it. These perturbations are described by the coupling of D3-branes and close string states. Remember that close string states are dual to small perturbation of the background by the state-operator correspondence.

2. $Ng_s$ large: the stack creates a big concentration of energy per length such that, even at small string coupling, it is capable of deforming the background, and not just perturbing it. Supersymmetry fixes completely the background configuration at leading order in $\alpha '$. This background configuration is know as the $3$-black brane solution.

The idea of large N duality is to conjecture that both descriptions are complementary descriptions of a same physical system. In particular, in the decoupling limit, where the degrees of freedom of the stack decouples from gravity.

In the first description above the decoupling limit is obvious (truncate down to some open strings states), while in the second is not so. In fact in the second description the decoupling limit is understood as a near horizon limit. The horizon is infinitely far from the observers sitting at the beginning of the throat, and so this limit is like going very deep in the throat. The space-time there looks like an $AdS_{5}\times S^{5}$. Then by the large N duality conjecture, this means that the open strings of a stack of D3-branes, in the decoupling limit, are dual to Type IIB superstring on the $AdS_{5}\times S^{5}$.

Now, in order to get concrete formulas for this duality it is interesting to take a system of $N+n$ D3-branes, with $n$ much smaller than $N$. Then moving the $n$ $D3$-branes away from the stack of $N$ $D3$-branes will not change the background deformation in the second description (the one for $Ng_s$ large) since $ng_S$ is much smaller than $Ng_s$. These are know as D3-brane probes.