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.

  • $\begingroup$ AdS is a solution to Einstein’s equations with no source (stress energy), but with a negative cosmological constant. $\endgroup$ – BRT Mar 21 '18 at 7:49
  • $\begingroup$ But I believe that the harmonic function R^4/u^4 is specific for a D3 brane. So, the metric mentioned in the text should be due to a D3 brane. $\endgroup$ – Angela Mar 21 '18 at 9:03
  • $\begingroup$ Yes, this is similar to how one obtains the Schwarzschild metric by solving the Einstein equations for a point particle. $\endgroup$ – BRT Mar 21 '18 at 17:28
  • $\begingroup$ More precisely, one uses $N>>1$ to view the probe brane as a D3 brane in the AdS background of the $N$ coincident D3 branes. $\endgroup$ – BRT Mar 21 '18 at 18:01
  • $\begingroup$ Ok, so, the metric is actually obtained by vacuum solutions with a negative cosmological constant. And this metric is interpreted to have been caused by the N coincident D3 branes. Another question is: since u is interpreted as energy, would this not mean that the probe D3 brane which is at u=infinity is of infinite mass? Sending something to u=infinity means it is acquiring infinite mass? So, its infinitely "heavier" than the N coincident D3 branes. What is wrong in this interpretation? $\endgroup$ – Angela Mar 21 '18 at 23:27

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

  • $\begingroup$ So are you saying it is the limit of some coincident dp branes? Is this what it means for it to be split? $\endgroup$ – user151266 Mar 27 '18 at 15:48
  • $\begingroup$ @Cows-lexuskyllaflexium Yes. The $AdS_5\times S^5$ geometry is realized as the limit of coincident D3-branes, as shown. The "split" refers to the splitting of the original N+1 D3-branes into the N D3-branes generating the background fields and the one D3-brane probe. $\endgroup$ – Kenny H Mar 27 '18 at 16:02

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.

  • $\begingroup$ Looking at it from a different perspective. The 5th dimension in $AdS_5$ is energy. When you take the ngs branes to near horizon, you are taking it to energy infinity. This process of higgsifying it and taking it to infinity, makes it acquire large mass. Why wont an extremely large energy string modify the space time around it? ngs was smaller energy than Ngs before higgsifying it, and not after. $\endgroup$ – Angela May 27 at 15:02
  • $\begingroup$ You should realize that there are two sourcers of gravity: The stack $N$ and the stack $n$. Both couples with gravity proportionally with the $g_s$. Keeping $ng_s$ small will guarantee that the stack of n D3-branes will not deform thr background. $\endgroup$ – Nogueira May 28 at 16:13
  • $\begingroup$ Now, the strings that interpolate the stack will become massive, but at the same time you should realize that the you are at the decoupling limit, so you are turning down the interaction between open and close strings in the SYM side of the duality. $\endgroup$ – Nogueira May 28 at 16:19
  • $\begingroup$ Considering the embedding of strings ending in the stack $n$ on the Ads space-time will be dual ro the interaction of theses opens strings solely with the open strings of the stack $N$, while considering the background response of the open strings that sttraches between the stacks will be dual to the interaction of the open strings living in the stack n with the close strings, but the decoupling limit turn this off. $\endgroup$ – Nogueira May 28 at 16:24
  • 1
    $\begingroup$ So, in short, are you saying that, though $ng_s$ acquires mass due to Higgs mechanism, it will not affect the $ADS_5$ space time since string tension $\alpha \rightarrow 0$. And $\alpha \rightarrow 0$ leads to negligible interaction between the bulk and brane ($ng_s$ branes). $\endgroup$ – Angela May 29 at 10:30

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