Worldvolume vs boundary in AdS/CFT

Question

The original AdS5/CFT4 correspondence is usually claimed to match near-horizon supergravity -rather IIB string theory- with a "boundary" super Yang-Mills theory at $T=0$, i.e. in a Minkowski spacetime without matter. But the stacked/parallel brane world volumes lie at the horizon of AdS$^5$, i.e. its center, $z=\infty$, $r=0$, and so does their low-energy worldvolume SYM theory describing low energy open strings ending on them. Now when talking of boundary physicists always mean $z=0$, $r=\infty$. But the boundary theory is not a priori the worldvolume's, this is very puzzling. The explanation I can make of this is that the boundary is actually a conformal boundary, which has a full representative slice at all scale factors $z\sim 1/r$, inverse radius. So one could actually view the "boundary" theory as sitting anywhere in AdS$^5$, on the equivalent of a Cauchy hypersurface except that instead of asking that all maximal timelike geodesics intersect it one asks that all scaling, i.e. holographic renormalization group trajectories extending from $z=\infty$ (IR) to $z=0$ (UV) intersect it. Conformal invariance would justify taking correlators anywhere in the bulk and scaling them according to their mass/conformal dimension. The arguments using the scalar or the graviton wave equation would be modified accordingly -taking boundary conditions $\phi(r,x)=\phi_r(x)\ne\phi_0(x)$ at $r\ne\infty$. The problem would be that at $z>0$, in the bulk, sources for worldvolume local operators would not correspond to local perturbations in the bulk, e.g. $\delta$-function sources would probably not correspond to $\delta$-function sources on the boundary. So to get the simple $\mathcal O\phi_0$ source term in the boundary action as precribed by GKPW we want to set the boundary condition at $z=0$ scale.

In his 1997 article Maldacena does not seem to place the CFT at the boundary but only initial conditions for the bulk which he does not precise how they affect the CFT. This was then clarified by Gubser, Klebanov, Polyakov; Witten, and recently by Harlow and others in between. It is also not clear to me that their prescription should hold without slight modifications at finite brane charges, $N<\infty$, finite momenta when considering multiple branes/wrappings.

So am I right thinking that it is just a historical convention -motivated by practical calculation considerations- that set the SYM to actually live at $z=0$? Am I right to find this extremely misleading, especially when combined with the potentially confusing issues on coordinates/notations for AdS?

References for this are:

Maldacena's original article,

various survey lectures on AdS/CFT -which surprisingly do not really clear up this issue in my mind,

on correlator correspondence, GKP arXiv:hep-th/9802109

Witten arXiv:hep-th/9802150

Thanks alot in advance.

EDIT: I was misunderstanding the notion of conformal boundary. It requires a compactification. So there is only $z=0$ as conformal boundary for AdS. But in the correspondence between CFT operators and SUGRA/string wavefunctions it is actually Cauchy hypersurface where one defines the wavefunction and its equations of motion define it in the whole AdS. So any fixed $z$ slice (a horosphere, e.g. y=cnst in the upper halfplane/halfspace model) may serve. In fact I looked better at the GKP paper and they use $z=R$ to set the boundary condition and they talk of worldvolume theory, not of boundary theory. They actually multiply the CFT operators by the bulk field values at $z=r$. I presume this avoids the need to rescale fields by $z^{d-\Delta}$, $\Delta$ the conformal dimension, as is usually done.

Answer 1

I struggle to get the essence of your question, regarding the historical convention, but note that the original form of the correspondence indeed places the duality at (I will use Poincare patch coordinates with the radial coordinate $r$) $r \to \infty$. We agree on this. Now, this $\infty$ is a bit arbitrary. How far is $\infty$ from the centre of the bulk. This is where the duality gets interesting. We agree that the duality is about string theory in the bulk and some field theory living in the boundary. This is the claim of Maldacena which is exact. To get useful results though we use, as almost always, the steepest descent method, at energies $E << \alpha'$, which gives IIB supergravity in the bulk, actually we get the following action for the bulk system

$$ S = S_{brane} + S_{bulk} + S_{interactions} $$

We want to get rid of the interactions and this happens at $\alpha' \to 0$ which can be translated to your desired $r \to \infty$. The $S_{bulk} = S_{DBI}$ of course is the one giving the SYM theory. The boundary is, exactly as you say, conformal boundary as can be seen from the bulk metric in appropriate coordinates (like the ones you also mention above). Note, that the boundary stops being conformal for other than $AdS_5/CFT_4$ and becomes asymptotically conformal for, say, $AdS_4/CFT_3$. Note that the program of holographic renormalisation, to my best knowledge, stops working for $Dp$-branes with $p \geq 5$ (at least the 3 methods I am aware of) - the case of $p=5$ is not even an $AdS$ space. Finally, what you say about the operators stop being local perturbations as we move towards the core of the bulk is true, and to the best of my knowledge progress in higher spin theories is working in this direction but I don't know about higher spin. Overall I would say that you are wrong about the fact that your $z=0$ being the boundary is due to historical reasons. It is just what it is required when you take the sugar limit to put it in simple words.

Have I answered at all your question?

Comments

Thanks alot for the answer. Actually you did not clear up my misunderstanding. But I read more. I have seen as I say above that GKP use $z=R$, the AdS "radius" to set boundary conditions. They also use the term world-volume theory which is less misleading than "boundary theory". The brane theory is actually point fields located at the horizon. The boundary fields are actually boundary conditions for bulk fields, which can be set at any z, determining through the equations of motion values everywhere in the bulk and then the source by which we multiply CFT operators. In conclusion the "boundary CFT" really is located at the horizon (at least for extremal black branes), not the boundary of AdS. It only takes its name from being determined by boundary conditions, not for describing fields at the boundary. The fact that boundary conditions are usually set at the conformal boundary is what must contribute to the confusion. I may still be mistaken, I'll read a little more. Tell me what you think.