# Worldvolume vs boundary in AdS/CFT

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

• Not a full answer, but first you can observe that geometrically AdS as a manifold has a boundary at z=0, not at z = $\infty$. – zzz Jul 14 '15 at 19:30
• As a manifold it is diffeomorphic to $R^d$ so it has no boundary -an open manifold since it is noncompact. It is a smooth algebraic subvariety (a smooth quadric) in R^{d+1} with the standard euclidian or lorentzian metric. It only has a conformal boundary, not situated anywhere actually. – plm Jul 14 '15 at 19:52
• Thanks. Actually what seems to matter is a cauchy hypersurface to determine boundray values for the wave equation. You can check the GKP paper where they actually use the AdS radius to set boundary conditions instead of the conformal boundary $z=0$. Also forget my comment about the conformal boundary being anywhere, I was wrong. – plm Jul 16 '15 at 1:57