In AdS/CFT, the story of renormalization has an elegant gravity dual. Regularizing the theory is done by putting a cutoff near the conformal boundary of AdS space, and renormalization is done by adding counterterms on that surface. Mathematically this is also interesting, since this utilizes the Lorentzian generalization of the Graham-Fefferman expansion.

But, in the spirit of “effective holography”, one ought to be able to do that in spacetimes which do not admit a conformal boundary. I am wondering if anyone has ever seen an attempt to systematically define holographic renormalization in such spaces, for example for p-branes ($p \neq 3$), the NS fivebrane, or the Sakai-Sugimoto model, etc. In such cases one can still take a cutoff surface at the UV of the theory, take the fields to be essentially non-fluctuating, but one does not have a conformal boundary and all the associated machinery.


I believe one has to distinguish two kinds of dualities. AdS/CFT, even in the context where it describes an RG flow (so not the pure AdS_5xS^5 case), is an exact duality to a four-dimensional theory, which interpolates between one well-defined conformal field theory in the UV and another conformal field theory in the IR. So holographic renormalization is in one-to-one correspondence with renormalization in the four-dimensional theory (that is to say, one can map the counterterms, and identify diff invariance with the renormalization group invariance of correlation functions). On the other hand, Sakai-Sugimoto is not a true duality, it only reduces in the IR to something like a four-dimensional theory (one would hope). The UV of the full Sakai-Sugimoto setup has nothing to do with the UV of QCD or any other four-dimensional theory. So in my opinion there is no reason that (whatever renormalization means in this context) it would resemble what we expect in QCD or any other RG flow in four dimensions.

  • $\begingroup$ I am not sure I fully agree. The cleanest case is a complete RG flow for field theory defined at all scales. But, most effective field theories are not defined at all scales, normally that does not prevent you from defining cut-off independent quantities in the IR. Of course, this is easier said than done in the holographic context, but it is entirely possible there are some papers discussing this which I’ve missed. $\endgroup$ – user566 Nov 1 '11 at 19:33
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    $\begingroup$ Yes you can do that, but above the scale of pion physics it won't be four-dimensional. And at the scale of pion physics there is nothing beyond Leutwyler+Gasser.The interesting thing about holographic RG is that you can see the onset of confinement and symmetry breaking in a controlled setup which mirrors four-dimensional physics. That's not the case in Sakai-Sugimoto (to my understanding). $\endgroup$ – Zohar Ko Nov 1 '11 at 20:02
  • $\begingroup$ Yeah, Sakai-Sugimoto may not be the best example, maybe Klebanov-Strassler is better place to start. $\endgroup$ – user566 Nov 1 '11 at 20:30
  • $\begingroup$ Yes, KS is much better. $\endgroup$ – Zohar Ko Nov 2 '11 at 15:12
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    $\begingroup$ Generally that would mean that there is no dual four-dimensional description in the UV, and my objection is in order. (In other words, in this context it is not clear what holographic RG is good for and what should it be compared to.) A cascade is a kind of a middle ground, where there is no ultimate UV fixed point, but also the departure from ordinary Wilsonian physics is not very significant. So in the case of a cascade I would think the idea of holographic RG should make sense. $\endgroup$ – Zohar Ko Nov 2 '11 at 19:14

Holographic renormalization for non-conformal branes, i.e. non-AdS/non-CFT systems, was systematically developed in this paper by Kanitscheider, Skenderis and Taylor. They even work it out for the example of the Witten model, which is the background of the Sakai-Sugimoto model.

The key principle that permits one to extend the formalism of holographic renormalization to non-conformal systems is the so-called generalized conformal structure. This can be understood as follows: if you extend conformal transformations in such a way that the coupling constant of the boundary Yang-Mills theory transforms like an operator of appropriate dimension, the theory possesses a (generalized) conformal invariance. This allows for an asymptotic Fefferman-Graham expansion and the construction of a renormalized action from which you can derive (renormalized) n-point functions.


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