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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.

Abhimanyu Pallavi Sudhir
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2 Answers2

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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.

Zohar Ko
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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.

Frederic Brünner
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