Holographic Renormalization in non-AdS/non-CFT

Question

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.

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Answer 1

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.

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Comments

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.

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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).

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Yeah, Sakai-Sugimoto may not be the best example, maybe Klebanov-Strassler is better place to start.

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Yes, KS is much better.

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Sorry to muddy the waters with the wrong example. My question is whether a systematic understanding of the issue exists in *any* context in which the space-time does not have a conformal boundary.

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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.

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Answer 2

I stumbled upon this old question and I thought maybe I could have something to say. 

As it happens to have done a lot of work in holographic renormalization it turns out it is indeed possible to do holography and thus  holographic renormalization for $AdS_{p+2}/\text{CFT}_{p+1}$ where $p+1$ is the dimension of the CFT using a generalized form of the Fefferman-Graham expansions. In specific holography in the methods I will discuss is possible to be performed because the bulk admits a conformally asymptotically $AdS$ geometry for $p =1,2,4$. For $p=5$ one gets $E^{5,1} \times \mathbb{R} \times S^3$ which is non-$AdS$ background it involves a linear dilaton and the qualitative picture of holography for both $D5$-branes and $NS5$-branes is different. Now, there are two methods to proceed. One is to take the action of the low energy SUGRA which involves a conformal factor and go to a string like frame perform holography there and then return to the original frame of the action. The other way is something called "generalized dilatation operator method". There one begins with the usual ADM foliation of the manifold, using the action in the original Einstein frame directly (thus more efficient already) in the radial $AdS$-like direction and finds asymptotic radial expansions the bulk fields near the boundary. The counterterm action that removes the divergences due to the infinite volume is nothing else than Hamilton's principal function $\mathcal{S}$. Then is not very complicated to find the holographic dictionary. Note that both methods fail to work for $p=6,7,8,9$. At least for the second methid, which I am familiar with, the asymptotic radial expansion fail to be correct. I am not sure on the progress for the cases of $p=6,7,8,9$.

By the way, let me mention that this process of the generalized dilatation operator method, in some sense, it is useful in the context of formulating the Hamiltonian problem in cases where the initial/final values of the fields asymptotically approach zero or infinity in problems that do not necessarily admit a holographic dual. It is a very useful concept indeed.