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The AdS/CFT correspondence in its prototypical case states that $\mathcal{N}=4$ Super Yang-Mills is dual to string theory in an $AdS_5 \times S^5$ background. Since this duality is of the type strong/weak it allowed us to probe both theories in situations that were previously inaccesible. Despite SYM not being the real QCD it is believed that there are certain properties of QCD in the large $N$ limit that are also present in some models of string theory (for example mesons and systems of $Dp$-branes that allow the study of deep inelastic scattering). There is also the scheme in warped AdS that studies the correspondence in warped AdS spaces in order to obtain a more "real" model of QCD.

In the other hand it is known that via compactifications in string theory one gets a large amount of gauge groups that can in principle acomodate the standard model of particles in it. So one would like to recover the "real world physics" by starting with a 10 d string theory and doing a suitable Calabi-Yau compactification.

So my questions are basically

  1. How can we conciliate both strategies in string theory? Are we in a compactified world or in a holographic one? Or is it the same?

  2. If we find the compactification that gets us to the standard model what happens with the point of view of the duality?

  3. If we find the real holographic model of QCD as some type of string theory, what happens with the compactification scheme?

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  • $\begingroup$ I'm not sure why you view the two aspects of string theory as somehow competing, or otherwise at tension with one another. Why can't it just be, for instance, that we life in a $4d$ world (with gravity!) which arises as a compactification of $10d$ [insert your favorite superstring theory], and which by a holographic principle is dual to a $3d$ gauge theory? Can you explain what you think the issue is, precisely? $\endgroup$ – Danu Oct 3 '16 at 18:07
  • $\begingroup$ If the 4d arises as a compactification I would expect it to be (3+1) plus all the gauge interactions of the standard model (including QCD) and hopefully new interactions/particles that would allow us further research in particle physics. In that scenario is far from clear that there is a duality between minkowsky x calabi-yau and a certain 3d gauge theory. But if I find the exact QCD with 10d holography I would expect it to brake some how in compactificating the 10d to 4d (cause we still live in a 4d world) $\endgroup$ – Jasimud Oct 3 '16 at 18:17
  • $\begingroup$ I don't think you understand what I'm suggesting. I'm saying that, if we arrive at an effective theory from compactifying, I don't see a reason why it's a problem that holography tells us that this should be dual to a $3d$ gauge theory. I'm talking about applying the holographic principle to the $4d$ theory, not the $10d$ theory. $\endgroup$ – Danu Oct 3 '16 at 18:58
  • $\begingroup$ Suppose I get the 'real 4d QCD' via some form of holography. People would still search for the compactification that gets us to the standard model including the 'real 4d QCD' that I've already found in holography. So what does it mean? QCD is "invariant" under the compactification? or if I find QCD via holography I can't get the same QCD in a compactification scheme?, is there any information that we can extract from holography models of QCD to help us with the compactification?, is QCD the only interaction compatible with holography? Why? $\endgroup$ – Jasimud Oct 3 '16 at 19:32
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Within string theory, the following two quests are not competing and only distantly related:

  1. Finding a UV completion of the Standard Model that incorporates gravity.
  2. Find holographic duals to gauge theories to understand their non-perturbative regimes.

In particular, QCD by itself is by itself a UV-complete theory (as long as we don't want to incorporate gravity). We search for a holographic dual because we want to understand the IR of QCD, which is strongly coupled. We do not wish to holographically describe a UV extension of QCD that includes gravity (which might be string theory including compactifications), because the holgraphic approach will not be suitable for studying gravity on the boundary. We really want to understand QCD itself, i.e. a specific 4-dimensional gauge theory with fundamental matter.

Furthermore, the question if our universe is holographic means the following: The UV extension of the Standard Model that includes gravity might have a dual description in terms of a lower dimensional gauge theory without gravity. In this sense string theory can both be a GUT and a holographic theory. The two are perfectly compatible.

In your question, you describe the opposite approach: Take the string theory extension of the Standard Model on the boundary and ask if this has a higher dimensional dual in the bulk. This is not expected as the boundary theory in holography does not incorporate gravity! Consequently, it only makes sense to ask if solutions of string theory (potentially with compactifications) have a lower dimensional gauge theory dual, not the other way around.

What you could possibly do is the following: take a background solution of string theory (with compactifications) and study dynamical QCD on that background (without dynamical gravity). Then you could ask the question if this theory has a holographic dual. However, it is probably not a very sensible thing to do as the compactified dimensions only become relevant at a scale where the QCD fields will couple to other dynamical fields (including gravity fluctuations).

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