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The AdS/CFT correspondence is a well-supported conjecture about the equivalence between an ordinary conformal field theory (CFT) and a theory of quantum gravity in asymptotically anti-de Sitter (AdS) spacetime. For the quantum-gravity theory to have anything resembling classical spacetime as a good approximation, the CFT needs to have some special properties. The conclusion (part IV) of Susskind's "Three Lectures on Complexity and Black Holes" (arXiv:1810.11563) says:

In the context of AdS/CFT only special CFTs give rise to bulk locality on scales much smaller than the AdS radius. Those systems require very strongly coupled large-N gauge theories. Most quantum field theories cannot be extrapolated to large coupling without encountering phase transitions or worse. Only supersymmetric QFTs have analyticity properties that ensure against such breakdown at large coupling. Thus, to my knowledge, only super-theories have sub-AdS locality. ... Of course the "real world" is not even approximately supersymmetric, yet it is very local on microscopic scales. This is a big puzzle...

The logic here seems to be (1) the CFT needs to be strongly coupled, (2) so it can't have a phase transition separating the strong and weak coupling regimes, and (3) supersymmetry ensures that no such phase transition exists. I'm comfortable with (1) and (3), but I don't understand (2). Couldn't a strongly-coupled CFT be healthy even if it has a phase transition separating the strong and weak coupling regimes? And if it is still healthy, then why would the existence a phase transition at some lower value of the coupling prevent the strongly-coupled version of the CFT from giving sub-AdS locality in the bulk?

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    $\begingroup$ I don't quite understand what "phase transition" or "phase diagram" is supposed to mean. One either has a strongly-coupled large-$N$ theory, or one doesn't. It is not clear to me why there needs to be a family of CFTs with varying coupling for AdS/CFT to make sense. If you do for some reason insist on a CFT with tunable coupling, however, this means you are asking for CFT with exactly marginal deformations. For CFT${}_d$ with $d>2$ the only known examples of this are supersymmetric. I am not 100% sure about $d=2$. $\endgroup$
    – Peter Kravchuk
    Commented Sep 30, 2021 at 14:16
  • $\begingroup$ Also: page 8 in arXiv:1311.0755 says "all known examples of CFT with a gap actually have superconformal symmetry." $\endgroup$
    – Chiral Anomaly
    Commented Oct 26, 2021 at 0:53

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A phase transition, as I've always used the term, does not say anything about whether you will or won't find a holographic CFT. In the Landau-Ginzburg paradigm, it is familiar that if you make the coupling constant too small, it will run to zero in the IR and lead to a free theory. If you instead raise the coupling to a zero of the beta function first, the IR phase will be different. In this sense, the "phase transition" is a good thing. The free CFT describing the unbroken phase is not holographic but the non-trivial CFT describing the critical point might be.

The excerpt sounds like it's trying extrapolate the $AdS_5 \times S^5$ lore a little too far. 3D superconformal gauge theories provide an example where the weak and strong coupling limits are both holographic. The situation is a little different because Chern-Simons levels have to be integers, but it has been studied in some nice recent works. ABJM theory with $k = 1$ is dual to 11D supergravity on $AdS_4 \times S^7$. If you instead take $k \to \infty$ in a `tHooft limit, you get IIA supergravity on $AdS_4 \times CP^3$.

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  • $\begingroup$ Thank you for calling my attention to the ABJM papers! $\endgroup$
    – Chiral Anomaly
    Commented Oct 26, 2021 at 0:51
  • $\begingroup$ By the way, in hindsight, I think I just misunderstood what Susskind meant. Now I think he just meant that QFTs which are sufficiently strongly coupled (for AdS/CFT) are hard to find without SUSY, because if we take a weakly coupled QFT and start trying to smoothly increase its coupling, we tend to run into a phase transition instead. (Is that a plausible interpretation of the excerpt?) Like you said, it's not that AdS/CFT can't use CFTs that have a phase transition somewhere, it's just phase transitions tend to obstruct naive approaches to reaching sufficiently strong coupling. $\endgroup$
    – Chiral Anomaly
    Commented Oct 26, 2021 at 0:52
  • $\begingroup$ When you have a running coupling, the CFT is the second order phase transition. But I agree that the naive approach usually breaks down long before we find one that has a sufficiently large gap. This might be what he meant. $\endgroup$
    – Connor Behan
    Commented Oct 26, 2021 at 4:10

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