Can AdS/CFT still give bulk locality on small scales if the CFT has a nontrivial phase diagram? 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?

Related:

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*Which CFTs have AdS/CFT duals?


*Supersymmetry in AdS/CFT
 A: 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 $AdS_4 \times CP^3$.
