One can obtain strongly coupled matter in AdS by adding branes in the bulk on which the matter is strongly coupled, or by considering RG flows which can flow to strongly coupled matter in the infrared, but such setups will either involve a relevant deformation of a CFT, or adding interfaces.

There can be additional intermediate regimes; see, e.g., [2].

The heavy operators, at least those whose dimensions scale linearly with $N$, are often referred to as black hole microstates.

For double-trace deformations with $\beta =0$, some operators (those who make up the double-trace deformation) can acquire $\mathcal{O}(N^0)$ anomalous dimensions. However, all other single-trace operators only acquire anomalous dimensions at $\mathcal{O}(N^{-1})$.

One may wonder why the case of $\mathsf{k}=2$ is not considered; it is more subtle. If the double-trace operator is made out of two protected single-trace components, which is likely always the case, then it turns out to have no effect at $\mathcal{O}(N^0)$. Its only potential effect at that order is on the two-point function of its constituents, but these are protected and hence their two-point function is not modified. Because of this, one may try to set $\beta =1$ for a double-trace deformation, but we expect higher order corrections in ${\lambda }_2$ to diverge in $N$.

This modulus appears for example in the symmetric orbifold of the $A$-series $\mathcal{N}=2$ minimal models $A_{k+1}$, with $k=1\text{mod}3$, and $k\ge 4$. It also appears in the symmetric orbifold of the $D$-series $\mathcal{N}=2$ minimal models $D_{k/2+2}$ whenever $k=4\text{mod}6$. There are also many other exactly marginal triple-trace and higher-trace operators for different choices of the constituent operators [15].

Note that the large-$N$ limit, (13) is fully determined by the conformal weight of the chiral primary $\mathbf{O}(z)$. Supersymmetry guarantees that the conformal weight of this field is protected on the conformal manifold. Consequently, (13) is valid (nonperturbatively) under any other exactly marginal deformations that preserve large-$N$ factorization.

For ${\mathrm{\Delta }}_{\mathbf{O}}+{\mathrm{\Delta }}_{\mathbf{O}}+{\mathrm{\Delta }}_{\chi }=d$ there is a divergence in Witten diagrams, analogous to those in extremal correlators [40]. We thank Shota Komatsu for his remarks on this issue.