The classical limit of AdS/CFT

[Demystifier]

The AdS/CFT correspondence is a correspondence of one quantum theory to another quantum theory. But what about the classical limit of these two theories? Is there a correspondence between the corresponding classical theories? If there is, what a precise form this classical-to-classical correspondence takes?

 

[atyy]

fzero commented on this in https://www.physicsforums.com/threa...-form-ways-it-could-fail.706159/#post-4479075. I don't understand it much, but here is his comment.

The bulk gravity is only ever classical in an extreme limit, but even so, the limit in which the rest of the theory is classical is actually different. Let's recall how parameters are related in the case of IIB on AdS5. The gauge theory $N$ is the 5-form flux and is related to the volume of the 5-sphere (which is in turn related to the AdS radius $R$), so

$$N \sim ( M_P R)^4.$$

The limit in which gravity is classical is one in which the radius of curvature is large in Planck units, so this is the limit of large $N$ in the gauge theory.

The gauge theory coupling on the other hand is directly related to the string coupling

$$ g^2 N \sim g_s N \sim ( M_s R)^4,$$

where the last expression uses the relationship between the string scale, string coupling, and Planck scale. Stringy corrections, which include bulk scalar and gauge interactions, involve the string scale, rather than the Planck scale. So there is a range of values for the string coupling, where gravity is classical, but quantum string interactions are important.

In the limit where both $$N, g^2 N$$ are large, everything in the bulk is classical.

 

[Demystifier]

Thanks atty, but I don't find in satisfying. For instance, one can certainly consider a classical gauge theory with small N (classical electrodynamics is an example), so I cannot accept the claim that classical limit is the large N limit. Indeed, the classical limit should be related to the Planck constant going to zero (in a suitable way), while the discussion above does not even mention the Planck constant.