# Gravity dual of N free scalars in 2D

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I have a very basic (and might be very naive) question. What should be the dual gravity description of $N$ (with $N>>1$) free scalars in two dimensions?

I was wondering whether it would be just pure $AdS_3$ because from the seminal work of Brown-Henneaux we know that $$c=\frac{3l}{2G_3}$$and for $N$ free scalars $c=N>>1$ which means $l>>G_3$. So "classical" gravity is good description. In that sense there is no inconsistency. I understand it is very naive argument.

Edit: Here is a related but more concrete question. For $AdS_5/CFT_4$ we know the statement of the conjecture is $\mathcal{N}=4$ SYM in 4D is equivalent to type IIB string theory in $AdS_5 \times S^5$. From there we can take different useful limits (e.g, $l>>l_s$ is SUGRA limit). What is the equivalent statement for $AdS_3/CFT_2$, at least in some 'useful' limit like large central charge etc.?

This post imported from StackExchange Physics at 2015-08-17 08:07 (UTC), posted by SE-user pinu

edited Aug 17, 2015
Related: physics.stackexchange.com/q/26955/2451 , physics.stackexchange.com/q/83025/2451 and links therein.

This post imported from StackExchange Physics at 2015-08-17 08:07 (UTC), posted by SE-user Qmechanic
@Qmechanic thanks a lot for those references. But can you elaborate on whether the statement ($N$ free scalars in 2D are dual to pure $AdS_3$) is completely wrong?

This post imported from StackExchange Physics at 2015-08-17 08:07 (UTC), posted by SE-user pinu

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It seems to me that the only argument given for the proposal in the question is that the family of CFTs given by N free scalars has a central charge going to infinity when N goes to infinity. It is a necessary condition : the expected CFT dual of some gravity theory on less and less curved spacetime has to be a CFT with bigger and bigger central charge. But it is clearly not enough.  In general, the CFT dual of a weakly coupled gravity theory has to be strongly coupled. If not, how a weakly coupled (and worse free) CFT could describe something as complicated as a black hole ? More precisely, for $AdS_3/CFT_2$, the partition function of the gravity theory is expected to be dominated by the trivial ($AdS_3$) topology at low temperature but by the BTZ black holes at high temperature, with a phase transition (the Hawking-Page transition) between the two regimes. In particular, this phase transition should be visible on the CFT partition function in the large central charge limit. Combined with modularity of the CFT partition function, this gives non-trivial restrictions on 2d CFT having gravity duals (see for example http://arxiv.org/abs/1405.5137 ) In particular, there is clearly no phase transition in N free scalars for N large and so it is excluded that N free scalars could be dual of a weakly coupled gravity theory on $AdS_3$.

answered Aug 17, 2015 by (5,140 points)
edited Aug 17, 2015 by 40227

Does this CFT of N free scalars in 2D have any gravity dual at all? If yes, then what is that  gravity theory?

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Please check the Gaberdiel Gopakumar conjecture. Free scalars is a particular case of W models, the dual being certain higher spin theory.

answered Aug 20, 2015 by (180 points)

Thank you for that nice reference!

I am still a bit confused. I am convinced that the gravity dual is certain higher spin theory. But suppose one is interested in leading behavior for some quantities (e.g, entanglement entropy of an interval in the 2d CFT) for which only the central charge is the only info that goes in, shouldn't pure $AdS_3$ be a good "approximation"? This is much like a particular limit one works in $AdS/CFT$ where one doesn't care about the full UV theory in the bulk as far as one is interested in $\mathcal{O}(N^2)$ quantities.

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