# "Light" states in critical $O(N)$ model in $2+1$ (and holography)

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Let me split the question in a few parts,

• Can someone give me a reference which explains the CFT properties of the critical $O(N)$ model in $2+1$? Like how are the CFT correlators (in a $1/N$ expansion?) and the central charge in that calculated?

• I guess that by "light" operators one means operators with conformal dimensions smaller than the central charge. Can someone give me a reference which explains as to what are these light states that exist in the critical $O(N)$ model and in what sense are they like a free field? (..as I have often heard being said..)

{..Are there other CFTs out there which have similar properties like the above and do for all of them the central charge grows with some $N$ (the same $N$ in whose large limit the system is having a lot of free-field like light states) ?...}

• For any CFT is it true that if it doesn't have "too many" light states then it is probably holographic? (...by "too many" I think it is meant that the number of states (primaries?) with scaling dimensions below the central charge don't grow exponentially in the central charge - may be they can grow at most polynomially?..)

This post imported from StackExchange Physics at 2014-08-11 14:51 (UCT), posted by SE-user user6818
For holography, it seems to be related to higher spin massless gauge fields in AdS4

This post imported from StackExchange Physics at 2014-08-11 14:51 (UCT), posted by SE-user Trimok
@Trimok You know of a good reference where the central charge and the anomalous dimensions of the critical $O(N)$ model is discussed?

This post imported from StackExchange Physics at 2014-08-11 14:51 (UCT), posted by SE-user user6818
Sorry , no. I tried to find a reference on the Web, without success.

This post imported from StackExchange Physics at 2014-08-11 14:51 (UCT), posted by SE-user Trimok
@Trimok I was wondering if one can at least say that at the large $N$ fixed point the central charge of the $O(N)$ model grows with N. Because large-N O(N) model is the same as the large-N spherical sigma model and hence if the former is critical then so is the later. And isn't it true that for a critical non-linear sigma model its central charge is given by the dimension of the target manifold? If that is true here we will get a $S^{N-1}$ and hence the central charge will grow as $N$?

This post imported from StackExchange Physics at 2014-08-11 14:51 (UCT), posted by SE-user user6818
I suppose you are quite right. I found a interesting formula for the central charge (formula $4.2$ page $11$) of this paper

This post imported from StackExchange Physics at 2014-08-11 14:51 (UCT), posted by SE-user Trimok
@Trimok Thanks a lot! This looks interesting! Do let me know if you find a reference which proves this fact that the dimension of the target manifold gives the central charge of the critical non-linear sigma model.

This post imported from StackExchange Physics at 2014-08-11 14:51 (UCT), posted by SE-user user6818
@Trimok How do they know this $c_{free} = D/(D-1)$?

This post imported from StackExchange Physics at 2014-08-11 14:51 (UCT), posted by SE-user user6818

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