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?..)

  • $\begingroup$ For holography, it seems to be related to higher spin massless gauge fields in AdS4 $\endgroup$
    – Trimok
    Nov 7, 2013 at 19:41
  • $\begingroup$ @Trimok You know of a good reference where the central charge and the anomalous dimensions of the critical $O(N)$ model is discussed? $\endgroup$
    – user6818
    Nov 13, 2013 at 22:25
  • $\begingroup$ Sorry , no. I tried to find a reference on the Web, without success. $\endgroup$
    – Trimok
    Nov 14, 2013 at 11:27
  • $\begingroup$ @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 $N$? $\endgroup$
    – user6818
    Nov 17, 2013 at 2:52
  • $\begingroup$ I suppose you are quite right. I found a interesting formula for the central charge (formula $4.2$ page $11$) of this paper $\endgroup$
    – Trimok
    Nov 18, 2013 at 9:16


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