# Virasoro primary for 2D Ising model at critical point

It is well known that 2D Ising model at critical point can be described by a 2D CFT. The CFT is identical to free Majorana fermions. It has three primary operators namely

1. The identity.

2. The Ising spin $\sigma$ with weight $(1/16, 1/16)$.

3. The energy density $\epsilon$ with weight $(1/2, 1/2)$.

As far as I understand these are global i.e, $SL(2,R)$ primaries. But 2D CFTs have larger symmetry group - Virasoro group. What are those Virasoro primaries?

• The Virasoro generators generate descendant fields/secondary operators. Aug 15, 2017 at 8:32
• Why do you think there are more primaries, and what is the distinction between "$\mathrm{SL}(2,\mathbb{R})$ primaries" and "Virasoro primaries" supposed to be? Do you perhaps mean quasi-primary instead of "Virasoro primary"? Aug 15, 2017 at 9:01

The three fields that you mention are indeed Virasoro primaries, i.e. they are eigenvectors of the Virasoro generators $L_0,\bar L_0$ and are killed by $L_{n>0},\bar L_{n>0}$. Therefore they are also $SL(2)$ primaries (in other words quasi-primaries), because in particular they are killed by $L_1, \bar L_1$, i.e. by those Virasoro annihilation modes that belong to the Lie algebra of $SL(2)$. These three fields are the only Virasoro primary fields in the model. However, there exist infinitely many other $SL(2)$ primary fields, because a number of Virasoro descendent fields are killed by $L_1,\bar L_1$ (but not by $L_{n>1},\bar L_{n>1}$).
In other words, there are fewer primary fields with respect to the larger algebra (Virasoro) than with respect to its subalgebra ($SL(2)$).
• Thanks! Actually I was thinking about expanding 4-pt correlator ($\sigma \sigma \sigma \sigma$, say) of 2D Ising CFT in these "SL(2) primary blocks" instead of Virasoro primary blocks (which is given in BPZ or the yellow book). Now this 4-pt fn will be a sum over infinite number of such blocks. Do you know if it has been worked out somewhere or can you suggest me how to go about it? Oct 6, 2017 at 16:20
• I am not sure, but Zamolodchikov's c-recursion for Virasoro blocks looks like expressing them in terms of $SL(2)$ blocks, i.e. hypergeometric functions. So it might give the solution. This c-recursion is reviewed for example in arXiv:1502.07742 . Of course your particular correlator is very special and is surely much simpler than the general case. Oct 6, 2017 at 19:55
• Thanks for the reference, looks very useful. I was wondering if I can just use the global blocks computed by Dolan and Osborn (hep-th : 0011040) for 2D CFT with infinite number of $\Delta$ and $l$ corresponding to infinite number of $SL(2)$ primaries.. Oct 7, 2017 at 14:27