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?

  • $\begingroup$ The Virasoro generators generate descendant fields/secondary operators. $\endgroup$
    – Qmechanic
    Aug 15, 2017 at 8:32
  • $\begingroup$ 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"? $\endgroup$
    – ACuriousMind
    Aug 15, 2017 at 9:01

1 Answer 1


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 ($SL(2)$).

  • $\begingroup$ 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? $\endgroup$ Oct 6, 2017 at 16:20
  • $\begingroup$ 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. $\endgroup$ Oct 6, 2017 at 19:55
  • $\begingroup$ 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.. $\endgroup$ Oct 7, 2017 at 14:27
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    $\begingroup$ Yes you do want the global blocks (aka SL(2) blocks aka hypergeometric functions). But the known and simple formula that you start with is a combination of Virasoro blocks. What Zamolodchikov's c-recursion does (I think) is to express each Virasoro block as a linear combination of infinitely many global blocks. There is a conservation of complexity: either you have a few complicated (Virasoro) blocks, or you have many simple (global) blocks. $\endgroup$ Oct 8, 2017 at 18:28

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