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The Virasoro algebra has a finite $sl(2)$ sub-algebra generated by $L_{-1}$, $L_0$ and $L_{+1}$. Let's consider a unitary highest weight representation of the Virasoro algebra with conformal weight $h>0$ and central charge $c>1$. How does this representation decompose under the $sl(2)$ sub-algebra?

It is clear that there is an invariant $sl(2)$ sub-module of weight $h$ consisting of the highest weight state $|h\rangle$ and the descendants $L_{-1}|h\rangle$, $L_{-1}L_{-1}|h\rangle$, $\dots$ However, there are many other descendants, such as $L_{-2}|h\rangle$, $L_{-3}|h\rangle$, etc. What $sl(2)$ modules will these states sit in? In particular, is the full $sl(2)$ representation completely reducible, so that it can be written as a sum of irreducible $sl(2)$ representations? Can anything general be said about the multiplicity of the $sl(2)$ representations appearing in the decomposition?

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Maybe this question belongs on math.SE, but there are hardly any questions about the Virasoro algebra at that site, while physics.SE has a whole bunch, so I think there is a bigger chance of getting an answer here. – Olof Aug 7 '13 at 13:19
Nah, I'd say the question should stay here, as these issues are of huge interest in theoretical physics, +1 BTW :-) – Dilaton Aug 7 '13 at 15:24
I did not understand a word of this reference, but the title seems to be related to your question... – Trimok Aug 7 '13 at 16:28
@Trimok: Thanks for the reference. I agree that it looks a bit dense but useful. – Olof Aug 7 '13 at 17:17
up vote 4 down vote accepted

This is indeed an interesting question. Define a quasi-primary state to be one that is annihilated by $L_1$ i.e., it is a highest-weight vector (state) of the $sl(2)$ subalgebra of the Virasoro algebra. Consider a generic Virasoro highest weight vector, $|h\rangle$, of weight $h$. The Verma module is constructed by acting on $|h\rangle$ by all combinations $L_{-n}$ for $n=1,2,3,\ldots$. By generic I mean none of the descendants are also highest weight vectors (aka null vectors). This is done for simplicity. It follows that the Verma module is irreducible.

Now we wish to decompose this Verma module into irreps of $sl(2)$. The first quasi-primary appears at level $0$ is $|h\rangle$ and its descendants are $L_{-1}^n|h\rangle$ for $n\geq1$. There is no quasi-primary at level 1. At level 2, there is $L_{-2}|h\rangle$ in addition to $(L_1)^2|h\rangle$. But it is not quasi-primary. But a simple calculation shows that $|\phi\rangle:=\left(L_{-2}-\tfrac32 (L_{-1})^2\right)|h\rangle$ is a quasi-primary. This along with its descendants $(L_{-1})^n |\phi\rangle$ is the second irrep of $sl(2)$. One can continue in this fashion at each level and look for quasi-primaries i.e., states annihilated by $L_1$. The statement in the reference mentioned by Trimok, if I understood it correctly, states that at level $(n+1)$, there are descendants that appear by the action of $L_{-1}$ on states at level $n$ and the remaining are necessarily quasi-primaries.

Recall that at level $n$, there are $p(n)$ states where $p(n)$ is the number of partitions of $n$. So it follows that there must be $(p(n+1)-p(n))$ quasi-primaries at level (n+1). I suspect that the proof is not too hard but I have not worked it out. Each quasi-primary is the highest weight vector for an infinite dimensional irrep of sl(2).

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Is there a systematic way to construct these quasi-primary (but virasoro decedent)? – Phys-Chan Apr 21 at 19:21
And it seems that the reference only consider that case that h=0, is that correct? – Phys-Chan Apr 21 at 20:18

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