It is clear that the Verma modules are indecomposable modules for the Virasoro algebra with the $L\_0$-weights bounded. I am wondering if the Verma modules exhaust all such indecomposable modules. Does there exist other types of indecomposable modules whose weights are bounded?


Any information and comments will be appreciated.



Indecomposable modules for Virasoro algebra whose weights are bounded
Yes there are other indecomposable modules where $L\_0$ is bounded from below. Logarithmic theories provide many examples.


The simplest example I can think of: take a Verma module generated by a primary state of conformal dimension $\Delta$, and formally take the derivative of this state with respect to $\Delta$. The derived state generates a logarithmic module (i.e. $L\_0$ is not diagonalizable) where $L\_0$ is still bounded from below.


This example works for any $\Delta$. For special values of $\Delta$ such that Verma modules have singular vectors, more complicated constructions are also possible, including non-logarithmic examples. For example, consider a primary state $v\_0$ of dimension zero, and a state $v\_1$ such that
$$
L\_{n\geq 2} v\_1 = 0 \quad , \quad L\_1 v\_1 = v\_0 \quad , \quad L\_0 v\_1 = v\_1
$$
Then the representation generated by $v\_1$ is not a Verma module, as it is strictly larger than the Verma module generated by $v\_0$. Since $v\_1$ is an $L\_0$ eigenvector, $L\_0$ is diagonalizable in that representation. And our representation is indecomposable thanks to our choice of dimension for $v\_0$, which prevents us from decomposing it using a subrepresentation generated by a linear combination of $v\_1$ and $L\_{-1}v\_0$.


