Free bosons and Virasoro null vectors

In a recent article, Manabe and Sulkowski have proposed a method for deriving Virasoro null vectors, starting with certain deformed matrix integrals. In this blog post I will look for a conformal field theory interpretation of this method.

 

Quick reminders on Virasoro null vectors.

 

A null vector of the Virasoro algebra is labelled by two integers $r,s\geq 1$, whose product is the level of the null vector. This null vector occurs in the Verma module with a specific conformal dimension $\Delta_{r,s}$, and it can be written as
$$|\chi_{r,s}\rangle = L_{r,s} |\Delta_{r,s}\rangle$$ where $|\Delta_{r,s}\rangle$ is the primary state of our Verma module, and $L_{r,s}$ is a level $rs$ creation operator.