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.