Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 306 / 4 / b / Solution

Let $|s⟩=L_{-1}|\chi ⟩$ with $L_n|\chi ⟩=0$ for every $n\ge 0$. Since $[L_0,L_{-1}]=L_{-1}$, $|s⟩$ has $L_0$ eigenvalue one. Moreover,and for $n>1$,It is therefore physical. It is a Virasoro descendant and hence spurious, while its norm isThus it is null.