Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 306 / 3 / a

The unintegrated closed-string operator $c\tilde{c}\mathcal{O}$ must have total conformal weight $(0,0)$ and ghost number $(1,1)$. Since $c$ and $\tilde{c}$ have weights $(-1,0)$ and $(0,-1)$, the matter operator $\mathcal{O}$ must be a Virasoro primary operator of weightEquivalently, the full operator must be a BRST-closed operator and not a BRST-exact operator. For a momentum-dependent tensor operator, these requirements impose its target-space mass-shell, transversality, and gauge-equivalence conditions.