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The three unintegrated vertices fix the residual Möbius symmetry of the sphere and saturate its three holomorphic and antiholomorphic ghost zero modes. Moving a fixed insertion changes the amplitude by a BRST-exact operator, whose expectation with physical BRST-closed vertices vanishes. The remaining integrated positions are dummy variables. Thus the amplitude is independent of the chosen coordinates $(z_i,{\bar{z}}_i)$.

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.