Möbius invariance of on-shell closed-string vertex integrals (${\alpha }^{\prime }{p}_i^2=4$)

Under $z\mapsto (az+b)/(cz+d)$ with $ad-bc=1$, the coordinate measure contributes $|c{z}_i+d{|}^{-4}$ at insertion $i$. The Koba-Nielsen factor contributes $|c{z}_i+d{|}^{-{\alpha }^{\prime }{\sum }_{j\ne i}{p}_i\cdot p_j}$. Four-momentum conservation makes this exponent ${\alpha }^{\prime }{p}_i^2$, so the factors cancel precisely on the displayed tachyon mass shell. The correlator alone is covariant; its product with all the coordinate measures is invariant.