= Möbius invariance of on-shell closed-string vertex integrals
{c}
{title2=$\alpha'p_i^2=4$}
Under $z\mapsto(az+b)/(cz+d)$ with $ad-bc=1$, the coordinate measure contributes $|cz_i+d|^{-4}$ at insertion $i$. The <Koba-Nielsen factor> contributes $|cz_i+d|^{-\alpha'\sum_{j\ne i}p_i\cdot p_j}$. <Four-momentum conservation> makes this exponent $\alpha'p_i^2$, 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.
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