= Sphere gauge fixing for four string vertices
At genus zero the residual <Möbius transformation> group fixes three marked insertion points, leaving one complex position to integrate. The corresponding <worldsheet ghost fields> give $|z_{ij}z_{jk}z_{ki}|^2$ for the three fixed points. For on-shell weight-$(1,1)$ <string vertex operators>, these ghost factors cancel the coordinate dependence of the fixed-point normalization, including the limit at infinity. The remaining <Koba-Nielsen factor> for four tachyons is the integrand of the <complex beta integral>, producing the <Virasoro–Shapiro amplitude>.
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