 Three-point worldsheet ghost correlator

On the Riemann sphere, three holomorphic reparameterization ghost zero modes correspond to the three infinitesimal Möbius transformations. Three $c$ insertions saturate their Grassmann integral. In a basis $1,z,z^2$, their integral is the Vandermonde determinant, giving $z_{12}{z}_{13}{z}_{23}$ up to an ordering-dependent sign. Including the antiholomorphic factor gives $|z_{12}{z}_{13}{z}_{23}{|}^2$, the Jacobian in sphere gauge fixing for four string vertices.