Global conformal Ward identities for chiral correlators

For primary operators of holomorphic weights $h_i$ and a Möbius transformation invariant vacuum, infinitesimal translations, dilations and special conformal transformations give ${\sum }_i{\partial }_{i}G=0$, ${\sum }_i(z_i{\partial }_i+h_i)G=0$, and ${\sum }_i(z_i^2{\partial }_i+2h_i{z}_i)G=0$. Solving gives a constant one-point function only at $h=0$, a two-point function $C{z}_{12}^{-2h}$ only for equal weights, and a three-point function $C{z}_{12}^{h_3-h_1-h_2}{z}_{13}^{h_2-h_1-h_3}{z}_{23}^{h_1-h_2-h_3}$. Three-point covariance imposes no further equality of weights; dynamical selection rules may still force $C=0$. For more points, arbitrary functions of conformal cross-ratios remain.