Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 51 / 3 / Solution

Let $G_n(z_1,\dots ,z_n)$ denote the ordered vacuum correlator. Inserting the vacuum-preserving Möbius transformation operator and its inverse givesFor $\gamma (z)=z+ϵv(z)$, the three vector fields $v=1,z,z^2$ yield the global conformal Ward identities for chiral correlatorsAll formulas below concern separated insertion points and a consistent local choice of branches for fractional powers.

For one insertion, the translation identity makes $G_1$ constant and the dilation identity gives $h_1{G}_1=0$. ThusFor two insertions, translation invariance gives $G_2=f(z_{12})$, where $z_{ij}=z_i-z_j$. Dilations imply $z_{12}{f}^{\prime }+(h_1+h_2)f=0$, so $f=C_{12}{z}_{12}^{-h_1-h_2}$. Substituting into the special conformal identity leaves $(h_1-h_2)z_{12}f=0$. Thereforewith $G_2=0$ when the weights differ.

For three insertions write $G_3=C_{123}{z}_{12}^{-a}{z}_{13}^{-b}{z}_{23}^{-c}$. Under a Möbius transformation, $\gamma (z_i)-\gamma (z_j)=z_{ij}\sqrt{{\gamma }^{\prime }(z_i){\gamma }^{\prime }(z_j)}$ with compatible local branches. The covariance condition therefore fixesSolving givesThis is the complete solution because Möbius transformations act transitively on ordered triples of distinct points: the ratio of any two covariant nonzero solutions has no remaining invariant variable and is constant. Three-point covariance imposes no equality or triangle inequality among the weights. Additional fusion rules or other symmetries may force $C_{123}=0$, but none follow from the stated covariance and vacuum assumptions alone.

For more than three generic points, three positions may be fixed by a Möbius transformation, leaving $n-3$ independent complex conformal cross-ratios. Correlators consequently have a covariant prefactor multiplied by an arbitrary function of these invariants. For example one convenient four-point form isThe Möbius invariance of the cross-ratio guarantees covariance for any $F$. Thus global conformal symmetry fixes one-, two- and three-point functions up to constants, but does not fix higher-point functions. Their remaining functions encode dynamics.