The reparametrization-ghost system possesses a U(1) "ghost number" symmetry (c carrying charge +1, b carrying charge -1) generated by a holomorphic current J(z) = -:bc:(z), whose operator product with the stress tensor reveals that J is only a quasi-primary: it acquires an anomalous, curvature-dependent non-conservation, ∂·J ∝ R, under a Weyl transformation, in contrast to a genuine conserved current in flat space. Integrating this anomalous divergence over a closed worldsheet of genus g, via the Gauss-Bonnet relation between curvature and Euler characteristic, produces a selection rule fixing the net ghost number of any nonvanishing path-integral correlator to 3(g-1) (equivalently: the number of unfixed conformal Killing vectors minus the number of moduli), a special case of the general index-theorem relation between an operator's and its adjoint's zero modes. This belongs to two-dimensional conformal field theory as applied to bosonic string quantization, where it fixes the required number and placement of vertex-operator and ghost insertions in string scattering amplitudes and motivates the subsequent BRST formalism for selecting physical, unitary string states.
The reparametrization-ghost system possesses a U(1) "ghost number" symmetry (c carrying charge +1, b carrying charge -1) generated by a holomorphic current J(z) = -:bc:(z), whose operator product wit…