Worldsheet ghost field ($b_{\alpha \beta },c^{\alpha }$)

Gauge fixing worldsheet diffeomorphism and Weyl symmetry introduces an anticommuting vector ghost $c^{\alpha }$ and a symmetric traceless antighost $b_{\alpha \beta }$. In complex coordinates they form holomorphic and antiholomorphic $bc$ systems.

For the traceless diffeomorphism operatorthe ghost action is