Worldsheet ghost action
= Worldsheet ghost action
For the traceless diffeomorphism operator
$$
(P_1c)_{\alpha\beta}=\nabla_\alpha c_\beta+\nabla_\beta c_\alpha-g_{\alpha\beta}\nabla_\gamma c^\gamma,
$$
the ghost action is
$$
S_{\rm gh}=\frac1{2\pi}\int d^2\sigma\,\sqrt g\,
b^{\alpha\beta}(P_1c)_{\alpha\beta}.
$$