I have a question concerning the Faddeev-Popov ghost boundary conditions in the path integral quantization of bosonic strings. My ghost action is:

$S_g= - \frac{i}{2\pi} \int d^2 \xi \sqrt{-h} \; b_{ab} \nabla^a c^b$

I can derive the equations of motion, but I cannot see the boundary conditions of the ghost and anti-ghost fields. They should be:

$c^+ = c^-$


$b_{++}=b_{--} $

for $\sigma = 0 , l$.
It always says, that those conditions arise from the boundary terms, when deriving the equations of motion. My boundary terms of the integration by parts are:

$ -\int \left. d\tau c^+ b_{++} \right|_{\sigma_i}^{\sigma_f} + \int \left.d\tau c^- b_{--} \right|_{\sigma_i}^{\sigma_f} + \int d\sigma \left. c^+ b_{++} \right|_{\tau_i}^{\tau_f} + \int \left. d\sigma c^- b_{--} \right|_{\tau_i}^{\tau_f}=0$


$\int \left. d\tau c^- \delta b_{--} -c^+ \delta b_{++}\right|_{\sigma_i}^{\sigma_f}=0$

Thank you very much for any help


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