In their paper 'Sewing Polyakov Amplitudes I: Sewing at a Fixed Conformal Structure' Carlip et al. start by considering the path integral $$Z_{\Sigma'}[\tilde{X}] = \int [dX]e^{-S[X]} $$ for the action $$ S = \frac{1}{2}\int \partial X \wedge \bar{\partial} X$$ of a scalar field $X$ on a disconneted Riemann surface $\Sigma'$ with boundaries $C_1$ and $C_2$, with $\tilde{X}$ the value of $X$ on the boudnaries $C_i$. They separate $X = \bar{X}+X'$, where $\bar{X}$ is the calssical piece satisfying $\partial^2\bar{X} = 0,\ \bar{X}|_{C_i} = \tilde{X}_i$ and $X'$ is the fluctuation satisfaying $X'|_{C_i}=0$, and claim $$Z_{\Sigma'}[\tilde{X}] = [\text{det} \partial^2]^{-1/2} e^{-S[\bar{X}]}\ \ \ \ \ \ \ \ \ \ \ \ \ (1) $$ where $$ S[\bar{X}] = \frac{1}{2}\sum_{ij}\int dx\int dx' \tilde{X}_i(x)\partial_{n_i}\partial_{n_j} G(x,x') \tilde{X}_j(x')\ \ \ \ \ \ \ \ \ \ \ \ \ (2)$$ where $G$ is the Dirichlet Green's functionand $\partial/\partial_n$ the normal derivative to the surface.
I see how $\bar{X}$ and $X'$ decouple in the action and $(1)$ is obtained but I am having trouble understanding the form of $(2)$, especially the double normal derivative on the Green's function.
Any insights would be appreciated. Thank you!