Calculation of effective action in <PERSON> frame
I want to do the calculation for the effective action in <PERSON> frame (see <PERSON>'s lecture notes Chapter 7.3), this action in string frame is given by \begin{align}\label{exp1} S=\frac{1}{2 \kappa_{0}^{2}} \int d^{26} X \sqrt{-G} e^{-2 \Phi}\left[\mathcal{R}-\frac{1}{12} H_{\mu \nu \lambda} H^{\mu \nu \lambda}+4 \partial_{\mu} \Phi \partial^{\mu} \Phi\right]. \end{align} With a rescaling of the metric $$\tilde{G}{\mu \nu}(X)=e^{-4 \bar{\Phi} /(D-2)} G{\mu \nu}(X)$$ where $\tilde{\Phi} = \Phi -\Phi_0$, this should yield the following expression for the effective action: $$ S_{Einstein}=\frac{1}{2 \kappa^{2}} \int d^{26} X \sqrt{-\tilde{G}}\left(\tilde{\mathcal{R}}-\frac{1}{12} e^{-\bar{\Phi} / 3} H_{\mu \nu \lambda} H^{\mu \nu \lambda}-\frac{1}{6} \partial_{\mu} \tilde{\Phi} \partial^{\mu} \tilde{\Phi}\right), $$ for $D=26$. Due to the change of the metric, the <PERSON> scalar also changes according to $$ \tilde{\mathcal{R}}=e^{-2 \omega}\left(\mathcal{R}-2(D-1) \nabla^{2} \omega-(D-2)(D-1) \partial_{\mu} \omega \partial^{\mu} \omega\right). 818
$$ with $\omega=-\frac{2 \tilde{\Phi}}{D-2}$. Now the introduction of $\kappa^{2}=\kappa_{0}^{2} e^{2 \Phi_{0}}$ and taking into account that the second and third term in square brackets above depend on powers of the inverse metric, gives me the following intermediate result \begin{align} S=\frac{1}{2 \kappa^{2}} \int d^{D} X \sqrt{-\tilde{G}} e^{-2 \omega} \left( \tilde{\mathcal{R}}e^{2 \omega} + 2(D-1) e^{2 \omega} \nabla^{2} \omega + (D-2)(D-1) e^{2 \omega} \partial_{\mu} \omega \partial^{\mu} \omega + \ \qquad \qquad + e^{6 \omega} \frac{1}{12} H_{\mu \nu \lambda} H^{\mu \nu \lambda} + 4 e^{2 \omega} \partial_{\mu} \tilde{\Phi} \partial^{\mu} \tilde{\Phi}\right), \end{align} where I plugged $$ \mathcal{R} = e^{2 \omega}\tilde{\mathcal{R}}+ 2(D-1) e^{2 \omega} \nabla^{2} \omega + (D-2)(D-1) e^{2 \omega}\partial_{\mu} \omega \partial^{\mu} \omega $$ into the expression for the action in string frame.
My problem now is that I have no idea how to get the $-\frac{1}{6} \partial_{\mu} \tilde{\Phi} \partial^{\mu} \tilde{\Phi} = -\frac{4}{D-2} \partial_{\mu} \tilde{\Phi} \partial^{\mu} \tilde{\Phi}$ term as stated in the lecture notes. I think the Laplacian term in my intermediate result can be discarded, as it is a boundary term. 818