Q: Calculation of effective action in Einstein frame I want to do the calculation for the effective action in Einstein frame (see Tong'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 Ricci scalar also changes according to
$$
\tilde{\mathcal{R}}=e^{-2 \omega}\left(\mathcal{R}-2(D-1) \nabla^{2} \omega-(D-2)(D-1) \partial_{\mu} \omega \partial^{\mu} \omega\right).
$$
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. Since the other terms do already match when $D=26$, I get
$$
-\frac{4}{D-2} \partial_{\mu} \tilde{\Phi} \partial^{\mu} \tilde{\Phi} = 
(D-2)(D-1) \partial_{\mu} \omega \partial^{\mu} \omega + 4 \partial_{\mu} \tilde{\Phi} \partial^{\mu} \tilde{\Phi}=(D-2)(D-1) \partial_{\mu} \omega \partial^{\mu} \omega + (D-2)^2 \partial_{\mu} \omega \partial^{\mu} \omega,
$$
which seems/is incorrect, unless I missed something trivial. Can someone point out to me, what I'm doing wrong here? Maybe I haven't understood how rescaling the metric changes the action completely. Also I have noticed that  if $(D-2)(D-1) \partial_{\mu} \omega \partial^{\mu} \omega$ would have opposite sign it would actually yield the correct result, but I don't see why this should be the case. Help is much appreciated!
