We know the action of topologically massive gravity in 3-dimentional spacetime is \begin{equation} \label{eq:EH} S=S_{\mathrm{EH}}+S_{\mathrm{CS}}=\frac{1}{\kappa^2}\int d^3x\sqrt{-g}R+\frac{1}{\mu}\int d^3x\varepsilon^{\lambda\mu\nu}\Gamma_{\lambda\sigma}^{\rho}\left(\partial_{\mu}\Gamma^{\sigma}_{\rho\nu}+\frac{2}{3}\Gamma^{\sigma}_{\mu\tau}\Gamma^{\tau}_{\nu\rho}\right) \end{equation}
Under conformal transformation, $g_{\mu\nu}\rightarrow\Phi^4 \bar{g}_{\mu\nu}\left(g^{\mu\nu}\rightarrow\Phi^{-4}\bar{g}^{\mu\nu}\right)$, Christoffel symbols become \begin{equation} \begin{aligned} \Gamma_{\mu\nu}^{\lambda}\left(g\right)&=\frac{1}{2}g^{\lambda\alpha}\left(\partial_\mu g_{\alpha\nu}+\partial_\nu g_{\alpha\mu}-\partial_\alpha g_{\mu\nu}\right)\\ &=\Gamma_{\mu\nu}^{\lambda}\left(\bar{g}\right)+2\left(\delta^{\lambda}_{\nu}\partial_{\mu}\ln\Phi+\delta^{\lambda}_{\mu}\partial_{\nu}\ln\Phi-g_{\mu\nu}\partial^{\lambda}\ln\Phi\right)\\ &=\Gamma_{\mu\nu}^{\lambda}\left(\bar{g}\right)+Z_{\mu\nu}^{\lambda} \end{aligned} \end{equation}
$Z_{\mu\nu}^{\lambda}$ is the difference between two Christoffel symbol, so it must be a tensor. The covariant derivatives of $Z_{\mu\nu}^{\lambda}$ is \begin{equation} D_{\alpha}Z_{\mu\nu}^{\lambda}=\partial_{\alpha}Z_{\mu\nu}^{\lambda}+\Gamma_{\alpha\beta}^{\lambda}Z_{\mu\nu}^{\beta}-\Gamma_{\alpha\mu}^{\beta}Z_{\beta\nu}^{\lambda}-\Gamma_{\alpha\nu}^{\beta}Z_{\mu\beta}^{\lambda} \end{equation}
Riemann curvature tensors become \begin{equation} \begin{aligned} R^{\lambda}_{\mu\rho\nu}\left(g\right)&=\partial_{\rho}\Gamma_{\mu\nu}^{\lambda}-\partial_{\nu}\Gamma_{\mu\rho}^{\lambda}+\Gamma^{\lambda}_{\rho\sigma}\Gamma^{\sigma}_{\mu\nu}-\Gamma^{\lambda}_{\nu\sigma}\Gamma^{\sigma}_{\mu\rho}\\ &=R^{\lambda}_{\mu\rho\nu}\left(\bar{g}\right)+D_{\rho}Z_{\mu\nu}^{\lambda}-D_{\nu}Z_{\mu\rho}^{\lambda}+Z^{\lambda}_{\rho\sigma}Z^{\sigma}_{\mu\nu}-Z^{\lambda}_{\nu\sigma}Z^{\sigma}_{\mu\rho} \end{aligned} \end{equation}
Let $\rho=\lambda$ and contract. We get Ricci tensor. Calculate this explicitly, \begin{equation} \begin{aligned} D_{\lambda}Z_{\mu\nu}^{\lambda}-D_{\nu}Z_{\mu\lambda}^{\lambda}&=-2\left(D_{\mu}D_{\nu}\ln\Phi+g_{\mu\nu}D^2\ln\Phi\right)\\ Z^{\lambda}_{\lambda\sigma}Z^{\sigma}_{\mu\nu}&=12\left(2D_\mu\ln\Phi\cdot D_\nu\ln\Phi-g_{\mu\nu}D_\sigma\ln\Phi\cdot D^\sigma\ln\Phi\right)\\ Z^{\lambda}_{\nu\sigma}Z^{\sigma}_{\mu\lambda}&=4\left(5D_\mu\ln\Phi\cdot D_\nu\ln\Phi-2g_{\mu\nu}D_\sigma\ln\Phi\cdot D^\sigma\ln\Phi\right) \end{aligned} \end{equation}
Ricci scalar becomes \begin{equation} R(g)=g^{\mu\nu}R_{\mu\nu}(g)=\Phi^{-4}R(\bar{g})-8\Psi^{-1}D_\mu D^\mu\Phi \end{equation}
So under conformal transformation, Einstein-Hilbert action becomes \begin{equation} \begin{aligned} S_{\mathrm{EH}}&=\frac{1}{\kappa^2}\int d^3x\sqrt{-\bar{g}}\Phi^6\left(\Phi^{-4}R\left(\bar{g}\right)-8\Phi^{-1} D_\mu D^\mu\Phi\right)\\ &=\frac{1}{\kappa^2}\int d^3x\sqrt{-\bar{g}}\left(\Phi^{2}R\left(\bar{g}\right)-8\bar{g}^{\mu\nu}\Phi D_\mu D_{\nu}\Phi\right)\\ &=\frac{1}{\kappa^2}\int d^3x\sqrt{-\bar{g}}\left(\Phi^{2}R\left(\bar{g}\right)+8\bar{g}^{\mu\nu}D_\mu\Phi D_{\nu}\Phi\right) \end{aligned} \end{equation}
but how gravitational Chern-Simons action change? \begin{equation} S_{\mathrm{CS}}=\frac{1}{\mu}\int d^3x\varepsilon^{\lambda\mu\nu}\Gamma_{\lambda\sigma}^{\rho}\left(\partial_{\mu}\Gamma^{\sigma}_{\rho\nu}+\frac{2}{3}\Gamma^{\sigma}_{\mu\tau}\Gamma^{\tau}_{\nu\rho}\right) \end{equation}
In Deser's article:enter link description here, it just tell me $S_{\mathrm{CS}}$ is conformal invariant. But how to calculate explicitly?
