CritPt / Challenge_1_main / Consider a quantum field theory with holographic dual. Under a Weyl…

# Problem setup:
Consider a quantum field theory with holographic dual. Under a Weyl transformation, the boundary metric transforms as $\gamma_{\mu\nu}^{(0)}\to{\cal B}^{-2}(x)\gamma_{\mu\nu}^{(0)}$. The Weyl anomaly ${\cal A}_k$ of the theory in 2k dimensions appears in the transformation of the partition function:
\begin{align}
Z[\gamma_{\mu\nu}^{(0)}]\to e^{-{\cal A}_k}Z[{\cal B}(x)^{-2}\gamma^{(0)}].
\end{align}
This can be computed by evaluating the on-shell action of the bulk gravitational theory.

The holographic Weyl anomaly in d≤8 can be express using the following quantities:
\begin{align}
P_{\mu\nu}={}&R^{(0)}_{\mu\nu}-\frac{R^{(0)}}{2(d-1)}\gamma_{\mu\nu}^{(0)}\,,\\
B_{\mu\nu}={}&\frac{1}{d-2}\big(\nabla^{(0)}_\rho\nabla_{(0)}^\rho  P_{\mu\nu}-\nabla^{(0)}_\rho\nabla^{(0)}_{\nu} P_{\mu}{}^{\rho}- W^{(0)}_{\rho\nu\mu\sigma} P^{\sigma\rho}\big)\,,\\
O_{\mu\nu}={}&\nabla_{(0)}^\lambda\nabla^{(0)}_\lambda B_{\mu\nu}-2W^{(0)}_{\rho\nu\mu\lambda}B^{\lambda\rho}-\frac{4}{d-2}B_{\mu\nu}P^\mu{}_\mu+\frac{2(d-4)}{(d-2)^2}\big(2P^{\rho\lambda}\nabla^{(0)}_\lambda C_{(\mu\nu)\rho}\\
&+\nabla^{(0)}_\lambda PC_{(\mu\nu)}{}^\lambda-C^{\rho}{}_{\mu}{}^{\lambda}C_{\lambda\nu\rho}+ \nabla_{(0)}^\lambda P^\rho{}_{(\mu}C_{\nu)\rho\lambda}-W^{(0)}_{\rho\mu\nu\lambda}P^{\lambda}{}_\sigma P^{\sigma\rho}\big)\,,\\
\Omega_{\mu\nu}={}&\nabla_{(0)}^\lambda\nabla^{(0)}_\lambda B_{\mu\nu}-2W^{(0)}_{\rho\nu\mu\lambda}B^{\lambda\rho}-4B_{\mu\nu}P^\mu{}_\mu+2(d-4)\big(2P^{\rho\lambda}\nabla^{(0)}_\lambda C_{(\mu\nu)\rho}\\
&+\nabla^{(0)}_\lambda PC_{(\mu\nu)}{}^\lambda-C^{\rho}{}_{\mu}{}^{\lambda}C_{\lambda\nu\rho}+ \nabla_{(0)}^\lambda P^\rho{}_{(\mu}C_{\nu)\rho\lambda}-W^{(0)}_{\rho\mu\nu\lambda}P^{\lambda}{}_\sigma P^{\sigma\rho}\big)+P_{\mu\rho}P^{\rho\sigma}P_{\sigma\nu},
\end{align}
where R_(μν)^((0)) is the Ricci tensor for the boundary metric γ_(μν)^((0)), ∇_(μ)^((0)) is the covariant derivative associated with γ_(ij)^((0)) on the boundary, W_(ρνμσ)^((0)) is the Weyl tensor on the boundary, and C_(μνρ)=∇_(ρ)^((0))P_(μν)−∇_(ν)^((0))P_(μρ).

The final expression will have the form
\begin{align}
{\cal A}_4=&-\frac{L^7}{8\pi G}\int d^8x\sqrt{-\det\gamma^{(0)}}X^{(4)}\ln {\cal B},
\end{align}
where X^((4)) may contain the following terms: tr(P⁴), tr(P³), tr(P³)tr(P), tr(BP), tr(BP²), tr(B²), tr(B²P), tr(OP), tr(OP²), tr(Ω), tr(ΩP).

# Main problem:
Determine the coefficients of these terms in X^((4)).

# Problem setup:
Consider a quantum field theory with holographic dual. Under a Weyl transformation, the boundary metric transforms as $\gamma_{\mu\nu}^{(0)}\to{\cal B}^{-2}(x)\gamma_{\mu\nu}^{(0)}$. The Weyl anomaly ${\cal A}_k$ of the theory in $2k$ dimensions appears in the transformation of the partition function:
\begin{align}
Z[\gamma_{\mu\nu}^{(0)}]\to e^{-{\cal A}_k}Z[{\cal B}(x)^{-2}\gamma^{(0)}].
\end{align}
This can be computed by evaluating the on-shell action of the bulk gravitational theory.

The holographic Weyl anomaly in $d\leqslant8$ can be express using the following quantities:
\begin{align}
P_{\mu\nu}={}&R^{(0)}_{\mu\nu}-\frac{R^{(0)}}{2(d-1)}\gamma_{\mu\nu}^{(0)}\,,\\
B_{\mu\nu}={}&\frac{1}{d-2}\big(\nabla^{(0)}_\rho\nabla_{(0)}^\rho  P_{\mu\nu}-\nabla^{(0)}_\rho\nabla^{(0)}_{\nu} P_{\mu}{}^{\rho}- W^{(0)}_{\rho\nu\mu\sigma} P^{\sigma\rho}\big)\,,\\
O_{\mu\nu}={}&\nabla_{(0)}^\lambda\nabla^{(0)}_\lambda B_{\mu\nu}-2W^{(0)}_{\rho\nu\mu\lambda}B^{\lambda\rho}-\frac{4}{d-2}B_{\mu\nu}P^\mu{}_\mu+\frac{2(d-4)}{(d-2)^2}\big(2P^{\rho\lambda}\nabla^{(0)}_\lambda C_{(\mu\nu)\rho}\\
&+\nabla^{(0)}_\lambda PC_{(\mu\nu)}{}^\lambda-C^{\rho}{}_{\mu}{}^{\lambda}C_{\lambda\nu\rho}+ \nabla_{(0)}^\lambda P^\rho{}_{(\mu}C_{\nu)\rho\lambda}-W^{(0)}_{\rho\mu\nu\lambda}P^{\lambda}{}_\sigma P^{\sigma\rho}\big)\,,\\
\Omega_{\mu\nu}={}&\nabla_{(0)}^\lambda\nabla^{(0)}_\lambda B_{\mu\nu}-2W^{(0)}_{\rho\nu\mu\lambda}B^{\lambda\rho}-4B_{\mu\nu}P^\mu{}_\mu+2(d-4)\big(2P^{\rho\lambda}\nabla^{(0)}_\lambda C_{(\mu\nu)\rho}\\
&+\nabla^{(0)}_\lambda PC_{(\mu\nu)}{}^\lambda-C^{\rho}{}_{\mu}{}^{\lambda}C_{\lambda\nu\rho}+ \nabla_{(0)}^\lambda P^\rho{}_{(\mu}C_{\nu)\rho\lambda}-W^{(0)}_{\rho\mu\nu\lambda}P^{\lambda}{}_\sigma P^{\sigma\rho}\big)+P_{\mu\rho}P^{\rho\sigma}P_{\sigma\nu},
\end{align}
where $R^{(0)}_{\mu\nu}$ is the Ricci tensor for the boundary metric $\gamma_{\mu\nu}^{(0)}$, $\nabla^{(0)}_\mu$ is the covariant derivative associated with $\gamma^{(0)}_{ij}$ on the boundary, $W^{(0)}_{\rho\nu\mu\sigma}$ is the Weyl tensor on the boundary, and $C_{\mu\nu\rho}=\nabla^{(0)}_\rho P_{\mu\nu}-\nabla^{(0)}_\nu P_{\mu\rho}$.

The final expression will have the form
\begin{align}
{\cal A}_4=&-\frac{L^7}{8\pi G}\int d^8x\sqrt{-\det\gamma^{(0)}}X^{(4)}\ln {\cal B},
\end{align}
where $X^{(4)}$ may contain the following terms: $\text{tr}(P^4)$, $\text{tr}(P^3)$, $\text{tr}(P^3)\text{tr}(P)$, $\text{tr}(BP)$, $\text{tr}(BP^2)$, $\text{tr}(B^2)$, $\text{tr}(B^2P)$, $\text{tr}(OP)$, $\text{tr}(OP^2)$, $\text{tr}(\Omega)$, $\text{tr}(\Omega P)$.

# Main problem:
Determine the coefficients of these terms in $X^{(4)}$.