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Refering to the paper "Replica Wormholes and the Entropy of Hawking Radiation" by Almheiri et al. in arXiv:1991.12333. The authors consider Jackiw-Teitelboim gravity theory describing a nearly AdS2 spacetime coupled to CFT matter, where the dilaton background in Eq. (3.5) of the paper (before considering conical singularities in the problem) $$\phi=-\frac{2\pi\phi_r}{\beta}\frac{1}{\tanh \frac{\pi(y+\bar{y})}{\beta}}\quad \text{for}\quad ds^2=\frac{4\pi^2}{\beta^2}\frac{dy\:d\bar{y}}{\sinh^2\frac{\pi(y+\bar{y})}{\beta}},$$ ($\phi_r$ as a constant) is solution to the equations of motion $$(\phi+\phi_b)(R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R)=8\pi G_N \langle T_{\mu\nu}\rangle+(\nabla_\mu\nabla_\nu-g_{\mu\nu}\Box)\phi+g_{\mu\nu}\phi$$ which is true for $\langle T_{\mu\nu}\rangle=0$ ($\phi_b$ as a constant). However, the set-up also includes CFT matter, i.e. $$I_{\text{total}}=I_{grav}\,[g_{\mu\nu},\:\phi]+\int d^2y\,\sqrt{-g}\;\mathcal{L}_{CFT}.$$ Later, the authors consider contributions from conical singularities to the energy momentum tensor, but it seems that the CFT action doesn't contribute to $\langle T_{\mu\nu}\rangle$ at any time. Could someone please explain why doesn't the CFT matter contribute to the solution in Eq. (3.5) and thereafter? Is it perphaps by the central charge $c\gg1$ limit that they consider?

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