Asymptotic charge conservation states that Eq. (4) is independent of the choice of $\pm$.

One may also consider timelike sources passing through $i^0$, contributing additionally to $Q^{\pm }$ through an $H_3$ bulk-to-boundary propagator.

The mean of $q$ is required to vanish, corresponding to a vanishing global charge.

Rényi entropies decompose similarly.

${}^R\rho [0]$ is prepared with two copies of the Euclidean path integral in Fig. 4, extending the range of $i\tau$ to $2\pi$.

In Eq. (27), we have switched to a different $\epsilon$ regulator from that inherited from $A^{\mathrm{CS}}$ in Eq. (12). Following [39, 40], ${}^R{A}^{\mathrm{E}}$ is defined to be an exactly Milne-static solution with electric field lines penetrating $\partial {}^R\mathrm{\Sigma }$ as shown in Fig. 3, except now $\partial {}^R\mathrm{\Sigma }$ is placed at a cutoff ${\epsilon }^2$ in the Fefferman-Graham coordinate $\rho =2u/r$ along the Euclidean AdS ${}^R\mathrm{\Sigma }$. When $\epsilon \to 0$, the electric field lines bunch up to $\rho ={\epsilon }^2\to 0$, similar to Fig. 3. This static ${}^R{A}^{\mathrm{E}}$ agrees with $A^{\log }$ (with the old $\epsilon$ removed) in $R$ up to rescaling and a gauge transformation [36].