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Almost every paper mentioning Brown and Henneaux's matching of asymptotic symmetries of AdS$_3$ with the Virasoro algebra of a $1{+}1$-dimensional CFT summarizes their results in the formula $$c=\frac{3R}{2G},$$ whereby the central charge $c$ is expressed in terms of the AdS radius $R$ and the gravitational constant $G$.

However, Brown and Henneaux's original paper doesn't contain this formula. Instead, it derives an asymptotic algebra of the form \begin{equation} \{J[L_n],J[L_m]\}^\star = (n-m) J[L_{n+m}] + 2\pi i \sigma R n (n^2 - 1) \delta_{n,-m} \end{equation} for charges $J[L_n]$ (on page 222), which looks very close to the standard Virasoro algebra \begin{equation} [L_n,L_m] = (n-m) L_{n+m} + \frac{c}{12} n (n^2 - 1) \delta_{n,-m} \ . \end{equation} But I don't see how this form would relate the central charge in any way to $G$ nor figure out where the factor $3/2$ comes from. Can anyone help me figure out the missing steps?

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