# Modern form of Brown-Henneaux formula

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?