Help with the understanding of boundary conditions on $AdS_3$

Question

So I am trying to reproduce results in this article, precisely the 3rd chapter 'Virasoro algebra for AdS$_3$'. I have the metric in this form:

$$ds^2=-\left(1+\frac{r^2}{l^2}\right)dt^2+\left(1+\frac{r^2}{l^2}\right)^{-1}dr^2+r^2d\phi^2$$

And I have the boundary conditions. So if I'm correct, I should find the most general diffeomorphism, by solving $\mathcal{L}_\xi g_{\mu\nu}=\mathcal{O}(h_{\mu\nu})$, where $h_{\mu\nu}$ are the boundary conditions (subleading terms).

So, if I'm doing things right, I get 5 equations. Because the $t\phi$ term of Lie derivative vanishes. Now, I should use the power expansion of $\xi$, as given in the paper, and solve these 5 differential equations or?

I'm not certain if I'm on a right path, so any advice is welcome...

Answer 1

Initially, you have 6 equations given by (16) and (17). Now you insert both the expressions for the components of the metric and the power series expansion and determine its coefficients in such a way that the equations are satisfied order by order.