# Help with the understanding of boundary conditions on $AdS_3$

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...

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– Qmechanic Jan 28 '13 at 15:51
Related question, but I cannot seem to check if it's the correct way of solving this :\ – dingo_d Jan 28 '13 at 17:11

I have a question about expansion. In the thesis, Porfyriadis explains how one needs to do this, but I'm still having issues with the series expansion. I get that the expansion is used so that the $\mathcal{O}(r^n)$ parts at the ends of ODE's are canceled, which should give me equations with coefficients. But how do I expand this? I'm stuck at this step. – dingo_d Aug 24 '13 at 10:12
Precisely, how should I solve this: $\frac{2}{l^2}\sum_n\xi^r_nr^{n+1}+2\sum_n\xi^t_{n,t} r^n+\frac{2}{l^2}\sum_n\xi^t_{n,t}r^{n+2}=\mathcal{O}(r)$ Do I start going backwards from n=0,-1,-2,-3,... etc? – dingo_d Aug 24 '13 at 10:24