# 0085 Asymptotic symmetry of AdS3 In this note, let us review the derivation of the [[0060 Asymptotic symmetry|asymptotic symmetry group]] of AdS$_3$. ## Refs - [[Rsc0004 CompereFiorucci Ch2 3d gravity]]: detailed derivations - most general BC - [[2016#Grumiller, Riegler]] ## Obtain the symmetry generators - Brown-Henneaux BC: fix $g^{(0)}_{\mu\nu}=\eta_{\mu\nu}$ in [[0011 Fefferman-Graham expansion|FG]] => residual gauge group gets restricted to [[0060 Asymptotic symmetry|asymptotic symmetry]] - FG gauge preserving: $\mathcal{L}_{\zeta} g_{\rho a} = \mathcal{L}_{\zeta} g_{\rho\rho}=0$ - => $\partial_{\rho} \xi^{b}=-g^{a b} \partial_{a} R \Rightarrow \xi^{b}\left(\rho, x^{a}\right)=V^{b}\left(x^{a}\right)-\int d \rho g^{a b} \partial_{a} R$ - BC preserving: $\mathcal{L}_{\zeta} g_{a b}=\mathcal{O}(e^\rho)$ - => $\mathcal{D}_{a} V_{b}+\mathcal{D}_{b} V_{a}=\mathcal{D}_{c} V^{c} \eta_{a b}$ i.e. conformal Killing equation - solve => - $\xi^{(+)}=V^{+}\left(x^{+}\right) \partial_{+}-\frac{1}{2} \partial_{+} V^{+} \partial_{\rho}+\int d \rho g^{+-} \partial_{+} \partial_{+} V^{+} \partial_{-}$ - $\xi^{(-)}=V^{-}\left(x^{-}\right) \partial_{-}-\frac{1}{2} \partial_{-} V^{-} \partial_{\rho}+\int d \rho g^{+-} \partial_{+} \partial_{-} V^{-} \partial_{-}$ - first two terms leading, third subleading ## Symmetry algebra - $i\left[\xi_{m}^{(+)}, \xi_{n}^{(+)}\right]=(m-n) \xi_{m+n}^{(+)}$ - i.e. ==Witt algebra for each chiral subalgebra== (Witt = centreless algebra of circle diffeomorphisms) - N.b. subscripts $m$ label Fourier components - global subalgebra - see [[0121 Global symmetries of spacetime|global spacetime symmetries]] - $m=-1,0,1$ (can check it's closed: $1+0=1, 1+(-1)=0$, etc.) - isomorphic to $sl(2,\mathbb{R}) \simeq so(2,1)$ - two of these make ==$sl(2,\mathbb{R})\oplus sl(2,\mathbb{R}) \simeq so(2,2)$== i.e. symmetry *algebra* (not group) of global AdS$_3$ ## Charge algebra - $i\left\{\mathcal{L}_{m}^{(+)}, \mathcal{L}_{n}^{(+)}\right\} =(m-n) \mathcal{L}_{m+n}^{(+)}+m^{3} \delta_{m+n, 0} \frac{c}{12}$ with $c=\frac{3\ell}{2G}$ - do a shift of the zero mode $\tilde{\mathcal{L}}_{m}^{(+)}=\mathcal{L}_{m}^{(+)}+\delta_{m, 0} N, N \in \mathbb{R}$ - choose $N=c/24$ <=> shift of the mass by $c/12=1/8G$, i.e. mass of global AdS - shifted charge algebra: $i\left\{\tilde{\mathcal{L}}_{m}^{(+)}, \tilde{\mathcal{L}}_{n}^{(+)}\right\}=(m-n) \tilde{\mathcal{L}}_{m+n}^{(+)}+\frac{c}{12}\left(m^{2}-1\right) m \delta_{m+n, 0}$