# Compere and Fiorucci Chapter 2: 3d gravity \[Links: [arXiv](https://arxiv.org/abs/1801.07064), [PDF](https://arxiv.org/pdf/1801.07064.pdf)\] ## 2.1 Overview of properties - solution space - in vacuum -> constant curvature - non-trivial topology + boundary -> still contain black holes and infinite dimensional asymptotic symmetry - no bulk degrees of freedom - $R_{\mu \nu \alpha \beta}=W_{\mu \nu \alpha \beta}+\frac{2}{n-2}\left(g_{\alpha[\mu} R_{\nu] \beta}+R_{\alpha[\mu} g_{\nu] \beta}\right)-\frac{2}{(n-1)(n-2)} R g_{\alpha[\mu} g_{\nu] \beta}$ ($n=3$ in 3d gravity) - outside sources, $R_{\mu\nu}=0$ -> Weyl tensor contains the gravitational effects - -> in 3d, $R_{\mu\nu}$ has $\frac{1}{2}\times 3\times 4 =6$ which equals the number of independent degrees of freedom in $R_{abcd}$ - => no gravitational degrees of freedom - homogeneity - vacuum equation: $G_{\mu \nu}+\Lambda g_{\mu \nu}=R_{\mu \nu}+\left(\Lambda-\frac{R}{2}\right) g_{\mu \nu}=0$ - => $R=6\Lambda$ - => (using the Riemann tensor decomposition above but now without the Weyl tensor) - => $R_{\alpha \beta \mu \nu} \approx \Lambda\left(g_{\alpha \mu}g_{\beta \nu}-g_{\alpha \nu} g_{\beta \mu}\right)$ ## 2.2 AAdS - global AdS - asymptotically AdS - [[0086 Banados-Teitelboim-Zanelli black hole]] - [[0099 Quotient method in AdS3]] - phase space - ![[Rsc0004_phase.png|200]] - $M<0$ and $J=0$: conical defects (particles) - $M<-1/8G$: conical excess and should be discarded (unbounded from below) - $M<0$ and $J\ne0$: spinning particles with mass $M$ and angular momentum $J$ - boundary conditions - Brown-Henneaux BC - fix boundary metric in [[0011 Fefferman-Graham expansion]] - [[0085 Asymptotic symmetry of AdS3]] - phase space (from FG point of view) - charges - use knowledge from [[Rsc0005 CompereFiorucci Ch1 Surface charges]] - summary in [[0085 Asymptotic symmetry of AdS3]] ## Related topics - [[0073 AdS3-CFT2]]