# Chapter 19: Conserved charges in AdS Links: [arXiv](https://arxiv.org/abs/1211.6347) ## Summary - talks about [[0209 Holographic renormalisation]] and [[0060 Asymptotic symmetry]] - introduces [[0001 AdS-CFT]] without string theory - c.f. [[Rsc0010 Hong Liu Lectures on holography]] ## Conventions - spacetime dimension = $d+1$ - $K_{i j}=-\left(\mathscr{L}_{n} h_{i j}\right) / 2$ a minus sign from standard <!-- ## Typos - 19.61 has opposite signs between $a_{(0)}$ and $a_{(2)}$ but (19.65) has same signs (which is correct) --> ## 19.1 Introduction - why AdS 1. natural and simple - maximally symmetric 2. richer structure - all multipole moments of a given field in AdS decay at the same rate at infinity - -> not just global charge important: local charge density of tells the story of all multipoles 3. AdS/CFT - historic - conserved charges for AdS were first found in [[AbbottDeser1982]] and the energy was argued to be positive ## 19.2 AlAdS ### 19.2.4 [[0011 Fefferman-Graham expansion|FG expansion]] ### 19.2.5 Diffeomorphisms and symmetries ### 19.2.6 Gravity with matter ## 19.3 Variational principle and charges ### 19.3.3 Variational principle for AlAdS gravity - renormalised action - $S_\text{ren}=\lim_{\epsilon\rightarrow0}(S_\text{reg}+S_\text{ct})$ - renormalised stress tensor - $T_{\text {bndy }}^{i j}=\lim _{\epsilon \rightarrow 0}\left(\frac{\ell}{\epsilon}\right)^{d+2}\left(\tau^{i j}+\tau_{c t}^{i j}\right)$ - turns out that this is related to the $d$-th order metric in the FG expansion - odd $d$: $T_{\text {bndy }}^{i j}=\frac{d \ell^{d-1}}{2 \kappa} \gamma^{(d)i j}$ - even $d$: more complicated - i.e. the only free data in FG are $\gamma^{(0)}$ and $T_\text{bndy}$ ### 19.3.4 Conserved charges for AlAdS gravity ## 19.5 Algebra of boundary observables and AdS/CFT - summary so far - charges $Q[\xi]$ are conserved and generate [[0060 Asymptotic symmetry]] $\xi$ under [[0150 Peierls bracket]] - charges are equivalent to Hamiltonian charges - why boundary stress tensor 1. they contain useful information such as in [[0228 Fluid-gravity correspondence]] 2. AdS/CFT (see below) - $\mathcal{A}_\text{bndy}$, the algebra of boundary observables - just the algebra generated by $T_\text{bndy}^{ij}$ and $\Phi_\text{bndy}$ (matter field) under [[0150 Peierls bracket]] - when the boundary observables admit a conformal Killing field $\hat{\xi}$, the corresponding transformation is generated by the associated $Q[\xi]$ - -> if $\hat{\xi}$ is chosen to be everywhere timelike, we would have $\mathcal{A}_\text{bndy}$ **closed under time evolution**!