# Holographic renormalisation (and holographic anomaly) In [[0001 AdS-CFT|holography]], one is usually interested in computing quantities of the CFT from the bulk, such as the expectation value of the stress tensor as a simple example. In doing so, one often encounters divergences due to the infinite volume of AdS. They need to be appropriately subtracted off -- with counter terms. This is referred to as **holographic renormalisation**. An interesting feature of CFTs in even dimensions is the existence of the conformal anomaly. The conformal boundary of AdS is also defined up to a conformal factor. Happily, the corresponding bulk computation is sensitive to the choice of the conformal frame when the bulk dimension is odd due to the presence of a log term that is allowed in the asymptotic expansion of the dynamic field of interest (e.g. the [[0011 Fefferman-Graham expansion|FG expansion]] of the metric), and this reproduces what we expect for the CFT, i.e., dependence on the conformal frame. This computes the anomaly in the bulk, and it is known as the **holographic anomaly**. It is closely related to holographic renormalisation as the latter is the setup needed to realise the former. ## Refs - anomaly - [[1998#Henningson, Skenderis (Jun)]] and [[HenningsonSkenderis199812]] - explicit treatment in [[0002 3D gravity|3d gravity]] - [[2000#Krasnov]]: relates action to 2d [[0562 Liouville theory|Liouville CFT]] - complete treatment - [[2020#de Haro, Skenderis, Solodukhin]] - reviews - lectures [[Skenderis2002]] - [[Rsc0020 Chapter 19 Conserved charges in AdS by FischettiKellyMarolf]] - generalisation - [[2024#Arenas-Henriquez, Diaz, Rivera-Betancour]]: Weyl-covariant approach (in 3D) ## Example expressions - from [[Rsc0019 Dong String 2 lecture on AdS-CFT]] - AdS${}_3$: $S_{C T}=-\frac{1}{8 \pi G} \int_{\partial A d S} d^{2} x \sqrt{\tilde \gamma}$ - AdS${}_4$: $S_{C T}=-\frac{1}{4 \pi G} \int_{\partial A d S} d^{3} x \sqrt{\tilde \gamma}\left(1-\frac{R\left[\tilde\gamma\right]}{4}\right)$ - AdS${}_5$: $S_{C T}=-\frac{3}{4 \pi G} \int_{\partial A d S} d^{4} x \sqrt{\tilde \gamma}\left(1-\frac{R\left[\tilde\gamma\right]}{12}\right)$ - sign seems wrong for the second term - $d=n+1=6$ counter-term and below - [[EmparanJohnsonMyers1999]][](https://arxiv.org/pdf/hep-th/9903238.pdf): - $I_{\mathrm{ct}}=\frac{1}{8 \pi G} \int_{\partial \mathcal{M}} d^{n} x \sqrt{h}$\left[\frac{n-1}{l}+\frac{l}{2(n-2)} \mathcal{R}+\frac{l^{3}}{2(n-4)(n-2)^{2}}\left(\mathcal{R}_{a b} \mathcal{R}^{a b}-\frac{n}{4(n-1)} \mathcal{R}^{2}\right)+\ldots\right]$ - (conventions seems modern) - [[2011#Blackman. McDermott, van Raamsdonk]] ## Properties - **order of each term**: each term in the CT expression is designed to cancel divergences at a particular order. For example, $\mathcal{R}$ goes like $1/r^2$ so it is obvious which order each term corresponds to just from the expression - **anomaly term is magical**: If one change the (bulk) conformal factor in a way that changes the boundary metric, then the log term ensures that the expression changes, correspond to the effect of conformal anomaly in the CFT; however, if two coordinates are related by $z = w + O(w^2)$ then the log term $\log z=\log (w(1+O(w)))=\log w + O(w)$ does not change, which matches with the fact that the boundary metric is unchanged ## Other approaches - Hamiltonian approach - [[2004#Papadimitriou, Skenderis]] - Hamilton-Jacobi approach - [[DeBoerVerlindeVerlinde1999]][](https://arxiv.org/pdf/hep-th/9912012.pdf) - [[McNees2005]][](https://arxiv.org/abs/hep-th/0512297) - new counter term - restores an issue that odd dimensions partially break diffeomorphism invariance - [[2024#Climent, Emparan, Magan, Sasieta, Lopez]] - a purely extrinsic approach in the sense that all the counter terms depend only on the field and its radial derivative ## Flat space analogues - gravity - [[2005#Mann, Marolf]] - [[2024#Hao]] - QED - [[2019#Freidel, Hopfmuller, Riello]] - $D\ge6$ - renormalised charges give QED soft theorem - near null infinity ## de Sitter analogue - [[2023#Bzowski, McFadden, Skenderis]] ## In higher derivative gravity or gauge theory - [[1999#Nojiri, Odintsov]] - quadratic curvature, $d=2,4$, only anomaly - [[BlauNarainGava1999]] - quadratic curvature - only did anomaly, not full CT - only did $d=4$ - Hamilton-Jacobi method - [[2001#Fukuma, Matsuura]] - [[2001#Fukuma, Matsuura, Sakai]] main - [[2002#Fukuma, Matsuura, Sakai]] - [[RajagopalStergiouZhu2015]][](https://arxiv.org/pdf/1508.01210.pdf) - $d=4,6$ - quadratic plus scalar - higher dim. [[0089 Chern-Simons theory]] - [[BanadosSchwimmerTheisen2004]][](https://arxiv.org/abs/hep-th/0404245) - [[2005#Banados, Olea, Theisen]] - higher dim. Lovelock CS - [[CvetkovicMiskovicSimic2017]][](https://arxiv.org/abs/1705.04522)