# AdS/CFT The AdS/CFT correspondence is a conjectured duality between a quantum gravity theory in $d+1$ dimensions in an asymptotically anti-de Sitter (AdS) spacetime, aka the bulk, and a [[0481 Conformal field theory|conformal field theory]] (CFT) in $d$ dimensions, commonly considered as the boundary. It can be summarised as$\mathcal{Z}_{AdS}=\mathcal{Z}_{CFT},\qquad \mathcal{H}_{AdS}=\mathcal{H}_{CFT},$i.e., the matching of the partition functions and of the Hilbert spaces. The matching of the partition is in some sense better understood and forms the basis of two holographic dictionaries (see below), while the matching between Hilbert spaces is less so. Since it is too big a topic to talk about here, I will use this page as a catalogue page for listing some related subjects. ## Reviews and lectures - [[Rsc0010 Hong Liu Lectures on holography]] - Daniel Harlow's Jerusalem lecture notes and [[2023#Harlow (Review)]] - Tom Hartman's lecture notes - [[Rsc0020 Chapter 19 Conserved charges in AdS by FischettiKellyMarolf]] - [[1999#Aharony, Gubser, Maldacena, Ooguri, Oz]] - [nLab](https://ncatlab.org/nlab/show/AdS-CFT) - Rangamani's TASI lectures in 2021 - original papers - [[1997#Maldacena]] - [[1998#Witten (Feb)]] ## Two dictionaries - GKPW [[1998#Gubser, Klebanov, Polyakov]] and [[1998#Witten (Feb)]] - BDHM [[1998#Banks, Douglas, Horowitz, Martinec]] - equivalence [[1999#Giddings (a)]] and [[2011#Harlow, Stanford]] ## Alternative proposals - [[2024#Aoki, Balog, Shimada]]: using just conformal symmetry ## Limits - large t'Hooft coupling - $\frac{l_\text{AdS}^{d-1}}{G_N} \sim \left(\frac{l_\text{AdS}}{l_p}\right)^{d-1} \sim N^2$ ## Specific dictionary entries - dimension of CFT operators $\leftrightarrow$ mass of particles in supergravity - [[0301 Entanglement entropy|entanglement entropy]] $\leftrightarrow$ [[0007 RT surface|RT surface]] - CFT states $\leftrightarrow$ [[0231 Bulk solutions for CFTs on non-trivial geometries|bulk solutions]] - [[0060 Asymptotic symmetry|asymptotic symmetries]] in AdS $\leftrightarrow$ global symmetries of CFT - much more! ## Explicit examples ![[0123_MukundPairs.png]] (picture from Mukund's TASI lecture) ![[A0002 Salt and pepper shaker.png]]