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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{AdS-CFT} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{duality_in_string_theory}{}\paragraph*{{Duality in string theory}}\label{duality_in_string_theory} [[!include duality in string theory -- contents]] \hypertarget{quantum_field_theory}{}\paragraph*{{Quantum field theory}}\label{quantum_field_theory} [[!include functorial quantum field theory - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{Idea}{Idea}\dotfill \pageref*{Idea} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{AdS5CFT4}{$AdS_5 / CFT_4$ -- Horizon limit of D3-branes}\dotfill \pageref*{AdS5CFT4} \linebreak \noindent\hyperlink{AdS7CFT6}{$AdS_7 / CFT_6$ -- Horizon limit of M5-branes}\dotfill \pageref*{AdS7CFT6} \linebreak \noindent\hyperlink{AdS4CFT3}{$AdS_4 / CFT_3$ --Horizon limit of M2-branes}\dotfill \pageref*{AdS4CFT3} \linebreak \noindent\hyperlink{AdS3CFT2}{$AdS_3 / CFT_2$ -- Horizon limit of D1-D5 brane bound states}\dotfill \pageref*{AdS3CFT2} \linebreak \noindent\hyperlink{nonconformal_duals}{Non-conformal duals}\dotfill \pageref*{nonconformal_duals} \linebreak \noindent\hyperlink{horizon_limit_of_branes_for_arbitrary_}{Horizon limit of $Dp$-branes for arbitrary $p$}\dotfill \pageref*{horizon_limit_of_branes_for_arbitrary_} \linebreak \noindent\hyperlink{horizon_limit_of_ns5brane}{Horizon limit of NS5-brane}\dotfill \pageref*{horizon_limit_of_ns5brane} \linebreak \noindent\hyperlink{qcd_models}{QCD models}\dotfill \pageref*{qcd_models} \linebreak \noindent\hyperlink{further_gauge_theories_induced_by_compactification_and_twisting}{Further gauge theories induced by compactification and twisting}\dotfill \pageref*{further_gauge_theories_induced_by_compactification_and_twisting} \linebreak \noindent\hyperlink{Checks}{Checks}\dotfill \pageref*{Checks} \linebreak \noindent\hyperlink{formalizations}{Formalizations}\dotfill \pageref*{formalizations} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \noindent\hyperlink{original_articles}{Original articles}\dotfill \pageref*{original_articles} \linebreak \noindent\hyperlink{introductions_and_surveys}{Introductions and surveys}\dotfill \pageref*{introductions_and_surveys} \linebreak \noindent\hyperlink{ReferencesAdS2CFT1}{On $AdS_2 / CFT_1$}\dotfill \pageref*{ReferencesAdS2CFT1} \linebreak \noindent\hyperlink{ReferencesAdS3CFT2}{On $AdS_3 / CFT_2$}\dotfill \pageref*{ReferencesAdS3CFT2} \linebreak \noindent\hyperlink{on__3}{On $AdS_4 / CFT_3$}\dotfill \pageref*{on__3} \linebreak \noindent\hyperlink{on__4}{On $AdS_5 / CFT_4$}\dotfill \pageref*{on__4} \linebreak \noindent\hyperlink{on__5}{On $AdS_7 / CFT_6$}\dotfill \pageref*{on__5} \linebreak \noindent\hyperlink{generalization_beyond_exact_ads__exact_cft}{Generalization beyond exact AdS / exact CFT}\dotfill \pageref*{generalization_beyond_exact_ads__exact_cft} \linebreak \noindent\hyperlink{Appications}{Applications to physics}\dotfill \pageref*{Appications} \linebreak \noindent\hyperlink{to_gravity}{To gravity}\dotfill \pageref*{to_gravity} \linebreak \noindent\hyperlink{to_the_quarkgluon_plasma}{To the quark-gluon plasma}\dotfill \pageref*{to_the_quarkgluon_plasma} \linebreak \noindent\hyperlink{to_particle_physics}{To particle physics}\dotfill \pageref*{to_particle_physics} \linebreak \noindent\hyperlink{to_fluid_dynamics}{To fluid dynamics}\dotfill \pageref*{to_fluid_dynamics} \linebreak \noindent\hyperlink{ToCondensedMatterPhysics}{To condensed matter physics}\dotfill \pageref*{ToCondensedMatterPhysics} \linebreak \noindent\hyperlink{applications_in_mathematics}{Applications in mathematics}\dotfill \pageref*{applications_in_mathematics} \linebreak \noindent\hyperlink{to_the_volume_conjecture}{To the volume conjecture}\dotfill \pageref*{to_the_volume_conjecture} \linebreak \noindent\hyperlink{to_deep_learning_in_neural_networks}{To deep learning in neural networks}\dotfill \pageref*{to_deep_learning_in_neural_networks} \linebreak \hypertarget{Idea}{}\subsection*{{Idea}}\label{Idea} The \emph{AdS-CFT correspondence} is at its basis the observation (\hyperlink{Witten98}{Witten 98, Section 2.4}) that the classical [[action functionals]] for various [[field (physics)|fields]] coupled to [[Einstein gravity]] on [[anti de Sitter spacetime]] are, when expressed as [[functions]] of the [[asymptotic boundary]]-values of the [[field (physics)|fields]], equal to the [[generating functions]] for the [[correlators]]/[[n-point functions]] of a [[conformal field theory]] on that asymptotic boundary. In extrapolation of these elementary computations, the AdS/CFT correspondence [[conjecture|conjecturally]] extends to a more general identification of [[states]] of [[gravity]] ([[quantum gravity]]) on asymptotically [[anti de Sitter spacetimes]] of [[dimension]] $d+1$ with [[correlators]]/[[n-point functions]] of [[conformal field theories]] on the [[asymptotic boundary]] of dimension $d$ (\hyperlink{GubserKlebanovPolyakov98}{Gubser-Klebanov-Polyakov 98 (12)}, \hyperlink{Witten98}{Witten 98, (2.11)}), such that [[perturbative quantum field theory|perturbation theory]] on one side of the correspondence relates to [[non-perturbative effects|non-perturbation]] on the other side. While this works to some extent quite generally (see e.g. \hyperlink{Natsuume15}{Natsuume 15} for review), the tightest form of the correspondence relates the [[large N limit]] of [[superconformal field theories]] ([[super Yang-Mills theories]]) on the [[asymptotic boundaries]] of [[near-horizon limits]] of $N$ coincident [[black brane|black]] [[M2-brane]]/[[D3-brane]]/[[M5-brane]] to corresponding sectors of the [[string theory]]/[[M-theory]] [[quantum gravity]] in the [[bulk]] [[spacetime]] away from the brane. A quick way to see that these [[supersymmetric]]-cases of AdS/CFT must be special is to observe that these are the only dimensions in which there are [[super anti de Sitter spacetime]]-enhancement of [[anti de Sitter spacetime]], matching the classification of simple [[superconformal geometry|superconformal symmetries]], see \href{supersymmetry#ClassificationSuperconformal}{there}. [[!include superconformal symmetry -- table]] Before the proposal for the actual matching rule of ADS/CFT (\hyperlink{GubserKlebanovPolyakov98}{Gubser-Klebanov-Polyakov 98 (12)}, \hyperlink{Witten98}{Witten 98, (2.11)}) it was by matching of [[BPS-states]] in these situations that a AdS/CFT correspondence was proposed in (\hyperlink{Maldacena97a}{Maldacena 97a}, \hyperlink{Maldacena97b}{Maldacena 97b}), and these articles are now widely regarded as the origin of the idea of the AdS/CFT correspondence, even though there have of been precursors: Generally, AdS/CFT correspondence is an incarnation of [[open/closed string duality]], which is based on the simple observation that the [[string scattering amplitude]] of a [[cylinder]]-shaped [[worldsheet]] has two equivalent interpretations: On the one hand, reading [[worldsheet]]-time along the [[circle]]-direction of the cylinder, it is a [[loop order|1-loop]] [[open string]]-[[vacuum amplitude]] for an [[open string]] attached to some [[D-brane]], while on the other hand, with [[worldsheet]]-time real in the orthogonal direction, it is a [[closed string]] [[loop order|tree-level]] amplitude describing a [[closed string]] emanating from/absorbed by that [[D-brane]]. Since the excitations of the [[open string]] are the quanta of the (``[[Chan-Paton gauge field|Chan-Paton]]'')-[[gauge fields]] on the [[D-brane]], while the excitations of the [[closed string]] include the [[gravitons]] witnessing the [[gravity|gravitational]] backreaction of the D-branes, this already indicates that there must be some close relation between the [[worldvolume]] [[super Yang-Mills theory]] on the [[D-brane]] and the [[quantum gravity]] of the ambient [[bulk]] [[spacetime]]. More specifically, it was shown in (\hyperlink{DuffSutton88}{Duff-Sutton 88}, see \hyperlink{Duff98}{Duff 98}, \hyperlink{Duff99}{Duff 99}) that the field theory of small perturbation of a [[Green-Schwarz sigma-model]] for a [[fundamental brane]] stretched over the [[asymptotic boundary]] of the AdS [[near horizon geometry]] of its own [[black brane]] incarnation is, after [[diffeomorphism]] [[gauge fixing]], a [[conformal field theory]]. This was further developed in \hyperlink{ClausKalloshProeyen97}{Claus-Kallosh-Proeyen 97}, \hyperlink{DGGGTT98}{DGGGTT 98}, \hyperlink{ClausKalloshKumarTownsend98}{Claus-Kallosh-Kumar-Townsend 98}, \hyperlink{PastiSorokinTonin99}{Pasti-Sorokin-Tonin 99}. See also at \emph{\href{Green-Schwarz+action+functional#AsPartOfTheAdSCFTCorrespodence}{super p-brane -- As part of the AdS-CFT correspondence}} For the archetypical case of AdS/CFT relating [[N=4 D=4 super Yang-Mills theory]] to [[type IIB string theory]] on [[super anti de Sitter spacetime]] $AdS_5 \times S^5$, fine detailed checks of the correspondence have been performed (\hyperlink{BeisertEtAl10}{Beisert et al. 10}, \hyperlink{Escobedo12}{Escobedo 12}), see the section \emph{\hyperlink{Checks}{Checks}} below. If one relaxes the various assumptions that go into this exact form of the correspondence ([[conformal invariance]], [[supersymmetry]], [[large N limit]], [[anti de Sitter spacetime|anti de Sitter geometry]]) there is still a correspondence, albeit less exact. Such approximate forms of the AdS/CFT correspondence are being argued to be of use for understanding of the [[quark-gluon plasma]] in [[quantum chromodynamics]] (\hyperlink{PolicastroSonStarinets01}{Policastro-Son-Starinets 01}) and for various models in [[solid state physics]] (see \hyperlink{HartnollLucasSachdev16}{Hartnoll-Lucas-Sachdev 16} and see at \emph{[[AdS-CFT in condensed matter physics]]}). More in detail, since the [[near horizon geometry]] of [[BPS state|BPS]] [[black branes]] is conformal to the [[Cartesian product]] of [[anti de Sitter spaces]] with the unit $n$-sphere around the brane, the [[cosmology]] of [[intersecting D-brane models]] realizes the [[observable universe]] on the [[asymptotic boundary]] of an \emph{approximately} [[anti de Sitter spacetime]] (see for instance \hyperlink{Kaloper04}{Kaloper 04}, \hyperlink{FlachiMinamitsuji09}{Flachi-Minamitsuji 09}). The basic structure is hence that of \emph{[[Randall-Sundrum models]]}, but details differ, such as notably in \emph{warped throat} geometries, see \hyperlink{Uranga05}{Uranga 05, section 18}. These warped throat models go back to \hyperlink{KlebanovStrassler00}{Klebanov-Strassler 00} which discusses aspects of [[confinement]] in [[Yang-Mills theory]] on conincident ordinary and \emph{[[fractional D-brane|fractional]]} [[D3-branes]] at the [[singularity]] of a warped [[conifold]]. See also \hyperlink{KlebanovWitten98}{Klebanov-Witten 98} \begin{quote}% snippet grabbed from \hyperlink{Uranga05}{Uranga 05, section 18} here: ``RS''=[[Randall-Sundrum model]]; ``KS''=\hyperlink{KlebanovStrassler00}{Klebanov-Strassler 00} \end{quote} In particular this means that AdS-CFT duality applies in \emph{some approximation} to intersecting D-brane models (e.g. \hyperlink{Soda10}{Soda 10}, \hyperlink{GHMO16}{GHMO 16}), thus allowing to compute, to some approximation, [[non-perturbative effects]] in the [[Yang-Mills theory]] on the intersecting branes in terms of [[gravity]] on the ambient warped throat $\sim$ [[anti de Sitter spacetime|AdS]] (\hyperlink{KlebanovStrassler00}{Klebanov-Strassler 00, section 6}) Such approximate version of [[AdS-CFT]] for gauge theories realized on intersecting D-branes are used for instance to estimate [[non-perturbative effects]] in [[QCD]], such as the [[shear viscosity]] of the [[quark-gluon plasma]] (\hyperlink{PolicastroSonStarinets01}{Policastro-Son-Starinets 01}). For more on this approximate relation see at \emph{[[AdS-QCD correspondence]]}. \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} The solutions to [[supergravity]] that preserve the maximum of 32 [[supersymmetries]] are (e.g. \href{BPS+state#HEGKS08}{HEGKS 08 (1.1)}) \begin{itemize}% \item $AdS_5 \times S^5$ in [[type II supergravity]] \item $AdS_7 \times S^4$ in [[11-dimensional supergravity]] \item $AdS_4 \times S^7$ in [[11-dimensional supergravity]] \end{itemize} as well as their [[Minkowski spacetime]] and plane wave limits. These are the main [[KK-compactifications]] for the following examples- \hypertarget{AdS5CFT4}{}\subsubsection*{{$AdS_5 / CFT_4$ -- Horizon limit of D3-branes}}\label{AdS5CFT4} [[type II string theory]] on 5d [[anti de Sitter spacetime]] (times a 5-sphere) is dual to [[N=4 D=4 super Yang-Mills theory]] on the [[worldvolume]] of a [[D3-brane]] at the [[asymptotic boundary]] (\hyperlink{Maldacena97a}{Maldacena 97, section 2}) (\hyperlink{AharonyGubserMaldacenaOoguriOz99}{Aharony-Gubser-Maldacena-Ooguri-Oz 99, section 3 and 4}) \hypertarget{AdS7CFT6}{}\subsubsection*{{$AdS_7 / CFT_6$ -- Horizon limit of M5-branes}}\label{AdS7CFT6} We list some of the conjectured statements and their evidence concerning the case of $AdS_7/CFT_6$-duality. The hypothesis (\hyperlink{Maldacena97a}{Maldacena 97, section 3.1}) (see (\hyperlink{AharonyGubserMaldacenaOoguriOz99}{Aharony-Gubser-Maldacena-Ooguri-Oz 99, section 6.1.1}) for a review) is that \begin{itemize}% \item the [[6d (2,0)-superconformal QFT]] on the [[worldvolume]] of $N$ coincident [[M5-branes]] \end{itemize} is holographically related to \begin{itemize}% \item [[11-dimensional supergravity]] [[Kaluza-Klein mechanism|reduced on]] the 4-[[sphere]] $S^4$ with $N$ units of [[flux]] of the [[supergravity C-field]] to [[7d supergravity]] on an asymptotically [[anti de Sitter spacetime]]. \end{itemize} In \begin{itemize}% \item [[Edward Witten]], \emph{Five-Brane Effective Action In M-Theory} J. Geom. Phys.22:103-133,1997 (\href{http://arxiv.org/abs/hep-th/9610234}{arXiv:hep-th/9610234}) \end{itemize} effectively this relation was already used to computed the 5-brane [[partition function]] in the abelian case from the states of abelian [[7d Chern-Simons theory]]. (The quadratic refinement of the [[supergravity C-field]] necessary to make this come out right is what led to [[Quadratic Functions in Geometry, Topology, and M-Theory|Hopkins-Singer 02]] and hence to the further mathematical development of [[differential cohomology]] and its application in physics.) In (\hyperlink{Witten98}{Witten 98, section 4}) this construction is argued for from within the framework of AdS/CFT, explicitly identifying the [[7d Chern-Simons theory]] here with the compactification of the 11-dimensional Chern-Simons term of the [[supergravity C-field]] in [[11-dimensional supergravity]], which locally is \begin{displaymath} \begin{aligned} S_{11d SUGRA, CS}(C_3) &= \int_{AdS_7} \int_{S^4} C_3 \wedge G_4 \wedge G_4 \\ & = N \, \int_{AdS_7} C_3 \wedge G_4 \end{aligned} \,. \end{displaymath} But in fact the [[quantum anomaly]] cancellation ([[Green-Schwarz mechanism|GS-type mechanism]]) for 11d sugra introduces a quantum correction to this Chern-Simons term (\href{\DLM}{DLM, equation (3.14)}), making it locally become \begin{displaymath} \begin{aligned} S(\omega,C_3) &= \int_{AdS_7} \int_{S^4} C_3 \wedge G_4 \wedge (G_4 + I_8(\omega)) \\ & = N \, \int_{AdS_7} \left( C_3 \wedge G_4 + \frac{1}{48} CS_{p_2}(\omega) - \frac{1}{12} CS_{\frac{1}{2}p_1}(\omega) \wedge tr(F_\omega \wedge \omega) \right) \end{aligned} \,, \end{displaymath} where now $\omega$ is the local 1-form representative of a [[spin connection]] and where $CS_{p_2}$ is a [[Chern-Simons form]] for the second [[Pontryagin class]] and $CS_{\frac{1}{2}p_1}$ for the first. That therefore not an abelian, but this \emph{nonabelian} [[higher dimensional Chern-Simons theory]] should be dual to the nonabelian [[6d (2,0)-superconformal QFT]] was maybe first said explicitly in (\hyperlink{LuWang}{LuWang 2010}). Its gauge field is hence locally and ignoring the flux quantization subtleties a pair consisting of the abelian 3-form field $C$ and a [[Spin group]] $Spin(6,1)$-valued [[connection on a bundle|connection]] (see \emph{[[supergravity C-field]] for global descriptions of such pairs). Or maybe rather $Spin(6,2)$ to account for the constraint that the configurations are to be asymptotic [[anti de Sitter spacetimes]] (in analogy to the well-understood situation in [[3d quantum gravity]], see there for more details).} Indeed, in (\hyperlink{SezginSundell}{SezginSundell 2002, section 7}) more detailed arguments are given that the 7-dimensional dual to the free 6d theory is a [[higher spin gauge theory]] for a higher spin [[gauge group]] extending the ([[superconformal group|super]]) [[conformal group]] $SO(6,2)$. A [[non-perturbative quantum field theory|non-perturbative]] description of this nonabelian [[7d Chern-Simons theory]] as a [[local prequantum field theory]] (hence defined [[non-perturbative quantum field theory|non-perturbatively]] on the global [[moduli stack]] of [[field (physics)|fields]] ([[twisted differential string structures]], in fact)) was discussed in (\hyperlink{FSS12a}{FSS 12a}, \hyperlink{FSS12b}{FSS 12b}). General discussion of [[boundary field theory|boundary]] [[local prequantum field theories]] relating higher Chern-Simons-type and higher WZW-type theories is in ([[schreiber:differential cohomology in a cohesive topos|dcct 13, section 3.9.14]]). Specifically, a characterization along these lines of the [[Green-Schwarz action functional]] of the [[M5-brane]] as a holographic [[infinity-Wess-Zumino-Witten theory - contents|higher WZW-type]] boundary theory of a 7d Chern-Simons theory is found in (\hyperlink{FSS13}{FSS 13}). Analogous discussion of the 6d theory as a higher WZW analog of a 7d Chern-Simons theory phrased in terms of [[extended quantum field theory]] is ([[4-3-2 8-7-6|Freed 12]]). \hypertarget{AdS4CFT3}{}\subsubsection*{{$AdS_4 / CFT_3$ --Horizon limit of M2-branes}}\label{AdS4CFT3} [[11d supergravity]]/[[M-theory]] on the asymptotic $AdS_4$ spacetime of an [[M2-brane]]. (\hyperlink{Maldacena97a}{Maldacena 97, section 3.2}, \hyperlink{AharonyGubserMaldacenaOoguriOz99}{Aharony-Gubser-Maldacena-Ooguri-Oz 99, section 6.1.2}, \href{KlebanovTorri10}{Klebanov-Torri 10}) \hypertarget{AdS3CFT2}{}\subsubsection*{{$AdS_3 / CFT_2$ -- Horizon limit of D1-D5 brane bound states}}\label{AdS3CFT2} [[D1-D5 brane system]] in [[type IIB string theory]] (\hyperlink{Maldacena97a}{Maldacena 97, section 4}) (\hyperlink{AharonyGubserMaldacenaOoguriOz99}{Aharony-Gubser-Maldacena-Ooguri-Oz 99, section 5}) see also at \emph{[[AdS3-CFT2 and CS-WZW correspondence]]} (\ldots{}) \hypertarget{nonconformal_duals}{}\subsubsection*{{Non-conformal duals}}\label{nonconformal_duals} \hypertarget{horizon_limit_of_branes_for_arbitrary_}{}\paragraph*{{Horizon limit of $Dp$-branes for arbitrary $p$}}\label{horizon_limit_of_branes_for_arbitrary_} (\hyperlink{AharonyGubserMaldacenaOoguriOz99}{Aharony-Gubser-Maldacena-Ooguri-Oz 99, section 6.1.3}) \hypertarget{horizon_limit_of_ns5brane}{}\paragraph*{{Horizon limit of NS5-brane}}\label{horizon_limit_of_ns5brane} (\hyperlink{AharonyGubserMaldacenaOoguriOz99}{Aharony-Gubser-Maldacena-Ooguri-Oz 99, section 6.1.4}) \hypertarget{qcd_models}{}\paragraph*{{QCD models}}\label{qcd_models} While all of the above horizon limits product [[super Yang-Mills theory]], one can consider certain limits of these in which they look like plain [[QCD]], at least in certain sectors. This leads to a discussion of holographic description of QCD properties that are actually experimentally observed. (\hyperlink{AharonyGubserMaldacenaOoguriOz99}{Aharony-Gubser-Maldacena-Ooguri-Oz 99, section 6.2}) See the \emph{\hyperlink{ToCondensedMatterPhysics}{References -- Applications -- In condensed matter physics}}. \hypertarget{further_gauge_theories_induced_by_compactification_and_twisting}{}\subsubsection*{{Further gauge theories induced by compactification and twisting}}\label{further_gauge_theories_induced_by_compactification_and_twisting} [[!include gauge theory from AdS-CFT -- table]] \hypertarget{Checks}{}\subsection*{{Checks}}\label{Checks} At the heart of the duality is the observation that the classical [[action functionals]] for various [[field (physics)|fields]] coupled to [[Einstein gravity]] on [[anti de Sitter spacetime]] are, when expressed as [[functions]] of the [[asymptotic boundary]] values of the [[field (physics)|fields]], equal to the [[generating functions]] for the [[correlators]]/[[n-point functions]] of a [[conformal field theory]] on that asymptotic boundary. These computations were laid out in \hyperlink{Witten98}{Witten 98, section 2.4 ``Some sample computation''}. These follow from elementary manipulation in [[differential geometry]] (involving neither [[supersymmetry]] nor [[string theory]]). For the more ambitious matching of the spectrum of the dilatation operator of [[N=4 D=4 super Yang-Mills theory]] to the corresponding spectrum of the [[Green-Schwarz superstring]] on the [[super anti de Sitter spacetime]] $AdS_5 \times S^5$ detailed checks are summarized in \hyperlink{BeisertEtAl10}{Beisert et al. 10}, \hyperlink{Escobedo12}{Escobedo 12} \begin{quote}% graphics grabbed from \hyperlink{Escobedo12}{Escobedo 12} \end{quote} Comparison to [[string scattering amplitudes]] beyond the planar SCFT limit: \hyperlink{ABP18}{ABP 18}. Numerical checks using [[lattice gauge theory]] are reviewed in \hyperlink{Joseph15}{Joseph 15}. Exact duality checks pertaining to the full stringy regime for $AdS_3/CFT_2$: \hyperlink{EberhardtGaberdiel19a}{Eberhardt-Gaberdiel 19a}, \hyperlink{EberhardtGaberdiel19b}{Eberhardt-Gaberdiel 19b}, \hyperlink{EberhardtGaberdielGopakumar19}{Eberhardt-Gaberdiel-Gopakumar 19}. \hypertarget{formalizations}{}\subsection*{{Formalizations}}\label{formalizations} The full formalization of AdS/CFT is still very much out of reach, but maybe mostly for lack of trying. But see \hyperlink{Anderson04}{Anderson 04}. One proposal for a formalization of a toy version in the context of [[AQFT]] is [[Rehren duality]]. However, it does not seem that this actually formalizes AdS-CFT, but something else. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[near-horizon geometry]] \item [[Fefferman-Graham ambient construction]] \item [[holographic entanglement entropy]] \item [[duality in physics]], [[duality in string theory]] \begin{itemize}% \item [[T-duality]], [[S-duality]], [[U-duality]] \item [[open/closed string duality]] \begin{itemize}% \item [[KLT relations]] \end{itemize} \end{itemize} \item [[black hole in anti de Sitter spacetime]] \item [[S-matrix theory]] \item [[anti de Sitter gravity]] \item [[Randall-Sundrum model]] \item [[Sachdev-Ye-Kitaev model]] \item [[p-adic AdS-CFT]] \end{itemize} [[!include table of branes]] \hypertarget{references}{}\subsection*{{References}}\label{references} \hypertarget{original_articles}{}\subsubsection*{{Original articles}}\label{original_articles} The original articles are \begin{itemize}% \item [[Juan Maldacena]], \emph{The Large N limit of superconformal field theories and supergravity}, Adv. Theor. Math. Phys. 2:231, 1998 (\href{http://arxiv.org/abs/hep-th/9711200}{hep-th/9711200}) \item [[Juan Maldacena]], \emph{Wilson loops in Large $N$ field theories}, Phys. Rev. Lett. \textbf{80} (1998) 4859 (\href{http://arxiv.org/abs/hep-th/9803002}{hep-th/9803002}) \end{itemize} The actual rule for matching [[bulk field theory|bulk]] [[states]] to [[generating functions]] for [[boundary field theory|boundary]] [[correlators]]/[[n-point functions]] is due to \begin{itemize}% \item [[Steven Gubser]], [[Igor Klebanov]], [[Alexander Polyakov]], around (12) of \emph{Gauge theory correlators from non-critical string theory}, Physics Letters B428: 105--114 (1998) (\href{http://arxiv.org/abs/hep-th/9802109}{hep-th/9802109}) \item [[Edward Witten]], around (2.11) of \emph{Anti-de Sitter space and holography}, Advances in Theoretical and Mathematical Physics 2: 253--291, 1998 (\href{http://arxiv.org/abs/hep-th/9802150}{hep-th/9802150}) \end{itemize} See also \begin{itemize}% \item Carlos Andrés Cardona Giraldo, \emph{Correlation functions in AdS/CFT correspondence}, 2012 (\href{inspirehep.net/record/1652794}{spire:1652794}, \href{https://digital.bl.fcen.uba.ar/download/tesis/tesis_n5179_CardonaGiraldo.pdf}{pdf}) \end{itemize} Discussion of how [[Green-Schwarz action functionals]] for super $p$-branes in AdS target spaces induce, after [[diffeomorphism]] [[gauge fixing]], superconformal field theory on the [[worldvolumes]] (see \emph{[[singleton representation]]}) goes back to \begin{itemize}% \item [[Mike Duff]], C. Sutton, \emph{The Membrane at the End of the Universe}, New Sci. 118 (1988) 67-71 (\href{http://inspirehep.net/record/268230?ln=en}{spire:268230}) \end{itemize} and was further developed in \begin{itemize}% \item Piet Claus, [[Renata Kallosh]], [[Antoine Van Proeyen]], \emph{M 5-brane and superconformal $(0,2)$ tensor multiplet in 6 dimensions}, Nucl.Phys. B518 (1998) 117-150 (\href{http://arxiv.org/abs/hep-th/9711161}{arXiv:hep-th/9711161}) \item [[Gianguido Dall'Agata]], Davide Fabbri, Christophe Fraser, [[Pietro Fré]], Piet Termonia, Mario Trigiante, \emph{The $Osp(8|4)$ singleton action from the supermembrane}, Nucl.Phys.B542:157-194, 1999 (\href{http://arxiv.org/abs/hep-th/9807115}{arXiv:hep-th/9807115}) \item Piet Claus, [[Renata Kallosh]], J. Kumar, [[Paul Townsend]], [[Antoine Van Proeyen]], \emph{Conformal Theory of M2, D3, M5 and `D1+D5' Branes}, JHEP 9806 (1998) 004 (\href{http://arxiv.org/abs/hep-th/9801206}{arXiv:hep-th/9801206}) \item [[Paolo Pasti]], [[Dmitri Sorokin]], Mario Tonin, \emph{Branes in Super-AdS Backgrounds and Superconformal Theories} (\href{http://arxiv.org/abs/hep-th/9912076}{arXiv:hep-th/9912076}) \end{itemize} Review is in \begin{itemize}% \item [[Mike Duff]], \emph{Anti-de Sitter space, branes, singletons, superconformal field theories and all that}, Int.J.Mod.Phys.A14:815-844,1999 (\href{https://arxiv.org/abs/hep-th/9808100}{arXiv:hep-th/9808100}) \item [[Mike Duff]], \emph{TASI Lectures on Branes, Black Holes and Anti-de Sitter Space} (\href{https://arxiv.org/abs/hep-th/9912164}{arXiv:hep-th/9912164}) \end{itemize} See also at \emph{\href{Green-Schwarz+action+functional#AsPartOfTheAdSCFTCorrespodence}{super p-brane -- As part of the AdS-CFT correspondence}}. Sketch of a derivation of AdS/CFT: \begin{itemize}% \item [[Horatiu Nastase]], \emph{Towards deriving the AdS/CFT correspondence} (\href{https://arxiv.org/abs/1812.10347}{arXiv:1812.10347}) \end{itemize} \hypertarget{introductions_and_surveys}{}\subsubsection*{{Introductions and surveys}}\label{introductions_and_surveys} Surveys and introductions include \begin{itemize}% \item [[Ofer Aharony]], [[Steven Gubser]], [[Juan Maldacena]], [[Hirosi Ooguri]], [[Yaron Oz]], \emph{Large $N$ Field Theories, String Theory and Gravity}, Phys.Rept.323:183-386,2000 (\href{http://arxiv.org/abs/hep-th/9905111}{arXiv:hep-th/9905111}) \item Michael T. Anderson, \emph{Geometric aspects of the AdS/CFT correspondence} (\href{https://arxiv.org/abs/hep-th/0403087}{arXiv:hep-th/0403087}) \item [[Horatiu Nastase]], \emph{Introduction to AdS-CFT} (\href{http://arxiv.org/abs/0712.0689}{arXiv:0712.0689}) \item [[Gaston Giribet]], \emph{Black hole physics and $AdS^3/CFT_2$}, lectures and proceedings of [[Croatian black hole school]]. \item Jens L. Petersen, \emph{Introduction to the Maldacena Conjecture on AdS/CFT}, Int.J.Mod.Phys. A14 (1999) 3597-3672, \href{http://arxiv.org/abs/hep-th/9902131}{hep-th/9902131} , \href{http://dx.doi.org/10.1142/S0217751X99001676}{doi} \item Jan de Boer, \emph{Introduction to AdS/CFT correspondence}, \href{http://www-library.desy.de/preparch/desy/proc/proc02-02/Proceedings/pl.6/deboer_pr.pdf}{pdf} \item [[Makoto Natsuume]], \emph{AdS/CFT Duality User Guide}, Lecture Notes in Physics 903, Springer 2015 (\href{https://arxiv.org/abs/1409.3575}{arXiv:1409.3575}) \item wikipedia: \href{http://en.wikipedia.org/wiki/AdS/CFT_correspondence}{AdS/CFT correspondence} \item an AdS/CFT \href{http://www.personal.uni-jena.de/~p5thul2/notes/adscft.html}{bibliography} \end{itemize} Further references include: \begin{itemize}% \item Gary T. Horowitz, [[Joseph Polchinski]], \emph{Gauge/gravity duality}, \href{http://arxiv.org/abs/gr-qc/0602037}{gr-qc/0602037} \item [[Edward Witten]], \emph{Three-dimensional gravity revisited}, \href{http://arxiv.org/abs/0706.3359}{arxiv/0706.3359} \item C.R. Graham, [[Edward Witten]], \emph{Conformal anomaly of submanifold observables in AdS/CFT correspondence}, \href{http://arxiv.org/abs/hep-th/9901021}{hepth/9901021}. \item [[Edward Witten]], \emph{AdS/CFT Correspondence And Topological Field Theory} (\href{http://arxiv.org/abs/hep-th/9812012}{arXiv:hep-th/9812012}) \end{itemize} Review of [[lattice gauge theory]]-numerics: \begin{itemize}% \item Anosh Joseph, \emph{Review of Lattice Supersymmetry and Gauge-Gravity Duality} (\href{https://arxiv.org/abs/1509.01440}{arXiv:1509.01440}) \end{itemize} Review of [[Yangian]] symmetry: \begin{itemize}% \item [[Alessandro Torrielli]], \emph{Yangians, S-matrices and AdS/CFT}, J.Phys.A44:263001,2011 (\href{http://arxiv.org/abs/1104.2474}{arXiv:1104.2474}) \end{itemize} \hypertarget{ReferencesAdS2CFT1}{}\subsubsection*{{On $AdS_2 / CFT_1$}}\label{ReferencesAdS2CFT1} Via [[Jackiw-Teitelboim gravity]]: \begin{itemize}% \item [[Ahmed Almheiri]], [[Joseph Polchinski]], \emph{Models of $AdS_2$ Backreaction and Holography}, J. High Energ. Phys. (2015) 2015: 14. (\href{https://arxiv.org/abs/1402.6334}{arXiv:1402.6334}) \end{itemize} \hypertarget{ReferencesAdS3CFT2}{}\subsubsection*{{On $AdS_3 / CFT_2$}}\label{ReferencesAdS3CFT2} An exact correspondence of the symmetric [[orbifold]] [[CFT]] of [[Liouville theory]] with a string theory on $AdS_3$ is claimed in: \begin{itemize}% \item [[Lorenz Eberhardt]], [[Matthias Gaberdiel]], \emph{String theory on $AdS_3$ and the symmetric orbifold of Liouville theory} (\href{https://arxiv.org/abs/1903.00421}{arXiv:1903.00421}) \item [[Lorenz Eberhardt]], [[Matthias Gaberdiel]], \emph{Strings on $AdS_3 \times S^3 \times S^3 \times S^1$} (\href{https://arxiv.org/abs/1904.01585}{arXiv:1904.01585}) \item [[Lorenz Eberhardt]], [[Matthias Gaberdiel]], [[Rajesh Gopakumar]], \emph{Deriving the $AdS_3/CFT_2$ Correspondence} (\href{https://arxiv.org/abs/1911.00378}{arXiv:1911.00378}) \end{itemize} based on \begin{itemize}% \item Shouvik Datta, [[Lorenz Eberhardt]], [[Matthias Gaberdiel]], \emph{Stringy $\mathcal{N} = (2,2)$ holography for $AdS_3$} JHEP 1801 (2018) 146 (\href{https://arxiv.org/abs/1709.06393}{arXiv:1709.06393}) \end{itemize} \hypertarget{on__3}{}\subsubsection*{{On $AdS_4 / CFT_3$}}\label{on__3} \begin{itemize}% \item [[Igor Klebanov]], Giuseppe Torri, \emph{M2-branes and AdS/CFT}, Int.J.Mod.Phys.A25:332-350,2010 (\href{http://arxiv.org/abs/0909.1580}{arXiv:0909.1580}) \end{itemize} \hypertarget{on__4}{}\subsubsection*{{On $AdS_5 / CFT_4$}}\label{on__4} \begin{itemize}% \item N. Beisert et al., \emph{Review of AdS/CFT Integrability, An Overview}, Lett. Math. Phys. vv, pp (2011), (\href{http://arxiv.org/abs/1012.3982}{arXiv:1012.3982}). \item Jorge Escobedo, \emph{Integrability in AdS/CFT: Exact Results for Correlation Functions}, 2012 (\href{http://inspirehep.net/record/1264432}{spire:1264432}) \end{itemize} Computing dual [[string scattering amplitudes]] by AdS/CFT beyond the [[planar limit]]: \begin{itemize}% \item [[Luis Alday]], [[Agnese Bissi]], [[Eric Perlmutter]], \emph{Genus-One String Amplitudes from Conformal Field Theory}, JHEP06(2019) 010 (\href{https://arxiv.org/abs/1809.10670}{arXiv:1809.10670}) \end{itemize} \hypertarget{on__5}{}\subsubsection*{{On $AdS_7 / CFT_6$}}\label{on__5} We list references specific to $AdS_7/CFT_6$. In \begin{itemize}% \item [[Edward Witten]], \emph{Five-Brane Effective Action In M-Theory} J. Geom. Phys.22:103-133,1997 (\href{http://arxiv.org/abs/hep-th/9610234}{arXiv:hep-th/9610234}) \item [[Edward Witten]], \emph{AdS/CFT Correspondence And Topological Field Theory} JHEP 9812:012,1998 (\href{http://arxiv.org/abs/hep-th/9812012}{arXiv:hep-th/9812012}) \end{itemize} it is argued that the [[conformal blocks]] of the [[6d (2,0)-superconformal QFT]] are entirely controled just by the effective [[higher dimensional Chern-Simons theory|7d Chern-Simons theory]] inside [[11-dimensional supergravity]], but only the abelian piece is discussed explicitly. The fact that this Chern-Simons term is in fact a \emph{nonabelian} [[higher dimensional Chern-Simons theory]] in $d = 7$, due the [[quantum anomaly]] cancellation, is clear from the original source, equation (3.14) of \begin{itemize}% \item [[Michael Duff]], [[James Liu]], [[Ruben Minasian]], \emph{Eleven Dimensional Origin of String/String Duality: A One Loop Test} (\href{http://arxiv.org/abs/hep-th/9506126}{arXiv:hep-th/9506126}) \end{itemize} but seems not to be noted explicitly in the context of $AdS_7/CFT_6$ before the references \begin{itemize}% \item H. L\"u{}, Yi Pang, \emph{Seven-Dimensional Gravity with Topological Terms} Phys.Rev.D81:085016 (2010) (\href{http://arxiv.org/abs/1001.0042}{arXiv:1001.0042}) \item H. Lu, Zhao-Long Wang, \emph{On M-Theory Embedding of Topologically Massive Gravity} Int.J.Mod.Phys.D19:1197 (2010) (\href{http://arxiv.org/abs/1001.2349}{arXiv:1001.2349}) \end{itemize} There is in fact one more quantization condition to be taken into account. Discussion of this nonabeloan [[7d Chern-Simons theory]] terms as a [[local prequantum field theory]] is in \begin{itemize}% \item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:7d Chern-Simons theory and the 5-brane|Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory]]}, Advances in Theoretical and Mathematical Physics (2014) (\href{http://arxiv.org/abs/1201.5277}{arXiv:1201.5277}) \end{itemize} and a corresponding non-perturbative discussion of the [[supergravity C-field]] that enters this Lagrangian is given in [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:The moduli 3-stack of the C-field|The E8 moduli 3-stack of the C-field]]} (\href{http://arxiv.org/abs/1202.2455}{arXiv:1202.2455}) Up to the further twists discussed there, this means that the gauge group of the effective 7d theory is some contraction of the [[Spin group]] $Spin(10,1)$. The asymptotic AdS condition suggests maybe that it should be $Spin(6,2)$. In fact, in \begin{itemize}% \item [[Ergin Sezgin]], P. Sundell, \emph{Massless Higher Spins and Holography} (\href{http://arxiv.org/abs/hep-th/0205131}{hep-th/0205131}) \end{itemize} arguments are given that the 7d theory is a [[higher spin gauge theory]] extension of $SO(6,2)$. More on the relation between the [[M5-brane]] and supergravity on $AdS_7 \times S^4$ and arguments for the $SO(5)$ [[R-symmetry]] group on the 6d theory from the 7d theory are given in \begin{itemize}% \item [[Alexei Nurmagambetov]], I. Y. Park, \emph{On the M5 and the AdS7/CFT6 Correspondence} (\href{http://arxiv.org/abs/hep-th/0110192}{arXiv:hep-th/0110192}) \end{itemize} See also \begin{itemize}% \item M. Nishimura, Y. Tanii, \emph{Local Symmetries in the AdS}7/CFT\_6 Correspondence\_, Mod. Phys. Lett. A14 (1999) 2709-2720 (\href{http://arxiv.org/abs/hep-th/9910192}{arXiv:hep-th/9910192}) \end{itemize} An explicit relalization of the [[Green-Schwarz action functional]] of the [[M5-brane]] as a boundary field theory to the fermionic Chern-Simons term in the [[11-dimensional supergravity]] action functional is given in \begin{itemize}% \item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:The brane bouquet|Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields]]} (\href{http://arxiv.org/abs/1308.5264}{arXiv:1308.5264}) \end{itemize} \hypertarget{generalization_beyond_exact_ads__exact_cft}{}\subsubsection*{{Generalization beyond exact AdS / exact CFT}}\label{generalization_beyond_exact_ads__exact_cft} Discussion for [[cosmology]] of [[intersecting D-brane models]] (ambient $\sim$ [[anti de Sitter spacetimes]] with the $\sim$ conformal intersecting branes at the [[asymptotic boundary]]) includes the following (see also at \emph{[[Randall-Sundrum model]]}): \begin{itemize}% \item [[Igor Klebanov]], [[Matthew Strassler]], \emph{Supergravity and a Confining Gauge Theory: Duality Cascades and $\chi^{SB}$-Resolution of Naked Singularities}, JHEP 0008:052, 2000 (\href{https://arxiv.org/abs/hep-th/0007191}{arXiv:hep-th/0007191}) \item [[Igor Klebanov]], [[Edward Witten]], \emph{Superconformal Field Theory on Threebranes at a Calabi-Yau Singularity}, Nucl.Phys.B536:199-218, 1998 (\href{https://arxiv.org/abs/hep-th/9807080}{arXiv:hep-th/9807080}) \item Nemanja Kaloper, \emph{Origami World}, JHEP 0405 (2004) 061 (\href{https://arxiv.org/abs/hep-th/0403208}{arXiv:hep-th/0403208}) \item [[Angel Uranga]], section 18 of \emph{TASI lectures on String Compactification, Model Building, and Fluxes}, 2005 (\href{http://cds.cern.ch/record/933469/files/cer-002601054.pdf}{pdf}) \item Antonino Flachi, Masato Minamitsuji, \emph{Field localization on a brane intersection in anti-de Sitter spacetime}, Phys.Rev.D79:104021, 2009 (\href{https://arxiv.org/abs/0903.0133}{arXiv:0903.0133}) \item Jiro Soda, \emph{AdS/CFT on the brane}, Lect.Notes Phys.828:235-270, 2011 (\href{https://arxiv.org/abs/1001.1011}{arXiv:1001.1011}) \item Shunsuke Teraguchi, around slide 21 \emph{String theory and its relation to particle physics}, 2007 (\href{http://phys.cts.ntu.edu.tw/ppp7/talks/PPP7_Shunsuke_Teraguchi.pdf}{pdf}) \item Gianluca Grignani, Troels Harmark, Andrea Marini, Marta Orselli, \emph{The Born-Infeld/Gravity Correspondence}, Phys. Rev. D 94, 066009 (2016) (\href{https://arxiv.org/abs/1602.01640}{arXiv:1602.01640}) \end{itemize} \hypertarget{Appications}{}\subsubsection*{{Applications to physics}}\label{Appications} \hypertarget{to_gravity}{}\paragraph*{{To gravity}}\label{to_gravity} Discussion of [[event horizons]] of [[black holes]] in terms of AdS/CFT (the ``[[firewall problem]]'') is in \begin{itemize}% \item Kyriakos Papadodimas, Suvrat Raju, \emph{An Infalling Observer in AdS/CFT} (\href{http://arxiv.org/abs/1211.6767}{arXiv:1211.6767}) \end{itemize} To [[black hole]] interiors: \begin{itemize}% \item [[Juan Maldacena]], \emph{Toy models for black holes II}, talk at PiTP 2018 \emph{From QBits to spacetime} (\href{https://video.ias.edu/PiTP/2018/0726-JuanMaldacena}{recording}) \end{itemize} \begin{quote}% The [[SYK model]] gives us a glimpse into the interior of an [[extremal black hole]]\ldots{}That's the feature of SYK that I find most interesting\ldots{}It is a feature this model has, that I think no other model has \end{quote} To [[symmetries]] in gravity: \begin{itemize}% \item [[Daniel Harlow]], [[Hirosi Ooguri]], \emph{Constraints on symmetry from holography}, Phys. Rev. Lett. 122, 191601, 2019 (\href{https://arxiv.org/abs/1810.05337}{arXiv:1810.05337}, \href{https://doi.org/10.1103/PhysRevLett.122.191601}{doi:10.1103/PhysRevLett.122.191601}) \end{itemize} \hypertarget{to_the_quarkgluon_plasma}{}\paragraph*{{To the quark-gluon plasma}}\label{to_the_quarkgluon_plasma} Applications of AdS-CFT to the [[quark-gluon plasma]] of [[QCD]]: Expositions and reviews include \begin{itemize}% \item Pavel Kovtun, \emph{Quark-Gluon Plasma and String Theory}, RHIC news (2009) (\href{http://www.bnl.gov/rhic/news/091107/story2.asp}{blog entry}) \item Makoto Natsuume, \emph{String theory and quark-gluon plasma} (\href{http://arxiv.org/abs/hep-ph/0701201}{arXiv:hep-ph/0701201}) \item [[Steven Gubser]], \emph{Using string theory to study the quark-gluon plasma: progress and perils} (\href{http://arxiv.org/abs/0907.4808}{arXiv:0907.4808}) \item Francesco Biagazzi, A. l. Cotrone, \emph{Holography and the quark-gluon plasma}, AIP Conference Proceedings 1492, 307 (2012) (\href{https://doi.org/10.1063/1.4763537}{doi:10.1063/1.4763537}, \href{http://cp3-origins.dk/content/movies/2013-01-14-bigazzi.pdf}{slides pdf}) \item Brambilla et al., section 9.2.2 of \emph{QCD and strongly coupled gauge theories: challenges and perspectives}, Eur Phys J C Part Fields. 2014; 74(10): 2981 (\href{https://link.springer.com/article/10.1140%2Fepjc%2Fs10052-014-2981-5}{doi:10.1140/epjc/s10052-014-2981-5}) \end{itemize} Holographic discussion of the [[shear viscosity]] of the quark-gluon plasema goes back to \begin{itemize}% \item [[Giuseppe Policastro]], D.T. Son, A.O. Starinets, \emph{Shear viscosity of strongly coupled N=4 supersymmetric Yang-Mills plasma}, Phys. Rev. Lett.87:081601, 2001 (\href{http://arxiv.org/abs/hep-th/0104066}{arXiv:hep-th/0104066}) \end{itemize} Other original articles include: \begin{itemize}% \item Hovhannes R. Grigoryan, Paul M. Hohler, Mikhail A. Stephanov, \emph{Towards the Gravity Dual of Quarkonium in the Strongly Coupled QCD Plasma} (\href{http://arxiv.org/abs/1003.1138}{arXiv:1003.1138}) \item Brett McInnes, \emph{Holography of the Quark Matter Triple Point} (\href{http://arxiv.org/abs/0910.4456}{arXiv:0910.4456}) \end{itemize} \hypertarget{to_particle_physics}{}\paragraph*{{To particle physics}}\label{to_particle_physics} \begin{itemize}% \item [[Joseph Polchinski]], [[Matthew Strassler]], \emph{Hard Scattering and Gauge/String Duality}, Phys. Rev. Lett. 88:031601, 2002, (\href{http://lanl.arxiv.org/abs/hep-th/0109174}{arXiv:hep-th/0109174}) \end{itemize} For more see at \emph{[[AdS/QCD correspondence]]}. \hypertarget{to_fluid_dynamics}{}\paragraph*{{To fluid dynamics}}\label{to_fluid_dynamics} Application to [[fluid dynamics]] -- see also at \emph{[[fluid/gravity correspondence]]}: \begin{itemize}% \item [[Sayantani Bhattacharyya]], [[Veronika Hubeny]], [[Shiraz Minwalla]], [[Mukund Rangamani]], \emph{Nonlinear Fluid Dynamics from Gravity}, JHEP 0802:045, 2008 (\href{https://arxiv.org/abs/0712.2456}{arXiv:0712.2456}) \end{itemize} \hypertarget{ToCondensedMatterPhysics}{}\paragraph*{{To condensed matter physics}}\label{ToCondensedMatterPhysics} On [[AdS-CFT in condensed matter physics]]: Textbook account \begin{itemize}% \item Sean A. Hartnoll, Andrew Lucas, [[Subir Sachdev]], \emph{Holographic quantum matter}, MIT Press 2018 (\href{https://arxiv.org/abs/1612.07324}{arXiv:1612.07324}, \href{https://mitpress.ublish.com/book/holographic-quantum-matter}{publisher}) \end{itemize} Further reviews include the following: \begin{itemize}% \item A S T Pires, \emph{Ads/CFT correspondence in condensed matter} (\href{http://arxiv.org/abs/1006.5838}{arXiv:1006.5838}) \item [[Subir Sachdev]], \emph{Condensed matter and AdS/CFT} (\href{http://arxiv.org/abs/1002.2947}{arXiv:1002.2947}) \item Yuri V. Kovchegov, \emph{AdS/CFT applications to relativistic heavy ion collisions: a brief review} (\href{http://arxiv.org/abs/1112.5403}{arXiv:1112.5403}) \item Alberto Salvio, \emph{Superconductivity, Superfluidity and Holography} (\href{http://arxiv.org/abs/1301.0201}{arXiv:1301.0201}) \item \emph{\href{http://igfae.usc.es/~holoquark2018/}{Holography and Extreme Chromodynamics}}, Santiago de Compostela, July 2018 \end{itemize} \hypertarget{applications_in_mathematics}{}\subsubsection*{{Applications in mathematics}}\label{applications_in_mathematics} \hypertarget{to_the_volume_conjecture}{}\paragraph*{{To the volume conjecture}}\label{to_the_volume_conjecture} Suggestion that the statement of the [[volume conjecture]] is really [[AdS-CFT duality]] combined with the [[3d-3d correspondence]] for [[M5-branes]] [[wrapped brane|wrapped]] on [[hyperbolic 3-manifolds]]: \begin{itemize}% \item Dongmin Gang, [[Nakwoo Kim]], Sangmin Lee, Section 3.2\_Holography of 3d-3d correspondence at Large $N$\emph{, JHEP04(2015) 091 (\href{https://arxiv.org/abs/1409.6206}{arXiv:1409.6206})} \item Dongmin Gang, [[Nakwoo Kim]], around (21) of: \emph{Large $N$ twisted partition functions in 3d-3d correspondence and Holography}, Phys. Rev. D 99, 021901 (2019) (\href{https://arxiv.org/abs/1808.02797}{arXiv:1808.02797}) \end{itemize} \hypertarget{to_deep_learning_in_neural_networks}{}\paragraph*{{To deep learning in neural networks}}\label{to_deep_learning_in_neural_networks} On the [[deep learning]] algorithm on [[neural networks]] as analogous to the [[AdS/CFT correspondence]]: \begin{itemize}% \item Yi-Zhuang You, Zhao Yang, Xiao-Liang Qi, \emph{Machine Learning Spatial Geometry from Entanglement Features}, Phys. Rev. B 97, 045153 (2018) (\href{https://arxiv.org/abs/1709.01223}{arxiv:1709.01223}) \item W. C. Gan and F. W. Shu, \emph{Holography as deep learning}, Int. J. Mod. Phys. D 26, no. 12, 1743020 (2017) (\href{https://arxiv.org/abs/1705.05750}{arXiv:1705.05750}) \item J. W. Lee, \emph{Quantum fields as deep learning} (\href{https://arxiv.org/abs/1708.07408}{arXiv:1708.07408}) \item [[Koji Hashimoto]], Sotaro Sugishita, Akinori Tanaka, Akio Tomiya, \emph{Deep Learning and AdS/CFT}, Phys. Rev. D 98, 046019 (2018) (\href{https://arxiv.org/abs/1802.08313}{arxiv:1802.08313}) \end{itemize} [[!redirects AdS/CFT]] [[!redirects AdS/CFT correspondence]] [[!redirects AdS-CFT correspondence]] [[!redirects AdS-CFT correspondence]] [[!redirects AdS-CFT duality]] [[!redirects AdS/CFT duality]] [[!redirects AdS7-CFT6]] [[!redirects AdS7-CFT6 duality]] [[!redirects AdS5-CFT4]] [[!redirects AdS5-CFT4 duality]] [[!redirects AdS4-CFT3]] [[!redirects AdS4-CFT3 duality]] \end{document}