Contents

Idea

General idea

The AdS-CFT correspondence at its heart is the observation (Witten 98, Section 2.4) that the classical action functionals for various fields coupled to Einstein gravity on anti de Sitter spacetime are, when expressed as functions of the asymptotic boundary-values of the fields, of the form of generating functions for correlators/n-point functions of a conformal field theory on that asymptotic boundary, in a large N limit.

This is traditionally interpreted as a concrete realization of a vague “holographic principle” according to which quantum gravity in bulk spacetimes is controlled, in one way or other, by “boundary field theories” on effective spacetime boundaries, such as event horizons. The original and main motivation for the holographic principle itself was the fact that the apparent black hole entropy in Einstein gravity scales with the area of the event horizon instead of the black hole’s bulk volume (which is not even well-defined), suggesting that gravity encodes or is encoded by some boundary field theory associated with horizons; an idea that, in turn, seems to find a concrete realization in open/closed string duality in the vicinity of, more generally, black branes. The original intuition about holographic black hole entropy has meanwhile found remarkably detailed reflection in the (mathematically fairly rigorous) analysis of holographic entanglement entropy, specifically via holographic tensor networks, which turn out to embody key principles of the AdS/CFT correspondence in the guise of quantum information theory, with concrete applications such as to quantum error correcting codes.

The AdS/CFT correspondence itself crucially involves the exceptional isomorphism between the isometry group of anti de Sitter spacetime $AdS_{d+1}$ (the anti de Sitter group) and the conformal group of Minkowski spacetime of dimension $d$: the connected component of both is the special orthogonal group $SO(d,2)$. But the AdS/CFT correspondence is deeper and more subtle than this group theory underlying it, in particular in how it puts fields and states on the gravity side in correspondence with sources and correlators on the field theory side, respectively.

In extrapolation of these elementary computations, the AdS/CFT correspondence 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 $d$ (Gubser-Klebanov-Polyakov 98 (12), Witten 98, (2.11)), such that perturbation theory on one side of the correspondence relates to non-perturbation on the other side.

While this works to some extent quite generally (see e.g. Natsuume 15 for review), allowing applications such as AdS/CFT in condensed matter physics and AdS/CFT in quantum chromodynamics, the tightest form of the correspondence relates the 1/N expansion of superconformal field theories (super Yang-Mills theories) on the asymptotic boundaries of near-horizon limits of $N$ coincident black M2-branes/D3-branes/M5-branes to corresponding sectors of the string theory/M-theory quantum gravity in the bulk spacetime away from the brane.

Before the proposal for the actual matching rule of ADS/CFT (Gubser-Klebanov-Polyakov 98 (12), Witten 98, (2.11)) it was by matching of BPS-states in these situations that the existence of an AdS/CFT correspondence was proposed in Maldacena 97a, Maldacena 97b; these articles are now widely regarded as the origin of the idea of the AdS/CFT correspondence.

A quick way to see that the supersymmetric-cases of AdS/CFT for near horizon geometries of M2-branes, D3-branes and M5-branes must be special is to observe that these are the only dimensions in which there are super anti-de Sitter spacetime-enhancements of anti de Sitter spacetime, matching the classification of simple superconformal symmetries, see there:

It had already been observed by Bergshoeff, Duff, Pope & Sezgin and Duff & Sutton 1988 (see also Duff 1998, Duff 1999) 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 Claus, Kallosh & Proeyen 1997, DGGGTT 98, Claus-Kallosh, Kumar & Townsend 1998, Pasti, Sorokin & Tonin 1999. See also at super p-brane – As part of the AdS-CFT correspondence

More recently, 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 (Beisert et al. 10, Escobedo 12), see the section Checks below.

Thus regarded as a duality in string theory, the AdS/CFT correspondence is an incarnation of open/closed string duality, reflecting the fact that the physics on D-branes has two equivalent descriptions:

1) as a Yang-Mills-gauge theory coming from open strings attached to the brane

2) as a gravity theory coming from closed strings emitted/absorbed by the brane.

graphics grabbed from Schomerus 07, Figure 4, see also e.g. Peschanskia 09, Figure 1

This gives a vivid intuitive picture of the mechanism underlying the correspondence: An excitation of the gauge field on the brane goes along with an excitation of the field of gravity around the brane, and either is faithfully reflected in the other; at least in the suitable limits.

Small-$N$ corrections

The AdS/CFT correspondence has been widely discussed and is mostly understood by default only in the large $N$ limit and for large 't Hooft coupling, where the given gauge theory is dual to plain classical supergravity, which stands out as being particularly tractable and well-understood.

But it is expected (cf. AGMOO99 p. 60, Chester 2018, Chester & Perlmutter 2018 p. 2) that the duality still applies in the opposite large 1/N limit, now involving on the gravity-side corrections

1. from perturbative string theory (for small 't Hooft coupling, there are some checks of such stringy corrections) and

2. from putative M-theory (for the full non-perturbative large 1/N limit, which remains largely unexplored):

$\,$

(graphics adapted from SS22)

Notice that for real-world applications such as to the confinement/mass gap-problem of quantum chromodynamics, the value of $N$ typically is indeed small (the number of colors in quantum chromodynamics is $N_c = 3$) so that the string theory/M-theory-corrections to the AdS/QCD correspondence are going to be crucial for the full discussion of these applications:

In lack of a full formulation of M-theory (see M-theory – The open problem) approximate forms of the AdS/CFT correspondence away from the case of conformal invariance, supersymmetry, large N limit and/or exact anti de Sitter geometry are being argued to be of use for understanding quantum chromodynamics (for instance the quark-gluon plasma (Policastro-Son-Starinets 01, but most notably confined hadron-spectra – the AdS/QCD correspondence) and for various models in solid state physics (the AdS-CFT in condensed matter physics, see e.g. Hartnoll-Lucas-Sachdev 16).

More in detail, since the near horizon geometry of 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 approximately anti de Sitter spacetime (see for instance Kaloper 04, Flachi-Minamitsuji 09). The basic structure is hence that of Randall-Sundrum models, but details differ, such as notably in warped throat geometries, see Uranga 05, section 18.

These warped throat models go back to Klebanov-Strassler 00 which discusses aspects of confinement in Yang-Mills theory on conincident ordinary and fractional D3-branes at the singularity of a warped conifold. See also Klebanov-Witten 98

snippet grabbed from Uranga 05, section 18

here: “RS”=Randall-Sundrum model; “KS”=Klebanov-Strassler 00

In particular this means that AdS-CFT duality applies in some approximation to intersecting D-brane models (e.g. Soda 10, 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 $\sim$ AdS (Klebanov-Strassler 00, section 6)

Matching single trace observables to string excitation

The single trace operators/observables in conformal field theories such as super Yang-Mills theories play a special role in the AdS-CFT correspondence: They correspond to single string excitations on the AdS-supergravity side of the correspondence, where, curiously, the “string of characters/letters” in the argument of the trace gets literally mapped to a superstring in spacetime (see the references below).

From Polyakov 02, referring to gauge fields and their single trace operators as letter and words, respectively:

The picture which slowly arises from the above considerations is that of the space-time gradually disappearing in the regions of large curvature. The natural description in this case is provided by a gauge theory in which the basic objects are the texts formed from the gauge-invariant words. The theory provides us with the expectation values assigned to the various texts, words and sentences.

These expectation values can be calculated either from the gauge theory or from the strongly coupled 2d sigma model. The coupling in this model is proportional to the target space curvature. This target space can be interpreted as a usual continuous space-time only when the curvature is small. As we increase the coupling, this interpretation becomes more and more fuzzy and finally completely meaningless.

From Berenstein-Maldacena-Nastase 02, who write $Z$ for the elementary field observables (“letters”) $\mathbf{\Phi}$ above:

In summary, the “string of $Z$s” becomes the physical string and that each $Z$ carries one unit of $J$ which is one unit of $p_+$. Locality along the worldsheet of the string comes from the fact that planar diagrams allow only contractions of neighboring operators. So the Yang Mills theory gives a string bit model where each bit is a $Z$ operator.

On the CFT side these BMN operators of fixed length (of “letters”) are usefully identified as spin chains which, with the dilatation operator regarded as their Hamiltonian, are integrable systems (Minahan-Zarembo 02, Beisert-Staudacher 03).

This integrability allows a detailed matching between

• single trace operators/BMN operators in D=4 N=4 super Yang-Mills theory

• the classical Green-Schwarz superstring on AdS5 $\times$ S5

under AdS/CFT duality (Beisert-Frolov-Staudacher-Tseytlin 03, …). For review see BBGK 04, Beisert et al. 10.

(…)

Checks

At the heart of the duality is the observation that the classical action functionals for various fields coupled to Einstein gravity on anti de Sitter spacetime are, when expressed as functions of the asymptotic boundary values of the 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 Witten 98, section 2.4 “Some sample computation”. These follow from elementary manipulation in differential geometry (involving neither supersymmetry nor string theory). A good exposition is in Hartnoll-Lucas-Sachdev 16, Section 1.6

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 Beisert et al. 10, Escobedo 12

graphics grabbed from Escobedo 12

Comparison to string scattering amplitudes beyond the planar SCFT limit: ABP 18.

Numerical checks using lattice gauge theory are reviewed in Joseph 15.

Exact duality checks pertaining to the full stringy regime for $AdS_3/CFT_2$: Eberhardt-Gaberdiel 19a, Eberhardt-Gaberdiel 19b, Eberhardt-Gaberdiel-Gopakumar 19. For more see the references there.

See also

• Umut Gursoy, Guim Planella Planas, Worldsheet from worldline [arXiv:2311.10142]

Examples

The solutions to supergravity that preserve the maximum of 32 supersymmetries are (e.g. HEGKS 08 (1.1))

• $AdS_5 \times S^5$ in type II supergravity

• $AdS_7 \times S^4$ in 11-dimensional supergravity

• $AdS_4 \times S^7$ in 11-dimensional supergravity

as well as their Minkowski spacetime and plane wave limits. These are the main KK-compactifications for the following examples-

$AdS_5 / CFT_4$ – Horizon limit of D3-branes

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

(Maldacena 97, section 2)

(Aharony-Gubser-Maldacena-Ooguri-Oz 99, section 3 and 4)

$AdS_7 / CFT_6$ – Horizon limit of M5-branes

We list some of the conjectured statements and their evidence concerning the case of $AdS_7/CFT_6$-duality.

The hypothesis of Maldacena 1997, section 3.1 (for review see (Aharony, Gubser, Maldacena, Ooguri & Oz 1999, section 6.1.1) is that

• the 6d (2,0)-superconformal QFT on the worldvolume of $N$ coincident M5-branes

is holographically related to

• 11-dimensional supergravityreduced 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.

In

• Edward Witten, Five-Brane Effective Action In M-Theory J. Geom. Phys.22:103-133,1997 (arXiv:hep-th/9610234)

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 Hopkins-Singer 02 and hence to the further mathematical development of differential cohomology and its application in physics.)

In (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

But in fact the quantum anomaly cancellation (GS-type mechanism) for 11d sugra introduces a quantum correction to this Chern-Simons term (DLM, equation (3.14)), making it locally become

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 nonabelian higher dimensional Chern-Simons theory should be dual to the nonabelian 6d (2,0)-superconformal QFT was maybe first said explicitly in (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 (see 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 Sezgin &Sundell 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 (super) conformal group $SO(6,2)$.

A non-perturbative description of this nonabelian 7d Chern-Simons theory as a local prequantum field theory (hence defined non-perturbatively on the global moduli stack of fields (twisted differential string structures, in fact)) was discussed in (FSS 12a, FSS 12b).

General discussion of boundary local prequantum field theories relating higher Chern-Simons-type and higher WZW-type theories is in (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 higher WZW-type boundary theory of a 7d Chern-Simons theory is found in (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 (Freed 12).

$AdS_4 / CFT_3$ –Horizon limit of M2-branes

11d supergravity/M-theory on the asymptotic $AdS_4$

spacetime of an M2-brane.

(Maldacena 97, section 3.2, Aharony-Gubser-Maldacena-Ooguri-Oz 99, section 6.1.2, Klebanov-Torri 10)

$AdS_3 / CFT_2$ – Horizon limit of D$p$-D($p+k$) brane bound states

(for more see at AdS3-CFT2 and CS-WZW correspondence)

D1-D5 brane system in type IIB string theory

(Maldacena 97, section 4)

(Aharony-Gubser-Maldacena-Ooguri-Oz 99, section 5)

D6-D8 brane bound state with D2-D4 brane bound state defects in massive type IIA string theory

(Dibitetto-Petri 17, …)

D4-D8 brane bound state with D2-D6 brane bound state defects in massive type IIA string theory

(Dibitetto-Petri 18, …)

$AdS_2 / CFT_1$

see at nearly AdS2/CFT1

Non-conformal duals

Horizon limit of $Dp$-branes for arbitrary $p$

(Aharony-Gubser-Maldacena-Ooguri-Oz 99, section 6.1.3)

Horizon limit of NS5-brane

(Aharony-Gubser-Maldacena-Ooguri-Oz 99, section 6.1.4)

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.

(Aharony-Gubser-Maldacena-Ooguri-Oz 99, section 6.2)

See the References – Applications – In condensed matter physics.

Further gauge theories induced by compactification and twisting

Formalizations

The full formalization of AdS/CFT is still very much out of reach, but maybe mostly for lack of trying.

But see 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.

Related concepts

• near-horizon geometry

• Fefferman-Graham ambient construction

• holographic entanglement entropy, ER = EPR

• single trace operator

• duality in physics, duality in string theory

  • T-duality, S-duality, U-duality

  • open/closed string duality

    • KLT relations

• black hole in anti de Sitter spacetime

• S-matrix theory

• anti de Sitter gravity

• Randall-Sundrum model

• Sachdev-Ye-Kitaev model

• flat space holography

• p-adic AdS-CFT

• dS/CFT correspondence?

References

Long before the modern formulation of the AdS/CFT correspondence, referenced below at

• The origin of modern AdS/CFT duality

there were developments that must be regarded at least as precursors, referenced at

• Polyakov gauge-string duality

• Microscopic AdS-CFT via p-brane sigma-models

The origin of modern AdS/CFT duality

The modern guise of AdS/CFT duality as a conjectured duality between string/M-theory near black branes and superconformal field theories on the worldvolume of these branes, at least in a suitable large-N limit, is due to:

• Juan Maldacena, The Large $N$ limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [hep-th/9711200, doi:10.4310/ATMP.1998.v2.n2.a1, inspire:451647]

• Juan Maldacena, Wilson loops in Large $N$ field theories, Phys. Rev. Lett. 80 (1998) 4859 [hep-th/9803002, doi:10.1103/PhysRevLett.80.4859]

An actual rule for matching, under this duality, bulk states to generating functions for boundary correlators/n-point functions was then given by

• Steven Gubser, Igor Klebanov, Alexander Polyakov, around (12) of: Gauge theory correlators from non-critical string theory, Physics Letters B 428 105-114 (1998) [hep-th/9802109, doi:10.1016/S0370-2693(98)00377-3]

• Edward Witten, around (2.11) of Anti-de Sitter space and holography, Advances in Theoretical and Mathematical Physics 2 (1998) 253-291 [hep-th/9802150, doi:10.4310/ATMP.1998.v2.n2.a2, spire:467400]

now often called “the holographic disctionary” or just “the dictionary” (e.g. Taylor 2023).

See also:

• Wikipedia, AdS/CFT correspondence

• Tom Banks, Michael Douglas, Gary Horowitz, Emil Martinec, AdS Dynamics from Conformal Field Theory (arXiv:hep-th/9808016, spire:474214)

• Carlos Andrés Cardona Giraldo, Correlation functions in AdS/CFT correspondence, 2012 (spire:1652794, pdf)

Sketch of a derivation of AdS/CFT:

• Horatiu Nastase, Towards deriving the AdS/CFT correspondence (arXiv:1812.10347)

• Ofer Aharony, Shai Chester, Erez Urbach, A Derivation of AdS/CFT for Vector Models (arXiv:2011.06328)

See also:

• Edward Witten, Three-dimensional gravity revisited, arxiv/0706.3359

• C. R. Graham, Edward Witten, Conformal anomaly of submanifold observables in AdS/CFT correspondence, hepth/9901021.

• Edward Witten, AdS/CFT Correspondence And Topological Field Theory (arXiv:hep-th/9812012)

Via exceptional field theory:

• Oscar Varela: Super-Chern-Simons spectra from Exceptional Field Theory, J. High Energ. Phys. 2021 283 (2021) [arXiv:2010.09743, doi:10.1007/JHEP04(2021)283]

Introductions and surveys

• Edward Witten, Baryons and Branes in Anti de Sitter Space talk at String98 (1998) [web]

  (by the title, apparently originally intended to touch on AdS-QCD, but de facto ending with focus on the problem of flat space holography)

• Alexander Polyakov, The wall of the cave, Int. J. Mod. Phys. A 14 (1999) 645-658 [arXiv:hep-th/9809057, doi:10.1142/S0217751X99000324]

  (early account with focus on AdS-QCD duality)

• John Schwarz, Introduction to M Theory and AdS/CFT Duality, in: Quantum Aspects of Gauge Theories, Supersymmetry and Unification Lecture Notes in Physics 525, Springer (1999) [arXiv:hep-th/9812037, doi:10.1007/BFb0104239]

  (early review with an eye towards M-theory)

• Jens L. Petersen, Introduction to the Maldacena Conjecture on AdS/CFT, Int. J. Mod. Phys. A 14 (1999) 3597-3672 [hep-th/9902131, doi:10.1142/S0217751X99001676]

• Ofer Aharony, Steven Gubser, Juan Maldacena, Hirosi Ooguri, Yaron Oz, Large $N$ Field Theories, String Theory and Gravity, Phys. Rept. 323 183-386 (2000) [doi:10.1016/S0370-1573(99)00083-6, arXiv:hep-th/9905111]

• Jan de Boer, Introduction to AdS/CFT correspondence, talk at SUSY02 (2002) [pdf, pdf]

• Michael T. Anderson, Geometric aspects of the AdS/CFT correspondence [arXiv:hep-th/0403087]

• Christopher P. Herzog, Igor R. Klebanov: AdS/CFT Correspondence, Encyclopedia of Mathematical Physics, Academic Press (2006) 174-183 [doi:10.1016/B0-12-512666-2/00245-5, pdf]

• Horatiu Nastase, Introduction to AdS-CFT [arXiv:0712.0689]

• Horatiu Nastase, Introduction to AdS/CFT correspondence, Cambridge University Press (2015) [cds:1984145, doi:10.1017/CBO9781316090954]

• Gary Horowitz, Joseph Polchinski, Gauge/gravity duality [gr-qc/0602037]

• Alberto Zaffaroni, Introduction to the AdS/CFT correspondence, LACES lectures (2009) [pdf, pdf]

• Hans Günter Dosch, A Practical Guide to AdS/CFT, lecture notes (2009) [pdf, pdf]

  (with aspects of holographic QCD)

• Joseph Polchinski, Introduction to Gauge/Gravity Duality (arXiv:1010.6134)

• Christopher Herzog, AdS/CFT, lecture notes (2015) [pdf, pdf]

• Makoto Natsuume, AdS/CFT Duality User Guide, Lecture Notes in Physics 903, Springer (2015) [arXiv:1409.3575, doi:10.1007/978-4-431-55441-7, webpage]

• Sebastian De Haro, Daniel R. Mayerson, Jeremy Butterfield, Conceptual Aspects of Gauge/Gravity Duality, Foundations of Physics 46 11 (2016) 1381-1425 [arXiv:1509.09231, doi:10.1007/s10701-016-0037-4]

• Johanna Erdmenger, Introduction to Gauge/Gravity Duality, PoS TASI2017 (2018) 001 [arXiv:1807.09872, doi:10.22323/1.305.0001]

  (focus on holographic entanglement entropy and AdS/CMT duality)

• Nirmalya Kajuri, ST4 Lectures on Bulk Reconstruction [arXiv:2003.00587]

• Francesco Benini: Brief Introduction to AdS/CFT SISSA lecture (2022) [pdf, pdf]

• Marika Taylor, Introduction to AdS/CFT and Holography – The holographic dictionary, lecture at pre-SUSY school 2023 [cern:1214657, slides: pdf, video: session 1, sessions 2, 3]

Review of Yangian symmetry:

• Alessandro Torrielli, Yangians, S-matrices and AdS/CFT, J. Phys. A 44 263001 (2011) [arXiv:1104.2474]

Lattice gauge theory computations

Review of lattice gauge theory-numerics for the AdS-CFT correspondence:

• Anosh Joseph, Review of Lattice Supersymmetry and Gauge-Gravity Duality (arXiv:1509.01440)

Using the KK-compactification of D=4 N=4 super Yang-Mills theory to the BMN matrix model for lattice gauge theory-computations in D=4 N=4 SYM and for numerical checks of the AdS-CFT correspondence:

• Masazumi Honda, Goro Ishiki, Sang-Woo Kim, Jun Nishimura, Asato Tsuchiya, Direct test of the AdS/CFT correspondence by Monte Carlo studies of N=4 super Yang-Mills theory, JHEP 1311 (2013) 200 (arXiv:1308.3525)

On single trace operators

The correspondence of single trace operators to superstring excitations under the AdS-CFT correspondence originates with these articles:

• Alexander Polyakov, Gauge Fields and Space-Time, Int. J. Mod. Phys. A17S1 (2002) 119-136 [arXiv:hep-th/0110196, doi:10.1142/S0217751X02013071]

• David Berenstein, Juan Maldacena, Horatiu Nastase, Strings in flat space and pp waves from $\mathcal{N} = 4$ Super Yang Mills, JHEP 0204 (2002) 013 (arXiv:hep-th/0202021)

  (whence “BMN operators”)

• Steven Gubser, Igor Klebanov, Alexander Polyakov, A semi-classical limit of the gauge/string correspondence, Nucl. Phys. B 636 (2002) 99-114 [arXiv:hep-th/0204051, doi:10.1016/S0550-3213(02)00373-5]

• Martin Kruczenski, Spiky strings and single trace operators in gauge theories, JHEP 0508:014, 2005 (arXiv:hep-th/0410226)

The identification of the relevant single trace operators with integrable spin chains is due to

• J. A. Minahan, Konstantin Zarembo, The Bethe-Ansatz for $N=4$ Super Yang-Mills, JHEP 0303 (2003) 013 (arXiv:hep-th/0212208)

• Niklas Beisert, Matthias Staudacher, The $\mathcal{N}=4$ SYM Integrable Super Spin Chain,

  Nucl. Phys. B670:439-463, 2003 (arXiv:hep-th/0307042)

which led to more detailed matching of single trace operators to rotating string excitations in

• Niklas Beisert, Sergey Frolov, Matthias Staudacher, Arkady Tseytlin, Precision Spectroscopy of AdS/CFT, JHEP 0310:037, 2003 (arXiv:hep-th/0308117)

Review includes

• A. V. Belitsky, Volker Braun, A. S. Gorsky, G. P. Korchemsky, Integrability in QCD and beyond, Int. J. Mod. Phys. A19:4715-4788, 2004 (arXiv:hep-th/0407232)

• Niklas Beisert, Luis Alday, Radu Roiban, Sakura Schafer-Nameki, Matthias Staudacher, Alessandro Torrielli, Arkady Tseytlin, et. al., Review of AdS/CFT Integrability: An Overview, Lett. Math. Phys. 99, 3 (2012) (arXiv:1012.3982)

On $AdS_3 / CFT_2$

(For more see the references at AdS3/CFT2.)

An exact correspondence of the symmetric orbifold CFT of Liouville theory with a string theory on $AdS_3$ is claimed in:

• Lorenz Eberhardt, Matthias Gaberdiel, String theory on $AdS_3$ and the symmetric orbifold of Liouville theory (arXiv:1903.00421)

• Lorenz Eberhardt, Matthias Gaberdiel, Strings on $AdS_3 \times S^3 \times S^3 \times S^1$ (arXiv:1904.01585)

• Lorenz Eberhardt, Matthias Gaberdiel, Rajesh Gopakumar, Deriving the $AdS_3/CFT_2$ Correspondence (arXiv:1911.00378)

• Andrea Dei, Lorenz Eberhardt, Correlators of the symmetric product orbifold (arXiv:1911.08485)

based on

• Shouvik Datta, Lorenz Eberhardt, Matthias Gaberdiel, Stringy $\mathcal{N} = (2,2)$ holography for $AdS_3$ JHEP 1801 (2018) 146 (arXiv:1709.06393)

See also

• Stefano Speziali, Spin 2 fluctuations in 1/4 BPS AdS3/CFT2 (arxiv:1910.14390)

• Lorenz Eberhardt, $AdS_3/CFT_2$ at higher genus (arXiv:2002.11729)

• Lorenz Eberhardt, Summing over Geometries in String Theory (arXiv:2102.12355)

On black$\;$D6-D8-brane bound states in massive type IIA string theory, with defect D2-D4-brane bound states inside them realizing AdS3-CFT2 “inside” AdS7-CFT6:

• Giuseppe Dibitetto, Nicolò Petri, 6d surface defects from massive type IIA, JHEP 01 (2018) 039 (arxiv:1707.06154)

• Nicolò Petri, section 6.5 of: Supersymmetric objects in gauged supergravities (arxiv:1802.04733)

• Nicolò Petri, Surface defects in massive IIA, talk at Recent Trends in String Theory and Related Topics 2018 (pdf)

• Giuseppe Dibitetto, Nicolò Petri, $AdS_3$ vacua and surface defects in massive IIA (arxiv:1904.02455)

• Yolanda Lozano, Niall T. Macpherson, Carlos Nunez, Anayeli Ramirez, $1/4$ BPS $AdS_3/CFT_2$ (arxiv:1909.09636)

• Yolanda Lozano, Niall T. Macpherson, Carlos Nunez, Anayeli Ramirez, Two dimensional $N=(0,4)$ quivers dual to $AdS_3$ solutions in massive IIA (arxiv:1909.10510)

• Yolanda Lozano, Niall T. Macpherson, Carlos Nunez, Anayeli Ramirez, $AdS_3$ solutions in massive IIA, defect CFTs and T-duality (arxiv:1909.11669)

• Kostas Filippas, Non-integrability on $AdS_3$ supergravity (arxiv:1910.12981)

On black$\;$D4-D8-brane bound states in massive type IIA string theory, with defect D2-D6-brane bound states inside them realizing AdS3-CFT2 “inside” AdS7-CFT6:

• Giuseppe Dibitetto, Nicolò Petri, Surface defects in the D4 − D8 brane system, JHEP 01 (2019) 193 (arxiv:1807.07768)

• Giuseppe Dibitetto, Nicolò Petri, $AdS_3$ vacua and surface defects in massive IIA (arxiv:1904.02455)

On $AdS_4 / CFT_3$

• Igor Klebanov, Giuseppe Torri, M2-branes and AdS/CFT, Int.J.Mod.Phys.A25:332-350,2010 (arXiv:0909.1580)

• Kazuo Hosomichi, M2-branes and AdS/CFT: A Review (arXiv:2003.13914)

• Silvia Penati, Exact Results in AdS4/CFT3 (arXiv:2004.00841)

• Manikantt Mummalaneni: AdS4/CFT3: ABJM Theory, Brane Geometry, Correlators and Mellin Space [arXiv:2408.11835]

On $AdS_5 / CFT_4$

• Beisert et. al. 10

• Jorge Escobedo, Integrability in AdS/CFT: Exact Results for Correlation Functions, 2012 (spire:1264432)

Computing dual string scattering amplitudes by AdS/CFT beyond the planar limit:

• Luis Alday, Agnese Bissi, Eric Perlmutter, Genus-One String Amplitudes from Conformal Field Theory, JHEP06(2019) 010 (arXiv:1809.10670)

On $AdS_7 / CFT_6$

We list references specific to $AdS_7/CFT_6$.

In

• Edward Witten, Five-Brane Effective Action In M-Theory J. Geom. Phys.22:103-133,1997 (arXiv:hep-th/9610234)

• Edward Witten, AdS/CFT Correspondence And Topological Field Theory JHEP 9812:012,1998 (arXiv:hep-th/9812012)

it is argued that the conformal blocks of the 6d (2,0)-superconformal QFT are entirely controled just by the effective 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 nonabelian higher dimensional Chern-Simons theory in $d = 7$, due the quantum anomaly cancellation, is clear from the original source, equation (3.14) of

• Michael Duff, James Liu, Ruben Minasian, Eleven Dimensional Origin of String/String Duality: A One Loop Test (arXiv:hep-th/9506126)

but seems not to be noted explicitly in the context of $AdS_7/CFT_6$ before the references

• H. Lü, Yi Pang, Seven-Dimensional Gravity with Topological Terms Phys.Rev.D81:085016 (2010) (arXiv:1001.0042)

• H. Lu, Zhao-Long Wang, On M-Theory Embedding of Topologically Massive Gravity Int.J.Mod.Phys.D19:1197 (2010) (arXiv:1001.2349)

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

• Alexei Nurmagambetov, I. Y. Park, On the M5 and the AdS7/CFT6 Correspondence, Phys. Lett. B524 (2002) 185-191 (arXiv:hep-th/0110192)

See also

• M. Nishimura, Y. Tanii, Local Symmetries in the $AdS_7/CFT_6$ Correspondence, Mod. Phys. Lett. A14 (1999) 2709-2720 (arXiv:hep-th/9910192)

Discussion of the $CFT_6$ in $AdS_7/CFT_6$ via conformal bootstrap:

• Shai Chester, Bootstrapping M-theory, PhD thesis, Princeton (2018) [pdf, ark:88435/dsp01kw52jb833]

• Nathan B. Agmon, Shai Chester, Silviu S. Pufu: Solving M-theory with the Conformal Bootstrap, J. High Energ. Phys. 2018 159 (2018). [arXiv:1711.07343, doi:10.1007/JHEP06(2018)159]

• Shai Chester, Eric Perlmutter: M-Theory Reconstruction from $(2,0)$ CFT and the Chiral Algebra Conjecture, J. High Energ. Phys. 2018 116 (2018) [arXiv:1805.00892, doi:10.1007/JHEP08(2018)116]

• Luis Alday, Shai Chester, Himanshu Raj, 6d $(2,0)$ and M-theory at 1-loop (arXiv:2005.07175)

In

• Ergin Sezgin, P. Sundell, Massless Higher Spins and Holography (hep-th/0205131)

arguments are given that the 7d theory is a higher spin gauge theory extension of $SO(6,2)$.

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 Randall-Sundrum model):

• Igor Klebanov, Matthew Strassler, Supergravity and a Confining Gauge Theory: Duality Cascades and $\chi^{SB}$-Resolution of Naked Singularities, JHEP 0008:052, 2000 (arXiv:hep-th/0007191)

• Igor Klebanov, Edward Witten, Superconformal Field Theory on Threebranes at a Calabi-Yau Singularity, Nucl.Phys.B536:199-218, 1998 (arXiv:hep-th/9807080)

• Nemanja Kaloper, Origami World, JHEP 0405 (2004) 061 (arXiv:hep-th/0403208)

• Angel Uranga, section 18 of TASI lectures on String Compactification, Model Building, and Fluxes, 2005 (pdf)

• Antonino Flachi, Masato Minamitsuji, Field localization on a brane intersection in anti-de Sitter spacetime, Phys.Rev.D79:104021, 2009 (arXiv:0903.0133)

• Jiro Soda, AdS/CFT on the brane, Lect.Notes Phys.828:235-270, 2011 (arXiv:1001.1011)

• Shunsuke Teraguchi, around slide 21 String theory and its relation to particle physics, 2007 (pdf)

• Gianluca Grignani, Troels Harmark, Andrea Marini, Marta Orselli, The Born-Infeld/Gravity Correspondence, Phys. Rev. D 94, 066009 (2016) (arXiv:1602.01640)

Applications to physics

To gravity

Discussion of event horizons of black holes in terms of AdS/CFT (the “firewall problem”) is in

• Kyriakos Papadodimas, Suvrat Raju, An Infalling Observer in AdS/CFT (arXiv:1211.6767)

To black hole interiors:

• Juan Maldacena, Toy models for black holes II, talk at PiTP 2018 From QBits to spacetime (recording)

The SYK model gives us a glimpse into the interior of an extremal black hole…That’s the feature of SYK that I find most interesting…It is a feature this model has, that I think no other model has

To symmetries in gravity:

• Daniel Harlow, Hirosi Ooguri, Constraints on symmetry from holography, Phys. Rev. Lett. 122, 191601, 2019 (arXiv:1810.05337, doi:10.1103/PhysRevLett.122.191601)

To the quark-gluon plasma

Applications of AdS-CFT to the quark-gluon plasma of QCD:

Expositions and reviews:

• Pavel Kovtun, Quark-Gluon Plasma and String Theory, RHIC news (2009) (blog entry)

• Makoto Natsuume, String theory and quark-gluon plasma (arXiv:hep-ph/0701201)

• Steven Gubser, Using string theory to study the quark-gluon plasma: progress and perils (arXiv:0907.4808)

• Francesco Biagazzi, A. l. Cotrone, Holography and the quark-gluon plasma, AIP Conference Proceedings 1492, 307 (2012) (doi:10.1063/1.4763537, slides pdf)

• Brambilla et al., section 9.2.2 of QCD and strongly coupled gauge theories - challenges and perspectives, Eur Phys J C Part Fields. 2014; 74(10): 2981 (arXiv:1404.3723, doi:10.1140/epjc/s10052-014-2981-5)

Holographic discussion of the shear viscosity of the quark-gluon plasema goes back to

• Giuseppe Policastro, D.T. Son, A.O. Starinets, Shear viscosity of strongly coupled N=4 supersymmetric Yang-Mills plasma, Phys. Rev. Lett.87:081601, 2001 (arXiv:hep-th/0104066)

Other original articles include:

• Hovhannes R. Grigoryan, Paul M. Hohler, Mikhail A. Stephanov, Towards the Gravity Dual of Quarkonium in the Strongly Coupled QCD Plasma (arXiv:1003.1138)

• Brett McInnes, Holography of the Quark Matter Triple Point (arXiv:0910.4456)

To particle physics

• Joseph Polchinski, Matthew Strassler, Hard Scattering and Gauge/String Duality, Phys. Rev. Lett. 88:031601, 2002, (arXiv:hep-th/0109174)

For more see at AdS/QCD correspondence.

To fluid dynamics

Application to fluid dynamics – see also at fluid/gravity correspondence:

• Sayantani Bhattacharyya, Veronika Hubeny, Shiraz Minwalla, Mukund Rangamani, Nonlinear Fluid Dynamics from Gravity, JHEP 0802:045, 2008 (arXiv:0712.2456)

To condensed matter physics

On AdS-CFT in condensed matter physics:

Textbook account

• Sean Hartnoll, Andrew Lucas, Subir Sachdev, Holographic quantum matter, MIT Press 2018 (arXiv:1612.07324, publisher)

Further reviews include the following:

• A S T Pires, Ads/CFT correspondence in condensed matter (arXiv:1006.5838)

• Subir Sachdev, Condensed matter and AdS/CFT (arXiv:1002.2947)

• Yuri V. Kovchegov, AdS/CFT applications to relativistic heavy ion collisions: a brief review (arXiv:1112.5403)

• Alberto Salvio, Superconductivity, Superfluidity and Holography (arXiv:1301.0201)

• Holography and Extreme Chromodynamics, Santiago de Compostela, July 2018

On holographic superconductors via AdS4/CFT3 duality (hence with the bulk theory being 11d supergravity aka M-theory):

The original article (according to ZLSS15, p. ix), regarding AdS4/CFT3-duality (hence with the bulk theory being 11d supergravity, whence “M-theory”) as a tool for understanding strongly coupled condensed matter physics (specifically superconductivity):

• Christopher P. Herzog, Pavel Kovtun, Subir Sachdev, Dam Thanh Son, Quantum critical transport, duality, and M-theory, Phys. Rev. D 75 (2007) 085020 [arXiv:hep-th/0701036, doi:10.1103/PhysRevD.75.085020]

and further discussion:

• Jerome P. Gauntlett, Julian Sonner, Toby Wiseman: Holographic superconductivity in M-Theory, J. High Energ. Phys. 2010 60 (2010) [arXiv:0907.3796, doi:10.1007/JHEP02(2010)060]

• Steven S. Gubser, Silviu S. Pufu, Fabio D. Rocha: Quantum critical superconductors in string theory and M-theory, Phys. Lett. B 683 (2010) 201-204 [arXiv:0908.0011, doi:10.1016/j.physletb.2009.12.017]

• Jerome Gauntlett, Julian Sonner, Toby Wiseman: Quantum Criticality and Holographic Superconductors in M-theory, J. High Energ. Phys. 2010 60 (2010) [arXiv:0912.0512, doi:10.1007/JHEP02(2010)060]

• Aristomenis Donos, Jerome P. Gauntlett, Julian Sonner, Benjamin Withers: Competing orders in M-theory: superfluids, stripes and metamagnetism, J. High Energ. Phys. 2013 108 (2013) [arXiv:1212.0871, doi:10.1007/JHEP03(2013)108]

• Aristomenis Donos, Jerome P. Gauntlett, Christiana Pantelidou: Semi-local quantum criticality in string/M-theory, J. High Energ. Phys. 2013 103 (2013) [doi:10.1007/JHEP03(2013)103, arXiv:1212.1462]

• Yu-Sen An, Li Li, Fu-Guo Yang, Run-Qiu Yang: Interior Structure and Complexity Growth Rate of Holographic Superconductor from M-Theory, J. High Energ. Phys. 2022 133 (2022) [arXiv:2205.02442, doi:10.1007/JHEP08(2022)133]

Applications in mathematics

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 on hyperbolic 3-manifolds:

• Dongmin Gang, Nakwoo Kim, Sangmin Lee, Section 3.2_Holography of 3d-3d correspondence at Large $N$, JHEP04(2015) 091 (arXiv:1409.6206)

• Dongmin Gang, Nakwoo Kim, around (21) of: Large $N$ twisted partition functions in 3d-3d correspondence and Holography, Phys. Rev. D 99, 021901 (2019) (arXiv:1808.02797)

To deep learning in neural networks

On the deep learning algorithm on neural networks as analogous to the AdS/CFT correspondence:

• Yi-Zhuang You, Zhao Yang, Xiao-Liang Qi, Machine Learning Spatial Geometry from Entanglement Features, Phys. Rev. B 97, 045153 (2018) (arxiv:1709.01223)

• W. C. Gan and F. W. Shu, Holography as deep learning, Int. J. Mod. Phys. D 26, no. 12, 1743020 (2017) (arXiv:1705.05750)

• J. W. Lee, Quantum fields as deep learning (arXiv:1708.07408)

• Koji Hashimoto, Sotaro Sugishita, Akinori Tanaka, Akio Tomiya, Deep Learning and AdS/CFT, Phys. Rev. D 98, 046019 (2018) (arxiv:1802.08313)

Philosophy of AdS/CFT

• Radin Dardashti, Richard Dawid, Sean Gryb, Karim Thébault (2020). On the Empirical Consequences of the AdS/CFT Duality. In Nick Huggett, Keizo Matsubara, and Christian Wüthrich (Eds.), Beyond Spacetime: The Foundations of Quantum Gravity (pp. 284-303). Cambridge: Cambridge University Press. (doi:10.1017/9781108655705.016, arXiv:1810.00949)