nLab
AdS-CFT

Contents

Context

Quantum field theory

Phyics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Contents

Idea

The AdS-CFT correspondence is at its basis 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, 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 conjecturally extends to a more general identification of states of gravity (quantum gravity) on asymptotically anti de Sitter spacetimes of dimension d+1d+1 with correlators/n-point functions of conformal field theories on the asymptotic boundary of dimension dd (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), 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 NN coincident 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 symmetries, see there.

ddNNsuperconformal super Lie algebraR-symmetryblack brane worldvolume
superconformal field theory
via AdS-CFT
A3A\phantom{A}3\phantom{A}A2k+1A\phantom{A}2k+1\phantom{A}AB(k,2)\phantom{A}B(k,2) \simeq osp(2k+1/4)A(2k+1/4)\phantom{A}ASO(2k+1)A\phantom{A}SO(2k+1)\phantom{A}
A3A\phantom{A}3\phantom{A}A2kA\phantom{A}2k\phantom{A}AD(k,2)\phantom{A}D(k,2)\simeq osp(2k/4)A(2k/4)\phantom{A}ASO(2k)A\phantom{A}SO(2k)\phantom{A}M2-brane
3d superconformal gauge field theory
A4A\phantom{A}4\phantom{A}Ak+1A\phantom{A}k+1\phantom{A}AA(3,k)𝔰𝔩(4/k+1)A\phantom{A}A(3,k)\simeq \mathfrak{sl}(4/k+1)\phantom{A}AU(k+1)A\phantom{A}U(k+1)\phantom{A}D3-brane
4d superconformal gauge field theory
A5A\phantom{A}5\phantom{A}A1A\phantom{A}1\phantom{A}AF(4)A\phantom{A}F(4)\phantom{A}ASO(3)A\phantom{A}SO(3)\phantom{A}
A6A\phantom{A}6\phantom{A}AkA\phantom{A}k\phantom{A}AD(4,k)\phantom{A}D(4,k) \simeq osp(8/2k)A(8/2k)\phantom{A}ASp(k)A\phantom{A}Sp(k)\phantom{A}M5-brane
6d superconformal gauge field theory

(Shnider 88, also Nahm 78, see Minwalla 98, section 4.2)

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 a AdS/CFT correspondence was proposed in (Maldacena 97a, 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 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 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 fields on the D-brane, while the excitations of the closed string include the gravitons witnessing the 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 (Duff-Sutton 88, Duff 98) 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 97, DGGGTT 98, Claus-Kallosh-Kumar-Townsend 98, Pasti-Sorokin-Tonin 99. See also at 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×S 5AdS_5 \times S^5, fine detailed checks of the correspondence have been performed (Beisert et al. 10, Escobedo 12), see the section 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 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 (Policastro-Son-Starinets 01) and for various models in solid state physics (see Hartnoll-Lucas-Sachdev 16 and see at AdS-CFT in condensed matter physics).

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 nn-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)

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 (Policastro-Son-Starinets 01). For more on this approximate relation see at AdS-QCD correspondence.

Examples

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

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

AdS 5/CFT 4AdS_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 6AdS_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 6AdS_7/CFT_6-duality.

The hypothesis (Maldacena 97, section 3.1) (see (Aharony-Gubser-Maldacena-Ooguri-Oz 99, section 6.1.1) for a review) is that

is holographically related to

In

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

S 11dSUGRA,CS(C 3) = AdS 7 S 4C 3G 4G 4 =N AdS 7C 3G 4. \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} \,.

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

S(ω,C 3) = AdS 7 S 4C 3G 4(G 4+I 8(ω)) =N AdS 7(C 3G 4+148CS p 2(ω)112CS 12p 1(ω)tr(F ωω)), \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} \,,

where now ω\omega is the local 1-form representative of a spin connection and where CS p 2CS_{p_2} is a Chern-Simons form for the second Pontryagin class and CS 12p 1CS_{\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 CC and a Spin group Spin(6,1)Spin(6,1)-valued connection (see supergravity C-field for global descriptions of such pairs). Or maybe rather Spin(6,2)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 (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 (super) conformal group SO(6,2)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 3AdS_4 / CFT_3 –Horizon limit of M2-branes

11d supergravity/M-theory on the asymptotic AdS 4AdS_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 2AdS_3 / CFT_2 – Horizon limit of D1-D5 brane bound states

D1-D5 brane system in type IIB string theory

(Maldacena 97, section 4)

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

see also at AdS3-CFT2 and CS-WZW correspondence

(…)

Non-conformal duals

Horizon limit of DpDp-branes for arbitrary pp

(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

gauge theory induced via AdS-CFT correspondence

M-theory perspective via AdS7-CFT6F-theory perspective
11d supergravity/M-theory
\;\;\;\;\downarrow Kaluza-Klein compactification on S 4S^4compactificationon elliptic fibration followed by T-duality
7-dimensional supergravity
\;\;\;\;\downarrow topological sector
7-dimensional Chern-Simons theory
\;\;\;\;\downarrow AdS7-CFT6 holographic duality
6d (2,0)-superconformal QFT on the M5-brane with conformal invarianceM5-brane worldvolume theory
\;\;\;\; \downarrow KK-compactification on Riemann surfacedouble dimensional reduction on M-theory/F-theory elliptic fibration
N=2 D=4 super Yang-Mills theory with Montonen-Olive S-duality invariance; AGT correspondenceD3-brane worldvolume theory with type IIB S-duality
\;\;\;\;\; \downarrow topological twist
topologically twisted N=2 D=4 super Yang-Mills theory
\;\;\;\; \downarrow KK-compactification on Riemann surface
A-model on Bun GBun_G, Donaldson theory

\,

gauge theory induced via AdS5-CFT4
type II string theory
\;\;\;\;\downarrow Kaluza-Klein compactification on S 5S^5
\;\;\;\; \downarrow topological sector
5-dimensional Chern-Simons theory
\;\;\;\;\downarrow AdS5-CFT4 holographic duality
N=4 D=4 super Yang-Mills theory
\;\;\;\;\; \downarrow topological twist
topologically twisted N=4 D=4 super Yang-Mills theory
\;\;\;\; \downarrow KK-compactification on Riemann surface
A-model on Bun GBun_G and B-model on Loc GLoc_G, geometric Langlands correspondence

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).

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×S 5AdS_5 \times S^5 detailed checks are summarized in Beisert et al. 10, Escobedo 12

graphics grabbed from Escobedo 12

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.

Table of branes appearing in supergravity/string theory (for classification see at brane scan).

branein supergravitycharged under gauge fieldhas worldvolume theory
black branesupergravityhigher gauge fieldSCFT
D-branetype IIRR-fieldsuper Yang-Mills theory
(D=2n)(D = 2n)type IIA\,\,
D0-brane\,\,BFSS matrix model
D2-brane\,\,\,
D4-brane\,\,D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane\,\,D=7 super Yang-Mills theory
D8-brane\,\,
(D=2n+1)(D = 2n+1)type IIB\,\,
D(-1)-brane\,\,\,
D1-brane\,\,2d CFT with BH entropy
D3-brane\,\,N=4 D=4 super Yang-Mills theory
D5-brane\,\,\,
D7-brane\,\,\,
D9-brane\,\,\,
(p,q)-string\,\,\,
(D25-brane)(bosonic string theory)
NS-branetype I, II, heteroticcircle n-connection\,
string\,B2-field2d SCFT
NS5-brane\,B6-fieldlittle string theory
D-brane for topological string\,
A-brane\,
B-brane\,
M-brane11D SuGra/M-theorycircle n-connection\,
M2-brane\,C3-fieldABJM theory, BLG model
M5-brane\,C6-field6d (2,0)-superconformal QFT
M9-brane/O9-planeheterotic string theory
M-wave
topological M2-branetopological M-theoryC3-field on G2-manifold
topological M5-brane\,C6-field on G2-manifold
solitons on M5-brane6d (2,0)-superconformal QFT
self-dual stringself-dual B-field
3-brane in 6d

References

Original articles

The original articles are

The actual rule for matching bulk states to generating functions for boundary correlators/n-point functions is due to

See also

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

Discussion of how Green-Schwarz action functionals for super pp-branes in AdS target spaces induce, after diffeomorphism gauge fixing, superconformal field theory on the worldvolumes (see singleton representation) goes back to

  • Mike Duff, C. Sutton, The Membrane at the End of the Universe, New Sci. 118 (1988) 67-71 (spire:268230)

  • Mike Duff, Anti-de Sitter space, branes, singletons, superconformal field theories and all that, Int.J.Mod.Phys.A14:815-844,1999 (arXiv:hep-th/9808100)

and was further developed in

See also at super p-brane – As part of the AdS-CFT correspondence.

Introductions and surveys

Surveys and introductions include

Further references include:

Review of Yangian symmetry includes

AdS 4/CFT 3AdS_4 / CFT_3

AdS 5/CFT 4AdS_5 / CFT_4

  • N. Beisert et al., Review of AdS/CFT Integrability, An Overview, Lett. Math. Phys. vv, pp (2011), (arXiv:1012.3982).

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

AdS 7/CFT 6AdS_7 / CFT_6

We list references specific to AdS 7/CFT 6AdS_7/CFT_6.

In

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=7d = 7, due the quantum anomaly cancellation, is clear from the original source, equation (3.14) of

but seems not to be noted explicitly in the context of AdS 7/CFT 6AdS_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)

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

and a corresponding non-perturbative discussion of the supergravity C-field that enters this Lagrangian is given in

Domenico Fiorenza, Hisham Sati, Urs Schreiber, The E8 moduli 3-stack of the C-field (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)Spin(10,1). The asymptotic AdS condition suggests maybe that it should be Spin(6,2)Spin(6,2).

In fact, in

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

More on the relation between the M5-brane and supergravity on AdS 7×S 4AdS_7 \times S^4 and arguments for the SO(5)SO(5) R-symmetry group on the 6d theory from the 7d theory are given in

  • A. J. Nurmagambetov, I. Y. Park, On the M5 and the AdS7/CFT6 Correspondence (arXiv:hep-th/0110192)

See also

  • M. Nishimura, Y. Tanii, Local Symmetries in the AdS7/CFT_6 Correspondence_, Mod. Phys. Lett. A14 (1999) 2709-2720 (arXiv:hep-th/9910192)

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

Generalization beyond exact AdS and/or 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 χ SB\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 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:

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 the quark-gluon plasma

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

Expositions and reviews include

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

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

To condensed matter physics

On AdS-CFT in condensed matter physics:

Textbook account

Further reviews include the following:

Last revised on November 27, 2018 at 01:51:55. See the history of this page for a list of all contributions to it.