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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 $d+1$ 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), 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 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.
$d$ | $N$ | superconformal super Lie algebra | R-symmetry | black brane worldvolume superconformal field theory via AdS-CFT |
---|---|---|---|---|
$\phantom{A}3\phantom{A}$ | $\phantom{A}2k+1\phantom{A}$ | $\phantom{A}B(k,2) \simeq$ osp$(2k+1/4)\phantom{A}$ | $\phantom{A}SO(2k+1)\phantom{A}$ | |
$\phantom{A}3\phantom{A}$ | $\phantom{A}2k\phantom{A}$ | $\phantom{A}D(k,2)\simeq$ osp$(2k/4)\phantom{A}$ | $\phantom{A}SO(2k)\phantom{A}$ | M2-brane 3d superconformal gauge field theory |
$\phantom{A}4\phantom{A}$ | $\phantom{A}k+1\phantom{A}$ | $\phantom{A}A(3,k)\simeq \mathfrak{sl}(4/k+1)\phantom{A}$ | $\phantom{A}U(k+1)\phantom{A}$ | D3-brane 4d superconformal gauge field theory |
$\phantom{A}5\phantom{A}$ | $\phantom{A}1\phantom{A}$ | $\phantom{A}F(4)\phantom{A}$ | $\phantom{A}SO(3)\phantom{A}$ | |
$\phantom{A}6\phantom{A}$ | $\phantom{A}k\phantom{A}$ | $\phantom{A}D(4,k) \simeq$ osp$(8/2k)\phantom{A}$ | $\phantom{A}Sp(k)\phantom{A}$ | 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 $AdS_5 \times S^5$, 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 $n$-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)
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.
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-
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
(Aharony-Gubser-Maldacena-Ooguri-Oz 99, section 3 and 4)
We list some of the conjectured statements and their evidence concerning the case of $AdS_7/CFT_6$-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
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 (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 $SO(6,2)$.
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).
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)
D1-D5 brane system in type IIB string theory
(Aharony-Gubser-Maldacena-Ooguri-Oz 99, section 5)
see also at AdS3-CFT2 and CS-WZW correspondence
(…)
(Aharony-Gubser-Maldacena-Ooguri-Oz 99, section 6.1.3)
(Aharony-Gubser-Maldacena-Ooguri-Oz 99, section 6.1.4)
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.
gauge theory induced via AdS-CFT correspondence
M-theory perspective via AdS7-CFT6 | F-theory perspective |
---|---|
11d supergravity/M-theory | |
$\;\;\;\;\downarrow$ Kaluza-Klein compactification on $S^4$ | compactificationon 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 invariance | M5-brane worldvolume theory |
$\;\;\;\; \downarrow$ KK-compactification on Riemann surface | double dimensional reduction on M-theory/F-theory elliptic fibration |
N=2 D=4 super Yang-Mills theory with Montonen-Olive S-duality invariance; AGT correspondence | D3-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 $Bun_G$, Donaldson theory |
$\,$
gauge theory induced via AdS5-CFT4 |
---|
type II string theory |
$\;\;\;\;\downarrow$ Kaluza-Klein compactification on $S^5$ |
$\;\;\;\; \downarrow$ 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 $Bun_G$ and B-model on $Loc_G$, geometric Langlands correspondence |
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 $AdS_5 \times S^5$ detailed checks are summarized in Beisert et al. 10, Escobedo 12
graphics grabbed from Escobedo 12
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).
The original articles are
Juan Maldacena, The Large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2:231, 1998 (hep-th/9711200)
Juan Maldacena, Wilson loops in Large $N$ field theories, Phys. Rev. Lett. 80 (1998) 4859 (hep-th/9803002)
The actual rule for matching bulk states to generating functions for boundary correlators/n-point functions is due to
Steven Gubser, Igor Klebanov, Alexander Polyakov, around (12) of Gauge theory correlators from non-critical string theory, Physics Letters B428: 105–114 (1998) (hep-th/9802109)
Edward Witten, around (2.11) of Anti-de Sitter space and holography, Advances in Theoretical and Mathematical Physics 2: 253–291, 1998 (hep-th/9802150)
See also
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 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
Piet Claus, Renata Kallosh, Antoine Van Proeyen, M 5-brane and superconformal $(0,2)$ tensor multiplet in 6 dimensions, Nucl.Phys. B518 (1998) 117-150 (arXiv:hep-th/9711161)
Gianguido Dall'Agata, Davide Fabbri, Christophe Fraser, Pietro Fré, Piet Termonia, Mario Trigiante, The $Osp(8|4)$ singleton action from the supermembrane, Nucl.Phys.B542:157-194, 1999 (arXiv:hep-th/9807115)
Piet Claus, Renata Kallosh, J. Kumar, Paul Townsend, Antoine Van Proeyen, Conformal Theory of M2, D3, M5 and ‘D1+D5’ Branes, JHEP 9806 (1998) 004 (arXiv:hep-th/9801206)
Paolo Pasti, Dmitri Sorokin, Mario Tonin, Branes in Super-AdS Backgrounds and Superconformal Theories (arXiv:hep-th/9912076)
See also at super p-brane – As part of the AdS-CFT correspondence.
Sketch of a derivation of AdS/CFT:
Surveys and introductions include
Ofer Aharony, S.S. Gubser, Juan Maldacena, H. Ooguri, Y. Oz, Large N Field Theories, String Theory and Gravity, Phys.Rept.323:183-386,2000 (arXiv:hep-th/9905111)
Michael T. Anderson, Geometric aspects of the AdS/CFT correspondence (arXiv:hep-th/0403087)
Mike Duff, TASI Lectures on Branes, Black Holes and Anti-de Sitter Space (arXiv:hep-th/9912164)
Horatiu Nastase, Introduction to AdS-CFT (arXiv:0712.0689)
Gaston Giribet, Black hole physics and $AdS^3/CFT_2$, lectures and proceedings of Croatian black hole school.
Jens L. Petersen, Introduction to the Maldacena Conjecture on AdS/CFT, Int.J.Mod.Phys. A14 (1999) 3597-3672, hep-th/9902131 , doi
Jan de Boer, Introduction to AdS/CFT correspondence, pdf
Makoto Natsuume, AdS/CFT Duality User Guide, Lecture Notes in Physics 903, Springer 2015 (arXiv:1409.3575)
wikipedia: AdS/CFT correspondence
an AdS/CFT bibliography
Further references include:
Gary T. Horowitz, Joseph Polchinski, Gauge/gravity duality, gr-qc/0602037
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)
Review of Yangian symmetry includes
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)
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
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)
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 $Spin(10,1)$. The asymptotic AdS condition suggests maybe that it should be $Spin(6,2)$.
In fact, in
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
See also
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
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)
Discussion of event horizons of black holes in terms of AdS/CFT (the “firewall problem”) is in
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
Applications of AdS-CFT to the quark-gluon plasma of QCD:
Expositions and reviews include
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 (doi:10.1140/epjc/s10052-014-2981-5)
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)
On AdS-CFT in condensed matter physics:
Textbook account
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
Last revised on December 28, 2018 at 05:03:07. See the history of this page for a list of all contributions to it.