Quantum field theory


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The AdS-CFT correspondence (or Maldacena duality ) is a class of cases for which there is strong evidence that it realizes the more general and more conjectural holographic duality:

the conjectural Ads/CFT correspondence asserts an identification of the states of quantum gravity given by string theory on an asymptotically anti-de Sitter spacetime with correlators of a superconformal Yang-Mills theory on the asymptotic boundary.

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
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)


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


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 hologrpahic 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


The full formalization of AdS/CFT is still very much out of reach.

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
D4-brane\,\,D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane\,\,D=7 super Yang-Mills theory
(D=2n+1)(D = 2n+1)type IIB\,\,
D1-brane\,\,2d CFT with BH entropy
D3-brane\,\,N=4 D=4 super Yang-Mills theory
(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\,
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
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


Original articles

The original articles are

The relevance of this was amplified in

  • Edward Witten, Anti-de Sitter space and holography, Advances in Theoretical and Mathematical Physics 2: 253–291, 1998, hep-th/9802150

Detailed 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) includes

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).

AdS 7/CFT 6AdS_7 / CFT_6

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


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


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


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 particle physics

To condensed matter physics

Application of AdS-CFT in condensed matter physics goes back to

A comprehensive textbook account is in

Further reviews include the following:

Last revised on September 9, 2018 at 04:41:04. See the history of this page for a list of all contributions to it.