nLab
AdS-CFT

Idea

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.

Examples

${\mathrm{AdS}}_5/{\mathrm{CFT}}_4$

type II string theory on 5d anti de Sitter spacetime is dual to N=4 D=4 super Yang-Mills theory.

${\mathrm{AdS}}_7/{\mathrm{CFT}}_6$

We list some of the conjectured statements and their evidence concerning the case of ${\mathrm{AdS}}_7/{\mathrm{CFT}}_6$-duality.

The hypothesis (Maldacena 97, section 3.1) is that

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

is holographically related to

• 11-dimensional supergravity reduced on the 4-sphere $S^4$ with $N$ units of flux of the supergravity C-field to 7d supergravity on an asymptotically anti de Sitter spacetime.

In (Witten 98, section 4) is is argued that the conformal blocks of the 6d theory are entirely given already by the states of (just) the effective 7-dimensional Chern-Simons term of the supergravity C-field, which locally is

$$\begin{array}{rl}{S}_{11d\mathrm{SUGRA},\mathrm{CS}}(C_3)& ={\int }_{{\mathrm{AdS}}_7}{\int }_{S^4}{C}_3\wedge G_4\wedge G_4\\ & =N{\int }_{{\mathrm{AdS}}_7}{C}_3\wedge G_4\end{array}.$$\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

$$\begin{array}{rl}S(\omega ,C_3)& ={\int }_{{\mathrm{AdS}}_7}{\int }_{S^4}{C}_3\wedge G_4\wedge (G_4+I_8(\omega ))\\ & =N{\int }_{{\mathrm{AdS}}_7}(C_3\wedge G_4+\frac{1}{48}{\mathrm{CS}}_{p_2}(\omega )-\frac{1}{12}{\mathrm{CS}}_{\frac{1}{2}{p}_1}(\omega )\wedge \mathrm{tr}(F_{\omega }\wedge \omega ))\end{array},$$\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 $\omega$ is the local 1-form representative of a spin connection and where ${\mathrm{CS}}_{p_2}$ is a Chern-Simons form for the second Pontryagin class and ${\mathrm{CS}}_{\frac{1}{2}{p}_1}$ for the first.

That therefore not an abelian, but this nonabelian higher dimensional Chern-Simons theory should be dual to the 6d theory was maybe first said explicitly in (LuWang 2010).

Its gauge field is hence locally a pair consisting of the abelian 3-form field $C$ and a Spin group $\mathrm{Spin}(6,1)$-valued connection (see supergravity C-field for global descriptions of such pairs). Or maybe rather $\mathrm{Spin}(6,2)$ 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) more detailed arguments are given that the 7-dimensional dual to the 6d theory is a higher spin gauge theory for a higher spin gauge group extending $\mathrm{SO}(6,2)$.

Formalizations

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.

Related concepts

• duality in physics, duality in string theory

  • T-duality, S-duality, U-duality

  • open/closed string duality

    • KLT relations

Table of branes appearing in supergravity/string theory

brane	in supergravity	charged under gauge field	has worldvolume theory
black brane	supergravity	higher gauge field	SCFT
D-brane	type II	RR-field	super Yang-Mills theory
$(D=2n)$	type IIA	$$	$$
D0-brane	$$	$$	BFSS matrix model
D2-brane	$$	$$	$$
D4-brane	$$	$$	D=5 super Yang-Mills theory with Khovanov homology observables
$(D=2n+1)$	type IIB	$$	$$
D1-brane	$$	$$	2d CFT with BH entropy
D3-brane	$$	$$	N=4 D=4 super Yang-Mills theory
D5-brane	$$	$$	$$
D7-brane	$$	$$	$$
NS-brane	type I, II, heterotic	circle n-connection	$$
string	$$	B2-field	2d SCFT
NS5-brane	$$	B6-field	little string theory
M-brane	11D SuGra/M-theory	circle n-connection	$$
M2-brane	$$	C3-field	ABJM theory, BLG model
M5-brane	$$	C6-field	6d (2,0)-superconformal QFT
topological M2-brane	topological M-theory	C3-field on G2-manifold
topological M5-brane	$$	C6-field on G2-manifold

References

Original articles

The original article is

• Juan Maldacena, The Large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2:231, 1998, hep-th/9711200; Wilson loops in Large N field theories, Phys. Rev. Lett. 80 (1998) 4859, hep-th/9803002

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

Introductions and surveys

Surveys and introductions include

• Ofer Aharony, S.S. Gubser, Juan Maldacena, H. Ooguri, Y. Oz, Large N Field Theories, String Theory and Gravity (arXiv:hep-th/9905111)

• Horatiu Nastase, Introduction to AdS-CFT (arXiv:0712.0689)

• Gaston Giribet, Black hole physics and ${\mathrm{AdS}}^3/{\mathrm{CFT}}_2$, 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

• wikipedia: AdS/CFT correspondence

• an AdS/CFT bibliography

Further references include:

• Gary T. Horowitz, Joseph Polchinski, Gauge/gravity duality, gr-qc/0602037

• S. S. Gubser, I. R. Klebanov, A. M. Polyakov, Gauge theory correlators from non-critical string theory, Physics Letters B428: 105–114 (1998), hep-th/9802109.

• 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)

${\mathrm{AdS}}_5/{\mathrm{CFT}}_4$

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

${\mathrm{AdS}}_7/{\mathrm{CFT}}_6$

We list references specific to ${\mathrm{AdS}}_7/{\mathrm{CFT}}_6$.

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 T. Liu, R. 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 ${\mathrm{AdS}}_7/{\mathrm{CFT}}_6$ 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. For more in that see here. 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 $\mathrm{Spin}(10,1)$. The asymptotic AdS condition suggests maybe that it should be $\mathrm{Spin}(6,2)$.

In fact, 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 $\mathrm{SO}(6,2)$.

More on the relation between the M5-brane and supergravity on ${\mathrm{AdS}}_7\times S^4$ and arguments for the $\mathrm{SO}(5)$ 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)

Applications

To gravity

Discussion of event horizons of black holes in terms of AdS/CFT is in

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

To condensed matter physics

Applications of AdS-CFT duality to condensed matter physics? are reviewed in

• 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)