# Contents

## 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

### $AdS_5 / CFT_4$ – 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

### $AdS_7 / CFT_6$ – 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 (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

\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{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 $CS_{p_2}$ 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) 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 $SO(6,2)$.

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 a7d 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 asymptotitc $AdS_4$ spacetime of an M2-brane.

### $AdS_3 / CFT_2$ – Horizon limit of D1-D5 brane bound states

D1-D5 brane system? in type IIB string theory

(…)

### Non-conformal duals

#### 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.

### Further gauge theories induced by compactification and twisting

gauge theory induced via AdS-CFT correspondence

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 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 $Bun_G$, Donaldson theory

$\,$

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

## 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.

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
$(D = 2n)$type IIA$\,$$\,$
D0-brane$\,$$\,$BFSS matrix model
D2-brane$\,$$\,$$\,$
D4-brane$\,$$\,$D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane$\,$$\,$
D8-brane$\,$$\,$
$(D = 2n+1)$type IIB$\,$$\,$
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
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 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

Further references include:

### $AdS_5 / CFT_4$

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

### $AdS_7 / CFT_6$

We list references specific to $AdS_7/CFT_6$.

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 $d = 7$, 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

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

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

• 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

### Applications

#### To gravity

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