functorial quantum field theory
Reshetikhin?Turaev model? / Chern-Simons theory
FQFT and cohomology
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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.
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 a7d Chern-Simons theory phrased in terms of extended quantum field theory is (Freed 12).
11d supergravity/M-theory on the asymptotitc $AdS_4$ 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 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.
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 |
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).
The original articles are
The relevance of this was amplified in
Detailed 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 includes
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)
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.
Surveys and introductions include
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
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)
Review of Yangian symmetry includes
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 of event horizons of black holes in terms of AdS/CFT (the “firewall problem”) is in
Application of AdS-CFT in condensed matter physics goes back to
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)