FQFT and cohomology
Types of quantum field thories
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.
We list some of the conjectured statements and their evidence concerning the case of -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
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 and a Spin group -valued connection (see supergravity C-field for global descriptions of such pairs). Or maybe rather 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 .
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).
see also at AdS3-CFT2 and CS-WZW correspondence
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.
|M-theory perspective via AdS7-CFT6||F-theory perspective|
|Kaluza-Klein compactification on||compactificationon elliptic fibration followed by T-duality|
|7-dimensional Chern-Simons theory|
|AdS7-CFT6 holographic duality|
|6d (2,0)-superconformal QFT on the M5-brane with conformal invariance||M5-brane worldvolume theory|
|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|
|topologically twisted N=2 D=4 super Yang-Mills theory|
|KK-compactification on Riemann surface|
|A-model on , Donaldson theory|
|gauge theory induced via AdS5-CFT4|
|type II string theory|
|Kaluza-Klein compactification on|
|5-dimensional Chern-Simons theory|
|AdS5-CFT4 holographic duality|
|N=4 D=4 super Yang-Mills theory|
|topologically twisted N=4 D=4 super Yang-Mills theory|
|KK-compactification on Riemann surface|
|A-model on and B-model on , geometric Langlands correspondence|
The full formalization of AdS/CFT is still very much out of reach.
|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|
|D0-brane||BFSS matrix model|
|D4-brane||D=5 super Yang-Mills theory with Khovanov homology observables|
|D1-brane||2d CFT with BH entropy|
|D3-brane||N=4 D=4 super Yang-Mills theory|
|(D25-brane)||(bosonic string theory)|
|NS-brane||type I, II, heterotic||circle n-connection|
|NS5-brane||B6-field||little string theory|
|D-brane for topological string|
|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|
|M9-brane/O9-plane||heterotic string theory|
|topological M2-brane||topological M-theory||C3-field on G2-manifold|
|topological M5-brane||C6-field on G2-manifold|
|solitons on M5-brane||6d (2,0)-superconformal QFT|
|self-dual string||self-dual B-field|
|3-brane in 6d|
The original articles are
The relevance of this was amplified in
Surveys and introductions include
Jan de Boer, Introduction to AdS/CFT correspondence, pdf
wikipedia: AdS/CFT correspondence
an AdS/CFT bibliography
Further references include:
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.
We list references specific to .
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.
but seems not to be noted explicitly in the context of 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.
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 . The asymptotic AdS condition suggests maybe that it should be .
In fact, in
arguments are given that the 7d theory is a higher spin gauge theory extension of .
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 of AdS-CFT duality to condensed matter physics go back to
Reviews include the following:
A S T Pires, Ads/CFT correspondence in condensed matter (arXiv:1006.5838)
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)