FQFT and cohomology
Types of quantum field thories
The classification of superconformal field theories predicts the existence of such a theory with -supersymmetry in dimension 6, which is such that it contains a self-dual higher gauge theory whose field configurations are connections on a 2-bundle (a circle 2-bundle with connection in the abelian case).
In (Claus-Kallosh-Proeyen 97) such has been derived, in the abelian case and to low order, as the small fluctuations of the Green-Schwarz sigma-model of the M5-brane around the embedding in the asymptotic boundary of the AdS-spacetime that is the near-horizon geometry of the black M5-brane.
In accord with this the AdS7-CFT6 correspondence predicts that the nonabelian 6d theory is the corresponding theory for coincident M5-branes.
In the non-abelian case this is expected (Witten 07) that the compactification of such theories are at the heart of the phenomenon that leads to S-duality in super Yang-Mills theory and further to geometric Langlands duality (Witten 09).
Due to the self-duality a characterization of these theories by an action functional is subtle. Therefore more direct descriptions are still under investigation (for instance SSW11). A review of recent developments is in (Moore11).
See AdS/CFT correspondence for more on this.
(graphics taken from (Workshop 14))
The AGT correspondence relates the partition function of -N=2 D=4 super Yang-Mills theory obtained by compactifying the M5-brane theory on a Riemann surface of genus with punctures to 2d Liouville theory on .
Famously the solutions to self-dual Yang-Mills theory in dimension 4 can be obtained as images of degree-2 cohomology classes under the Penrose-Ward twistor transform. Since the 6d QFT on the M5-brane contains a 2-form self-dual higher gauge field it seems natural to expect that it can be described by a higher analogy of the twistor transform. For references exploring this idea see at twistor space – References – Application to the self-dual 2-form field in 6d.
|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|
|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 first indication of a 6d theory with a self-dual 2-form field appears in section 1 of
Derivation of the abelian 6d theory to low order as the small fluctuations of the Green-Schwarz sigma-model of the M5-brane around a solution embedding as the asymptotic boundary of the AdS-spacetime near-horizon geometry of a black 5-brane is due to
at the University of California, Berkeley
See also the references and discussion at M5-brane.
and the resulting relation to the geometric Langlands correspondence is disucssed in
Relation to BFSS matrix model on tori:
A proposal for related higher nonabelian differential form data is made in
Since by transgression every nonabelian principal 2-bundle/gerbe gives rise to some kind of nonabelian 1-bundle on loop space it is clear that some parts (but not all) of the nonabelian gerbe theory on the 5-brane has an equivalent reformulation in terms of ordinary gauge theory on the loop space of the 5-brane and possibly for gauge group the loop group of the original gauge group.
Comments along these lines have been made in
In fact, via the strict 2-group version of the string 2-group there is a local gauge in which the loop group variables appear already before transgression of the 5-brane gerbe to loop space. This is discussed from a holographic point of view in
The nonabelian generalization of this 7d action functional that follows from taking the quantum corrections (Green-Schwarz mechanism and flux quantization) of the supergravity C-field into account is discussed in
Eric D'Hoker, John Estes, Michael Gutperle, Darya Krym,
Exact Half-BPS Flux Solutions in M-theory I Local Solutions (arXiv:0806.0605)
Exact Half-BPS Flux Solutions in M-theory II: Global solutions asymptotic to (arXiv:0810.4647)
Relation to extended TQFT is discussed in
David Ben-Zvi, Algebraic geometry of topological field theories, talk at Reimagining the Foundations of Algebraic Topology April 07, 2014 - April 11, 2014 (web video)