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).
It is expected (Witten) that such theories arise as the worldvolume theories of M5-branes and that their compactifications are at the heart of the phenomenon that leads to S-duality and hence geometric Langlands duality (Witten).
Due to the self-duality a characterization of these theories by an action functional is at best subtle, maybe impossible. 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.
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 .
|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|
|NS-brane||type I, II, heterotic||circle n-connection|
|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|
The first indication of a 6d theory with a self-dual 2-form field appears in section 1 of
See also the references and discussion at M5-brane.
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