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The classification of superconformal field theories predicts the existence of such a theory with $(2,0)$-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 07) 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 09).
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).
Under AdS7/CFT6 the 6d $(2,0)$-superconformal QFT is supposed to be dual to M-theory on anti de Sitter spacetime $AdS_7 \times S^4$.
See AdS/CFT correspondence for more on this.
The 5d $(2,0)$-SCFT has tensionless 1-brane configurations. From the point of view of the ambient 11-dimensional supergravity these are the boundaries of M2-branes ending on the M5-branes. (GGT)
(graphics taken from (Workshop 14))
The compactification of the 5-brane on a Riemann surface yields as worldvolume theory N=2 D=4 super Yang-Mills theory. See at N=2 D=4 SYM – Construction by compactification of 5-branes.
The AGT correspondence relates the partition function of $SU(2)^{n+3g-3}$-N=2 D=4 super Yang-Mills theory obtained by compactifying the $6d$ M5-brane theory on a Riemann surface $C_{g,n}$ of genus $g$ with $n$ punctures to 2d Liouville theory on $C_{g,n}$.
More generally, this kind of construction allows to describe the 6d (2,0)-theory as a “2d SCFT with values in 4d SYM”. See at AGT correspondence for more on this.
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.
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 |
Table of branes appearing in supergravity/string theory (for classification see at brane scan).
The first indication of a 6d theory with a self-dual 2-form field appears in section 1 of
Surveys include
Greg Moore, On the role of six‐dimensional $(2,0)$-theories in recent developments in Physical Mathematics , talk at Strings2011 (pdf slides)
Greg Moore, Applications of the six-dimensional (2,0) theories to Physical Mathematics, Felix Klein lectures Bonn (2012) (pdf)
Workshops include
at the University of California, Berkeley
See also the references and discussion at M5-brane.
The conformal structure of the 6d theory and its relation under compactification on a Riemann surface to electric-magnetic duality/S-duality in 4-dimensions is discussed in
and the resulting relation to the geometric Langlands correspondence is disucssed in
For more references on this see at N=2 D=4 super Yang-Mills theory the section References - Constructions from 5-branes.
Relation to BFSS matrix model on tori:
Discussion of the ADE classification of the 6d theories includes, after (Witten95)
Realization of the 6d theory in F-theory is discussed in (Heckmann-Morrison-Vafa 13).
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 basics of the relation of the 6d theory to a 7d theory under AdS-CFT is reviewed holographic duality
The argument that the abelian 7d Chern-Simons theory of a 3-connection yields this way the conformal blocks of the abelian self-dual higher gauge theory of the 6d theory on a single brane is due to
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)
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
See also
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 $AdS_7 \times S^4$ (arXiv:0810.4647)
Jerome Gauntlett, Joaquim Gomis, Paul Townsend, BPS Bounds for Worldvolume Branes (arXiv:hep-th/9711205)
P.S. Howe, Neil Lambert, Peter West, The Threebrane Soliton of the M-Fivebrane (arXiv:hep-th/9710033)
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)