nLab
6d (2,0)-supersymmetric QFT

Idea

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) 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).

Properties

Holographic dual

Under AdS7/CFT6 the 6d $(2,0)$-superconformal QFT is supposed to be dual to M-theory on anti de Sitter spacetime ${\mathrm{AdS}}_7\times S^4$.

See AdS/CFT correspondence for more on this.

Solitonic 1-branes

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)

Compactification on a Riemann surface

The AGT correspondence relates the partition function of $\mathrm{SU}(2{)}^{n+3g-3}$-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 $C_{g,n}$.

Related entries

Table of branes appearing in supergravity/string theory (for classification see at brane scan).

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
$(D=2n)$	type IIA	$$	$$
D0-brane	$$	$$	BFSS matrix model
D2-brane	$$	$$	$$
D4-brane	$$	$$	D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane	$$	$$
D8-brane	$$	$$
$(D=2n+1)$	type IIB	$$	$$
D1-brane	$$	$$	2d CFT with BH entropy
D3-brane	$$	$$	N=4 D=4 super Yang-Mills theory
D5-brane	$$	$$	$$
D7-brane	$$	$$	$$
D9-brane	$$	$$	$$
NS-brane	type I, II, heterotic	circle n-connection	$$
string	$$	B2-field	2d SCFT
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

References

General

The first indication of a 6d theory with a self-dual 2-form field appears in section 1 of

• Edward Witten, Some comments on string dynamocs (hepth/9507121)

The conformal structure of this 6d theory and its relation to electric-magnetic duality/S-duality in 4-dimensions is discussed in

• Edward Witten, Conformal Field Theory In Four And Six Dimensions in Ulrike Tillmann (ed.) Topology, geometry and quantum field theory LMS Lecture Note Series (2004) (arXiv:0712.0157)

• Edward Witten, Geometric Langlands From Six Dimensions (arXiv:0905.2720) .

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)

See also the references and discussion at M5-brane.

Models and special properties

A proposal for related higher nonabelian differential form data is made in

• Henning Samtleben, Ergin Sezgin, Robert Wimmer, (1,0) superconformal models in six dimensions (arXiv:1108.4060)

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

• Andreas Gustavsson, Selfdual strings and loop space Nahm equations (arXiv:0802.3456).

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

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, 7d Chern-Simons theory and the 5-brane

On the holographic dual

The basics of the relation of the 6d theory to a 7d theory under AdS-CFT is reviewed holographic duality

• Juan Maldacena, The Large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2:231, 1998, hep-th/9711200; Wilson loops in Large N field theories, Phys. Rev. Lett. 80 (1998) 4859, hep-th/9803002

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

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, 7d Chern-Simons theory and the 5-brane

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 ${\mathrm{AdS}}_7\times S^4$ (arXiv:0810.4647)

Solitonic 1-brane excitations

• 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

Relation to extended TQFT is discussed in

• Dan Freed, 4-3-2 8-7-6