nLab
6d (2,0)-supersymmetric QFT

Context

Quantum field theory

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Contents

Idea

The classification of superconformal field theories predicts the existence of such a theory with (2,0)(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).

Properties

Holographic dual

Under AdS7/CFT6 the 6d (2,0)(2,0)-superconformal QFT is supposed to be dual to M-theory on anti de Sitter spacetime AdS 7×S 4AdS_7 \times S^4.

See AdS/CFT correspondence for more on this.

Solitonic 1-branes

The 5d (2,0)(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 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+3g3SU(2)^{n+3g-3}-N=2 D=4 super Yang-Mills theory obtained by compactifying the 6d6d M5-brane theory on a Riemann surface C g,nC_{g,n} of genus gg with nn punctures to 2d Liouville theory on C g,nC_{g,n}.

Twistor space description

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-CFT6F-theory perspective
11d supergravity/M-theory
\;\;\;\;\downarrow Kaluza-Klein compactification on S 4S^4compactificationon 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 invarianceM5-brane worldvolume theory
\;\;\;\; \downarrow KK-compactification on Riemann surfacedouble dimensional reduction on M-theory/F-theory elliptic fibration
N=2 D=4 super Yang-Mills theory with Montonen-Olive S-duality invariance; AGT correspondenceD3-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 GBun_G, Donaldson theory

\,

gauge theory induced via AdS5-CFT4
type II string theory
\;\;\;\;\downarrow Kaluza-Klein compactification on S 5S^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 GBun_G and B-model on Log GLog_G, geometric Langlands correspondence

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

branein supergravitycharged under gauge fieldhas worldvolume theory
black branesupergravityhigher gauge fieldSCFT
D-branetype IIRR-fieldsuper Yang-Mills theory
(D=2n)(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)(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\,\,\,
(p,q)-string\,\,\,
(D25-brane)(bosonic string theory)
NS-branetype I, II, heteroticcircle n-connection\,
string\,B2-field2d SCFT
NS5-brane\,B6-fieldlittle string theory
M-brane11D SuGra/M-theorycircle n-connection\,
M2-brane\,C3-fieldABJM theory, BLG model
M5-brane\,C6-field6d (2,0)-superconformal QFT
M9-brane/O9-planeheterotic string theory
topological M2-branetopological M-theoryC3-field on G2-manifold
topological M5-brane\,C6-field on G2-manifold
solitons on M5-brane6d (2,0)-superconformal QFT
self-dual stringself-dual B-field
3-brane in 6d

References

General

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

Surveys include

See also the references and discussion at M5-brane.

Compactification to 4d super-Yang-Mills

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

  • Edward Witten, Geometric Langlands From Six Dimensions, in Peter Kotiuga (ed.) A Celebration of the Mathematical Legacy of Raoul Bott, AMS 2010 (arXiv:0905.2720)

For more references on this see at N=2 D=4 super Yang-Mills theory the section References - Constructions from 5-branes.

Models and special properties

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

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

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×S 4AdS_7 \times S^4 (arXiv:0810.4647)

Solitonic 1-brane excitations

Relation to extended TQFT

Relation to extended TQFT is discussed in

Revised on April 16, 2014 23:00:18 by Urs Schreiber (82.169.114.243)