Contents

supersymmetry

# Contents

## Idea

According to the classification of superconformal symmetry, there should exists superconformal field theories in 6 dimensions…

$d$$N$superconformal super Lie algebraR-symmetryblack brane worldvolume
superconformal field theory
$\phantom{A}3\phantom{A}$$\phantom{A}2k+1\phantom{A}$$\phantom{A}B(k,2) \simeq$ osp$(2k+1 \vert 4)\phantom{A}$$\phantom{A}SO(2k+1)\phantom{A}$
$\phantom{A}3\phantom{A}$$\phantom{A}2k\phantom{A}$$\phantom{A}D(k,2)\simeq$ osp$(2k \vert 4)\phantom{A}$$\phantom{A}SO(2k)\phantom{A}$M2-brane
3d superconformal gauge field theory
$\phantom{A}4\phantom{A}$$\phantom{A}k+1\phantom{A}$$\phantom{A}A(3,k)\simeq \mathfrak{sl}(4 \vert k+1)\phantom{A}$$\phantom{A}U(k+1)\phantom{A}$D3-brane
4d superconformal gauge field theory
$\phantom{A}5\phantom{A}$$\phantom{A}1\phantom{A}$$\phantom{A}F(4)\phantom{A}$$\phantom{A}SO(3)\phantom{A}$
$\phantom{A}6\phantom{A}$$\phantom{A}k\phantom{A}$$\phantom{A}D(4,k) \simeq$ osp$(8 \vert 2k)\phantom{A}$$\phantom{A}Sp(k)\phantom{A}$M5-brane
6d superconformal gauge field theory

(Shnider 88, also Nahm 78, see Minwalla 98, section 4.2)

…with $(2,0)$-supersymmetry, that contain 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 $N$ 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).

## 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 $AdS_7 \times S^4$.

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 and AGT correspondence

(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.

### 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

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 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 $Bun_G$, Donaldson theory

$\,$

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).

branein supergravitycharged under gauge fieldhas worldvolume theory
black branesupergravityhigher gauge fieldSCFT
D-branetype IIRR-fieldsuper Yang-Mills theory
$(D = 2n)$type IIA$\,$$\,$
D(-2)-brane$\,$$\,$
D0-brane$\,$$\,$BFSS matrix model
D2-brane$\,$$\,$$\,$
D4-brane$\,$$\,$D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane$\,$$\,$D=7 super Yang-Mills theory
D8-brane$\,$$\,$
$(D = 2n+1)$type IIB$\,$$\,$
D(-1)-brane$\,$$\,$$\,$
D1-brane$\,$$\,$2d CFT with BH entropy
D3-brane$\,$$\,$N=4 D=4 super Yang-Mills theory
D5-brane$\,$$\,$$\,$
D7-brane$\,$$\,$$\,$
D9-brane$\,$$\,$$\,$
(p,q)-string$\,$$\,$$\,$
(D25-brane)(bosonic string theory)
NS-branetype I, II, heteroticcircle n-connection$\,$
string$\,$B2-field2d SCFT
NS5-brane$\,$B6-fieldlittle string theory
D-brane for topological string$\,$
A-brane$\,$
B-brane$\,$
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
M-wave
topological M2-branetopological M-theoryC3-field on G2-manifold
topological M5-brane$\,$C6-field on G2-manifold
S-brane
SM2-brane,
membrane instanton
M5-brane instanton
D3-brane instanton
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

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

Construction from F-theory KK-compactification is reviewed in

General survey includes

### 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.

Relation to BFSS matrix model on tori:

Discussion of the ADE classification of the 6d theories includes, after (Witten95)

### Models and special properties

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.

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

• 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)

### Extended TQFT and quantum anomalies

Relation to extended TQFT and quantum anomalies (motivated via M5-brane lore) is discussed in

a summary of

Last revised on March 12, 2019 at 15:18:18. See the history of this page for a list of all contributions to it.