# Contents

## Idea

The classification of superconformal field theories predicts the existence of such a theory with $\left(2,0\right)$-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 $\left(2,0\right)$-superconformal QFT is supposed to be dual to M-theory on anti de Sitter spacetime ${\mathrm{AdS}}_{7}×{S}^{4}$.

See AdS/CFT correspondence for more on this.

### Solitonic 1-branes

The 5d $\left(2,0\right)$-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 $\mathrm{SU}\left(2{\right)}^{n+3g-3}$-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 ${C}_{g,n}$.

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
$\left(D=2n\right)$type IIA$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
D0-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$BFSS matrix model
D2-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
D4-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
D8-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
$\left(D=2n+1\right)$type IIB$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
D1-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$2d CFT with BH entropy
D3-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$N=4 D=4 super Yang-Mills theory
D5-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
D7-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
NS-branetype I, II, heteroticcircle n-connection$\phantom{\rule{thinmathspace}{0ex}}$
string$\phantom{\rule{thinmathspace}{0ex}}$B2-field2d SCFT
NS5-brane$\phantom{\rule{thinmathspace}{0ex}}$B6-fieldlittle string theory
M-brane11D SuGra/M-theorycircle n-connection$\phantom{\rule{thinmathspace}{0ex}}$
M2-brane$\phantom{\rule{thinmathspace}{0ex}}$C3-fieldABJM theory, BLG model
M5-brane$\phantom{\rule{thinmathspace}{0ex}}$C6-field6d (2,0)-superconformal QFT
topological M2-branetopological M-theoryC3-field on G2-manifold
topological M5-brane$\phantom{\rule{thinmathspace}{0ex}}$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

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

Surveys include

• Greg Moore, On the role of six‐dimensional $\left(2,0\right)$-theories in recent developments in Physical Mathematics , talk at Strings2011 (pdf slides)

### 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.

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 ${\mathrm{AdS}}_{7}×{S}^{4}$ (arXiv:0810.4647)

### Relation to extended TQFT

Relation to extended TQFT is discussed in

Revised on May 15, 2013 17:09:11 by Urs Schreiber (82.113.121.239)