# Contents

## Idea

The theory of 11-dimensional supergravity contains a higher gauge field – the supergravity C-field – that naturally couples to higher electrically charged 2-branes (membranes). By a process called double dimensional reduction, these are related to strings (Bergshoeff-Sezgin-Townsend, Duff, 80s).

When in (Witten95) it was argued that the 10-dimensional target space theories of the five types of superstring theories are all limiting cases of one single 11-dimensional target space theory that extends 11-dimensional supergravity, it was natural to guess that this supergravity membrane accordingly yields a 3-dimensional sigma-model that reduces in limiting cases to the string sigma-models.

But there are two aspects that make this idea a little subtle, even at this vague level: first, there is no good theory of the quantization of the membrane sigma-model, as opposed to the well understood quantum string. Secondly, Secondly, that hypothetical “theory extending 11-dimensional supergravity” (“M-theory”) has remained elusive enough that it is not clear in which sense the membrane would relate to it in a way analogous to how the string relates to its target space theories (which is fairly well understood).

Later, with the BFSS matrix model some people gained more confidence in the idea, by identifying the corresponding degrees of freedom in a special case.

## Properties

In some situations stacks of M2-branes are accurately described by ABJM theory of the BLG model.

Table of branes appearing in supergravity/string theory

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}}$$\phantom{\rule{thinmathspace}{0ex}}$D=5 super Yang-Mills theory with Khovanov homology observables
$\left(D=2n+1\right)$type IIB$\phantom{\rule{thinmathspace}{0ex}}$$\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

Early speculations trying to model the electron by a relativistic membrane are due to Paul Dirac:

• Paul Dirac, An Extensible Model of the Electron, Proc. Roy. Soc. A268, (1962) 57-67.

• Paul Dirac, Motion of an Extended Particle in the Gravita- tional Field, in Relativistic Theories of Gravitation, Proceedings of a Conference held in Warsaw and Jablonna, July 1962, ed. L. Infeld, P. W. N. Publishers, 1964, Warsaw, 163-171; discussion 171-175

• Paul Dirac, Particles of Finite Size in the Gravitational Field, Proc. Roy. Soc. A270, (1962) 354-356.

The Green-Schwarz action-type formulation of the supermembrane appears in

• E. Bergshoeff, E. Sezgin, and Paul Townsend, Supermembranes and eleven-dimensional supergravity, Phys. Lett. B189 (1987) 75–78. (pdf)

The existence of the M2-brane as an object related to string theory was proposed in

around the time when M-theory was envisioned in

More recent developments are discussed in

Formulations of multiple M2-branes on top of each other are given by the BLG model and the ABJM model. The relation of these to the above is discussed in section 3 of

### Dualities

The role of and the relation to duality in string theory of the membrane is discussed in the following articles.

Relation to T-duality:

• J.G. Russo, T-duality in M-theory and supermembranes (arXiv:hep-th/9701188)

• M.P. Garcia del Moral, J.M. Pena, A. Restuccia, T-duality Invariance of the Supermembrane (arXiv:1211.2434)

Relation to U-duality:

Revised on May 7, 2013 02:25:03 by Urs Schreiber (150.212.93.166)