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.

Related concepts

• string theory

• 11-dimensional supergravity, M-theory

• string, superstring

• D-brane

• M5-brane, 6d (2,0)-superconformal QFT

• M9-brane

• topological membrane

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

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

• Paul Townsend, The eleven-dimensional supermembrane revisited (arXiv:hep-th/9501068)

around the time when M-theory was envisioned in

• Edward Witten, String Theory Dynamics In Various Dimensions (arXiv)

More recent developments are discussed in

• P.S. Howe, E. Sezgin, The supermembrane revisited, (arXiv:hep-th/0412245)

• Igor A. Bandos, Paul Townsend, SDiff Gauge Theory and the M2 Condensate (arXiv:0808.1583)

• Jonathan Bagger, Neil Lambert, Sunil Mukhi, Constantinos Papageorgakis, Multiple Membranes in M-theory (arXiv:1203.3546)

• Nathan Berkovits, Towards Covariant Quantization of the Supermembrane (arXiv:hep-th/0201151)

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

• Horatiu Nastase, Constantinos Papageorgakis, Dimensional reduction of the ABJM model, JHEP 1103:094,2011 (arXiv:1010.3808)

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:

• Martin Cederwall, M-branes on U-folds (arXiv:0712.4287)

• M.P. Garcia del Moral, Dualities as symmetries of the Supermembrane Theory (arXiv)

• pdf