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.
|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|
|D0-brane||BFSS matrix model|
|D4-brane||D=5 super Yang-Mills theory with Khovanov homology observables|
|D1-brane||2d CFT with BH entropy|
|D3-brane||N=4 D=4 super Yang-Mills theory|
|NS-brane||type I, II, heterotic||circle n-connection|
|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|
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
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
P.S. Howe, E. Sezgin, The supermembrane revisited, (arXiv:hep-th/0412245)
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:
M.P. Garcia del Moral, Dualities as symmetries of the Supermembrane Theory (arXiv)