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 superstrings (Bergshoeff-Sezgin-Townsend 87).
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 (M-theory), 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 were 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 (Nicolai-Helling 98, Dasgupta-Nicolai-Plefka 02). See also at membrane matrix model.
In a more modern perspective, the M2-brane worldvolume theory appears under AdS4-CFT3 duality as a holographic dual of a 4-dimensional Chern-Simons theory. Indeed, its Green-Schwarz action functional is entirely controled by the super-Lie algebra 4-cocycle of super Minkowski spacetime given by the brane scan. This exhibits the M2-brane worldvolume theory as a 3-dimensional higher WZW model?.
There are two different incarnations of the M2-brane. On the one hand it is defined as a Green-Schwarz sigma model with target space a spacetime that is a solution to the equations of motion of 11-dimensional supergravity. One would call this the “fundamental” M2 in analogy with the “fundamental string”, if only there were an “M2-perturbation series” which however is essentially ruled out.
On the other hand the M2 also appears as a black brane, hence as a solution to the equations of motion of 11-dimensional supergravity with singularity that looks from outside like a charged 2 dimensional object.
For this is a 1/2 BPS state of 11d sugra.
(table taken from Blumenhagen-Lüst-Theisen “Basic concepts of string theory”)
More generally, one may classify those solutions of 11-dimensional supergravity of the form for some closed manifold , that are at least 1/2 BPS states. One finds (Medeiros-Figueroa 10) that all these are of the form , where is an orbifold of the 7-sphere by a finite subgroup of the special unitary group, i.e. a finite group in the ADE-classification
|Dynkin diagram||Platonic solid||finite subgroup of||finite subgroup of||simple Lie group|
|cyclic group||cyclic group||special unitary group|
|dihedron/hosohedron||dihedral group||binary dihedral group||special orthogonal group|
|tetrahedron||tetrahedral group||binary tetrahedral group||E6|
|cube/octahedron||octahedral group||binary octahedral group||E7|
|dodecahedron/icosahedron||icosahedral group||binary icosahedral group||E8|
Here for -supersymmetry then the action of on is via the canonical action of as in the quaternionic Hopf fibration (Medeiros-Figueroa 10), while for then there is an extra twist to the action (MFFGME 09):
|BPS state||Dynkin label||AdS-CFT-dual worldvolume theory|
|(group of order 2)||BLG model|
|(cyclic group)||ABJM model|
|(binary dihedral group)||?|
|(binary tetrahedral group)||?|
|(binary octahedral group)||?|
|(binary icosahedral group)||?|
|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|
|D6-brane||D=7 super Yang-Mills theory|
|D1-brane||2d CFT with BH entropy|
|D3-brane||N=4 D=4 super Yang-Mills theory|
|(D25-brane)||(bosonic string theory)|
|NS-brane||type I, II, heterotic||circle n-connection|
|NS5-brane||B6-field||little string theory|
|D-brane for topological string|
|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|
|M9-brane/O9-plane||heterotic string theory|
|topological M2-brane||topological M-theory||C3-field on G2-manifold|
|topological M5-brane||C6-field on G2-manifold|
|solitons on M5-brane||6d (2,0)-superconformal QFT|
|self-dual string||self-dual B-field|
|3-brane in 6d|
and its quantization was first explored in
The interpretation of the membrane as as an object related to string theory, hence as the M2-brane was proposed in
around the time when M-theory became accepted due to
Hermann Nicolai, Robert Helling, Supermembranes and M(atrix) Theory, Lectures given by H. Nicolai at the Trieste Spring School on Non-Perturbative Aspects of String Theory and Supersymmetric Gauge Theories, 23 - 31 March 1998 (arXiv:hep-th/9809103)
Meanwhile AdS-CFT duality was recognized in
where a dual description of the worldvolume theory of M2-brane appears in seciton 3.2. More on this is in
An account of the history as of 1999 is in
More recent review is in
The generalization of this to BPS sugra solutions of the form is due to
Discussion of the history includes
Other recent developments are discussed in
The role of and the relation to duality in string theory of the membrane is discussed in the following articles.
Relation to T-duality is discussed in:
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 is discussed in:
M.P. Garcia del Moral, Dualities as symmetries of the Supermembrane Theory (arXiv)