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 (Bergshoeff-Sezgin-Townsend 87). By double dimensional reduction, these turn into the superstrings of type IIA string theory (Duff-Howe-Inami-Stelle 87). (See at duality between M-theory and type IIA string theory.)
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, 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 dimensional 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.
See at Green-Schwarz sigma model and brane scan.
As a black brane solution to the equations of motion of 11-dimensional supergravity the M2 is the spacetime $\mathbb{R}^{2,1} \times (\mathbb{R}^8-\{0\})$ with pseudo-Riemannian metric being
where
for $(\alpha,\beta) \in \mathbb{R}^2 \setminus \{(0,0)\}$;
and the field strength of the supergravity C-field is
For $\alpha \beta \neq 0$ this is a 1/2 BPS state of 11d sugra.
In the above coordinates the metric is ill-defined at $r = - \beta^{1/6} \alpha$, but in fact it may be smoothly continued through this point (Duff-Gibbons-Townsend 94, section 3), which is a event horizon. An actual singularity is at $r = 0$.
The near horizon geometry of this spacetime is the Freund-Rubin compactification AdS4$\times$S7. For more on this see at AdS-CFT.
1/2 BPS black branes in supergravity: D-branes, F1-brane, NS5-brane, M2-brane, M5-brane
(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 $AdS_4 \times X_7$ for some closed manifold $X_7$, that are at least 1/2 BPS states. One finds (Medeiros-Figueroa 10) that all these are of the form $AdS_4 \times S^7/G_{ADE}$, where $S^7 / G_{ADE}$ is an orbifold of the 7-sphere (a spherical space form in the smooth case, see there) by a finite subgroup of SU(2) $G_{ADE} \hookrightarrow SU(2)$, i.e. a finite group in the ADE-classification
Dynkin diagram via McKay correspondence | Platonic solid | finite subgroups of SO(3) | finite subgroups of SU(2) | simple Lie group |
---|---|---|---|---|
$A_n$ | cyclic group $\mathbb{Z}_{n+1}$ | cyclic group $\mathbb{Z}_{n+1}$ | special unitary group | |
$D_{n+4}$ | dihedron, hosohedron | dihedral group $D_{n+2}$ | binary dihedral group $2 D_{n+2}$ | special orthogonal group |
$E_6$ | tetrahedron | tetrahedral group $T$ | binary tetrahedral group $2T$ | E6 |
$E_7$ | cube, octahedron | octahedral group $O$ | binary octahedral group $2O$ | E7 |
$E_8$ | dodecahedron, icosahedron | icosahedral group $I$ | binary icosahedral group $2I$ | E8 |
Here for $5 \leq \mathcal{N} \leq 8$-supersymmetry then the action of $G_{ADE}$ on $S^7$ is via the canonical action of $SU(2)$ as in the quaternionic Hopf fibration (Medeiros-Figueroa 10), while for $\mathcal{N} = 4$ then there is an extra twist to the action (MFFGME 09). See the table below.
The worldvolume QFT of black M2-branes is a 3d superconformal gauge field theory:
$d$ | $N$ | superconformal super Lie algebra | R-symmetry | black brane worldvolume superconformal field theory via AdS-CFT |
---|---|---|---|---|
$\phantom{A}3\phantom{A}$ | $\phantom{A}2k+1\phantom{A}$ | $\phantom{A}B(k,2) \simeq$ osp$(2k+1/4)\phantom{A}$ | $\phantom{A}SO(2k+1)\phantom{A}$ | |
$\phantom{A}3\phantom{A}$ | $\phantom{A}2k\phantom{A}$ | $\phantom{A}D(k,2)\simeq$ osp$(2k/4)\phantom{A}$ | $\phantom{A}SO(2k)\phantom{A}$ | M2-brane 3d superconformal gauge field theory |
$\phantom{A}4\phantom{A}$ | $\phantom{A}k+1\phantom{A}$ | $\phantom{A}A(3,k)\simeq \mathfrak{sl}(4/k+1)\phantom{A}$ | $\phantom{A}U(k+1)\phantom{A}$ | D3-brane 4d superconformal gauge field theory |
$\phantom{A}5\phantom{A}$ | $\phantom{A}1\phantom{A}$ | $\phantom{A}F(4)\phantom{A}$ | $\phantom{A}SO(3)\phantom{A}$ | |
$\phantom{A}6\phantom{A}$ | $\phantom{A}k\phantom{A}$ | $\phantom{A}D(4,k) \simeq$ osp$(8/2k)\phantom{A}$ | $\phantom{A}Sp(k)\phantom{A}$ | M5-brane 6d superconformal gauge field theory |
(Shnider 88, also Nahm 78, see Minwalla 98, section 4.2)
Specifically, worldvolume quantum field theory of M2-branes sitting at ADE singularities (as above) is supposed to be described by ABJM theory and, for the special case of $SU(2)$ gauge group, by the BLG model. See also at gauge enhancement.
$N$ Killing spinors on spherical space form $S^7/\widehat{G}$ | $\phantom{AA}\widehat{G} =$ | spin-lift of subgroup of isometry group of 7-sphere | 3d superconformal gauge field theory on back M2-branes with near horizon geometry $AdS_4 \times S^7/\widehat{G}$ |
---|---|---|---|
$\phantom{AA}N = 8\phantom{AA}$ | $\phantom{AA}\mathbb{Z}_2$ | cyclic group of order 2 | BLG model |
$\phantom{AA}N = 7\phantom{AA}$ | — | — | — |
$\phantom{AA}N = 6\phantom{AA}$ | $\phantom{AA}\mathbb{Z}_{k\gt 2}$ | cyclic group | ABJM model |
$\phantom{AA}N = 5\phantom{AA}$ | $\phantom{AA}2 D_{k+2}$ $2 T$, $2 O$, $2 I$ | binary dihedral group, binary tetrahedral group, binary octahedral group, binary icosahedral group | (HLLLP 08a, BHRSS 08) |
$\phantom{AA}N = 4\phantom{AA}$ | $\phantom{A}2 D_{k+2}$ $2 O$, $2 I$ | binary dihedral group, binary octahedral group, binary icosahedral group | (HLLLP 08b, Chen-Wu 10) |
Under AdS-CFT duality the M2-brane is given by AdS4-CFT3 duality. (Maldacena 97, section 3.2, Klebanov-Torri 10).
Table of branes appearing in supergravity/string theory (for classification see at brane scan).
The Green-Schwarz sigma-model-type formulation of the supermembrane (as in the brane scan) first appears in
and its quantization of the was first explored in
Mike Duff, T. Inami, Christopher Pope, Ergin Sezgin, Kellogg Stelle, Semiclassical Quantization of the Supermembrane, Nucl.Phys. B297 (1988) 515-538 (spire:247064)
Bernard de Wit, Jens Hoppe, Hermann Nicolai, On the Quantum Mechanics of Supermembranes, Nucl. Phys. B305 (1988) 545. (pdf, pdf)
Bernard de Wit, W. Lüscher, Hermann Nicolai, The supermembrane is unstable, Nucl. Phys. B320 (1989) 135.
The double dimensional reduction of the M2-brane to the Green-Schwarz superstring was observed 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
The interpretation related to the BFSS matrix model of D0-branes is discussed in some detail in
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)
Arundhati Dasgupta, Hermann Nicolai, Jan Plefka, An Introduction to the Quantum Supermembrane, Grav.Cosmol.8:1,2002; Rev.Mex.Fis.49S1:1-10, 2003 (arXiv:hep-th/0201182)
Gijs van den Oord, On Matrix Regularisation of Supermembranes, 2006 (pdf)
The back membrane solution of 11-dimensional supergravity was found in
Its regularity throught the event horizon is due to
The Horava-Witten-orientifold of the black M2, supposedly yielding the black heterotic string is discussed in
Zygmunt Lalak, André Lukas, Burt Ovrut, Soliton Solutions of M-theory on an Orbifold, Phys. Lett. B425 (1998) 59-70 (arXiv:hep-th/9709214)
Ken Kashima, The M2-brane Solution of Heterotic M-theory with the Gauss-Bonnet $R^2$ terms, Prog.Theor.Phys. 105 (2001) 301-321 (arXiv:hep-th/0010286)
Meanwhile AdS-CFT duality was recognized in
where a dual description of the worldvolume theory of M2-brane appears in section 3.2. More on this is in
An account of the history as of 1999 is in
More recent review is in
A detailed discussion of this black brane-realization of the M2 and its relation to AdS-CFT is in
The generalization of this to $\geq 1/2$ BPS sugra solutions of the form $AdS_4 \times X_7$ is due to
Paul de Medeiros, José Figueroa-O'Farrill, Sunil Gadhia, Elena Méndez-Escobar, Half-BPS quotients in M-theory: ADE with a twist, JHEP 0910:038,2009 (arXiv:0909.0163, pdf slides)
Paul de Medeiros, José Figueroa-O'Farrill, Half-BPS M2-brane orbifolds, Adv. Theor. Math. Phys. Volume 16, Number 5 (2012), 1349-1408. (arXiv:1007.4761, Euclid)
Discussion of the history includes
Other recent developments are discussed in
Paul Howe, Ergin Sezgin, The supermembrane revisited, (arXiv:hep-th/0412245)
Igor 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. See there for more pointers. The relation of these to the above is discussed in section 3 of
Discusson of boundary conditions in the ABJM model (for M2-branes ending on M5-branes) is in
A kind of double dimensional reduction of the ABJM model to something related to type II superstrings and D1-branes is discussed in
Discussion of the ABJM model in Horava-Witten theory and reducing to heterotic strings is in
Discussion of general phenomena of M-branes in higher geometry and generalized cohomology is in
Discussion from the point of view of Green-Schwarz action functional-∞-Wess-Zumino-Witten theory is 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:
Martin Cederwall, M-branes on U-folds (arXiv:0712.4287)
M.P. Garcia del Moral, Dualities as symmetries of the Supermembrane Theory (arXiv)
Discussion from the point of view of E11-U-duality and current algebra is in
Hirotaka Sugawara, Current Algebra Formulation of M-theory based on E11 Kac-Moody Algebra, International Journal of Modern Physics A, Volume 32, Issue 05, 20 February 2017 (arXiv:1701.06894)
Shotaro Shiba, Hirotaka Sugawara, M2- and M5-branes in E11 Current Algebra Formulation of M-theory (arXiv:1709.07169)
Last revised on May 17, 2018 at 04:17:34. See the history of this page for a list of all contributions to it.