nLab membrane

Idea

The next higher dimensional generalization of the concept of a relativistic particle and a relativistic string is called the relativistic membrane.

The concept of a bosonic membrane was considered already by Dirac 62 as a hypothetical model for the electron, then abandoned and eventually re-incarnated as the “super-membrane in 11d” due to Bergshoeff-Sezgin-Townsend 87 (whose “m” became the “M” in “M-theory”), namely the Green-Schwarz sigma-model for super p-branes corresponding to the brane scan-entry with $p=2$, $D=11$, now commonly known as the M2-brane.

In fact, according to the brane scan, a super 2-brane sigma model exists on superspacetimes of dimension 4, 5, 7, and 11:

• 4: super 2-brane in 4d

• 5: super 2-brane in 5d

• 7: …

• 11: M2-brane

Related concepts

• ´membrane instanton

References

Pre-history

Early speculations trying to unify the electron and the muon as two excitations of a single relativistic membrane:

• Paul Dirac, An Extensible Model of the Electron, Proc. Roy. Soc. A 268 (1962) 57-67 [jstor:2414316]

  (also proposing the Dirac-Born-Infeld action)

• Paul Dirac, The motion of an Extended Particle in the Gravitational Field, in: L. Infeld (ed.), Relativistic Theories of Gravitation, Proceedings of a Conference held in Warsaw and Jablonna, July 1962, P. W. N. Publishers, 1964, Warsaw, 163-171; discussion 171-175 [spire:1623740, article:pdf, full proceedings:pdf]

• Paul Dirac, Particles of Finite Size in the Gravitational Field, Proc. Roy. Soc. A 270 (1962) 354-356 [doi:10.1098/rspa.1962.0228]

Super-membrane/M2-brane as a sigma model

The Green-Schwarz sigma-model-type formulation of the super-membrane in 11d (as in the brane scan and in contrast to the black brane-solutions of 11d supergravity) first appears in:

• Eric Bergshoeff, Ergin Sezgin, Paul Townsend: Supermembranes and eleven-dimensional supergravity, Phys. Lett. B 189 (1987) 75-78 [doi:10.1016/0370-2693(87)91272-X, spire:248230]

Via the superembedding approach the equations of motion were obtained in

• Igor Bandos, Paolo Pasti, Dmitri Sorokin, Mario Tonin, Dmitry Volkov, Chapter 3 of Superstrings and supermembranes in the doubly supersymmetric geometrical approach, Nucl. Phys. B446:79-118, 1995 (arXiv:hep-th/9501113)

and the Lagrangian density in:

• Paul Howe, Ergin Sezgin, The supermembrane revisited, Class. Quant. Grav. 22 (2005) 2167-2200 [arXiv:hep-th/0412245, doi:10.1088/0264-9381/22/11/017]

The double dimensional reduction of the M2-brane to the Green-Schwarz superstring was observed in

• Michael Duff, Paul Howe, T. Inami, Kellogg Stelle, Superstrings in $D=10$ from Supermembranes in $D=11$, Phys. Lett. B 191 (1987) 70 [doi:10.1016/0370-2693(87)91323-2]

  also in: Michael Duff (ed.): The World in Eleven Dimensions 205-206 (1987) [spire:245249]

• Paul Townsend, The eleven-dimensional supermembrane revisited, Phys. Lett. B 350 (1995) 184-187 [arXiv:hep-th/9501068, doi:10.1016/0370-2693(95)00397-4]

around the time when M-theory became accepted due to

• Edward Witten, String Theory Dynamics In Various Dimensions (arXiv:hep-th/9503124)

On the early history of these developments:

• Eric A. Bergshoeff, Ergin Sezgin, Paul K. Townsend: A brief history of supermembranes [arXiv:2503.06234]

See also:

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

• Maria P. Garcia del Moral, C. Las Heras, P. Leon, J. M. Pena, Alvaro Restuccia, Fluxes, Twisted tori, Monodromy and $U(1)$ Supermembranes, J. High Energ. Phys. 2020 97 (2020) [arXiv:2005.06397, doi:10.1007/JHEP09(2020)097]

Discussion from the point of view of Green-Schwarz action functional-∞-Wess-Zumino-Witten theory:

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields (2013)

On possible structures in M2-brane dynamics and M2-M5-brane bound states which could be M-theory-lifts of the familiar integrability of the Green-Schwarz superstring on $AdS_5$ $\times$ $S^5$:

• Kirill Gubarev, Edvard Musaev, Integrability structures in string theory [arXiv:2301.06486]

Quantization of the M2-brane sigma-model to a matrix model

The Poisson bracket-formulation of the classical light-cone gauge Hamiltonian for the bosonic relativistic membrane and the corresponding matrix commutator regularization is due to:

• Jens Hoppe, Quantum theory of a massless relativistic surface and a two-dimensional bound state problem, MIT 1982 (dspace:1721.1/15717, pdf)

Some exact solutions:

• Jens Hoppe, Exact algebraic M(em)brane solutions $[$arXiv:2107.00569$]$

• Jens Hoppe, The fast non-commutative sharp drop $[$arXiv:2302.13146$]$

On the regularized light-cone gauge quantization of the Green-Schwarz sigma model for the M2-brane on (super) Minkowski spacetime, yielding the BFSS matrix model:

Original articles:

• Bernard de Wit, Jens Hoppe, Hermann Nicolai, On the Quantum Mechanics of Supermembranes, Nucl. Phys. B 305 (1988) 545-581 [doi:10.1016/0550-3213(88)90116-2, spire:261702, pdf, pdf]

Observation that the spectrum is continuous:

• Bernard de Wit, W. Lüscher, Hermann Nicolai, The supermembrane is unstable, Nucl. Phys. B 320 (1989) 135 [doi:10.1016/0550-3213(89)90214-9, spire:266584]

Review:

• Hermann Nicolai, Robert C. Helling, Supermembranes and M(atrix) Theory, In: Trieste 1998, Nonperturbative aspects of strings, branes and supersymmetry (1998) 29-74 (arXiv:hep-th/9809103, spire:476366)

• Jens Hoppe, Membranes and Matrix Models [arXiv:hep-th/0206192]

• Arundhati Dasgupta, Hermann Nicolai, Jan Plefka, An Introduction to the Quantum Supermembrane, Grav. Cosmol. 8 1 (2002) and Rev. Mex. Fis. 49S1 (2003) 1-10 [arXiv:hep-th/0201182, spire:582067]

• Gijs van den Oord, On Matrix Regularisation of Supermembranes MSc thesis (2006) (pdf)

• Meer Ashwinkumar, Lennart Schmidt, Meng-Chwan Tan, Section 2 of: Matrix Regularization of Classical Nambu Brackets and Super $p$-Branes (arXiv:2103.06666)

The generalization to pp-wave spacetimes (leading to the BMN matrix model):

• Keshav Dasgupta, Mohammad Sheikh-Jabbari, Mark Van Raamsdonk, Section 2 of: Matrix Perturbation Theory For M-theory On a PP-Wave, JHEP 0205:056, 2002 (arXiv:hep-th/0205185)

• Keshav Dasgupta, Mohammad Sheikh-Jabbari, Mark Van Raamsdonk, Section 2 of: Matrix Perturbation Theory For M-theory On a PP-Wave, JHEP 0205:056, 2002 (arXiv:hep-th/0205185)

See also

• Mike Duff, T. Inami, Christopher Pope, Ergin Sezgin, Kellogg Stelle, Semiclassical Quantization of the Supermembrane, Nucl.Phys. B297 (1988) 515-538 (spire:247064)

• Daniel Kabat, Washington Taylor, section 2 of: Spherical membranes in Matrix theory, Adv. Theor. Math. Phys. 2: 181-206, 1998 (arXiv:hep-th/9711078)

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

• Qiang Jia, On matrix description of D-branes (arXiv:1907.00142)

A new kind of perturbation series for the quantized super-membrane:

• Olaf Lechtenfeld, Hermann Nicolai, A perturbative expansion scheme for supermembrane and matrix theory (arXiv:2109.00346)

Relation to the string dilaton under double dimensional reduction:

• Krzysztof A. Meissner, Hermann Nicolai, Fundamental Membranes and the String Dilaton, J. High Energ. Phys. 2022 219 (2022) [doi:10.1007/JHEP09(2022)219, arXiv:2208.05822]

See also:

• So Katagiri: A Lorentz Covariant Matrix Model for Bosonic M2-Branes: Nambu Brackets and Restricted Volume-Preserving Deformations [arXiv2504.05940]