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 , , 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:
7: …
11: M2-brane
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]
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:
Via the superembedding approach the equations of motion were obtained in
and the Lagrangian density in:
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 from Supermembranes in , 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
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 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:
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 :
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:
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:
Observation that the spectrum is continuous:
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 -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:
Relation to the string dilaton under double dimensional reduction:
Last revised on June 16, 2023 at 11:32:37. See the history of this page for a list of all contributions to it.