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:
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. A268, (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 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 (spire:1623740)
Paul Dirac, Particles of Finite Size in the Gravitational Field, Proc. Roy. Soc. A270, (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:
The equations of motion of the super membrane are derived via the superembedding approach in
and the Lagrangian density for the super membrane is derived via the superembedding approach in
Discussion from the point of view of Green-Schwarz action functional-∞-Wess-Zumino-Witten theory is in
The double dimensional reduction of the M2-brane to the Green-Schwarz superstring was observed in
The interpretation of the super-membrane as an object related to string theory via double dimensional reduction, hence as the M2-brane was proposed in
around the time when M-theory became accepted due to
See also
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)
M.P. Garcia del Moral, C. Las Heras, P. Leon, J.M. Pena, A. Restuccia, Fluxes, Twisted tori, Monodromy and $U(1)$ Supermembranes (arXiv:2005.06397)
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:
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 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; Rev. Mex. Fis. 49S1:1-10, 2003 (arXiv:hep-th/0201182)
Gijs van den Oord, On Matrix Regularisation of Supermembranes, 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: