nLab quantization of M2-brane to matrix model -- references

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, PhD thesis, MIT (1982) [hdl:1721.1/15717, pdf]

Some exact solutions:

• Jens Hoppe: Exact algebraic M(em)brane solutions, Asian J. Math. 26 2 (2022) 253-256 [doi:10.4310/AJM.2022.v26.n2.a4, arXiv:2107.00569]

• Jens Hoppe: The fast non-commutative sharp drop, Physics Letters B 848 (2024) 138357 [doi:10.1016/j.physletb.2023.138357, arXiv:2302.13146]

Discussion of the residual (in light cone gauge) gauge symmetry of area-preserving diffeomorphisms:

• Emmanuel G. Floratos, John Iliopoulos: A note on the classical symmetries of the closed bosonic membranes, Physics Letters B 201 2 (1988) 237-240 [doi:10.1016/0370-2693(88)90220-1]

• Bernard de Wit, U. Marquard, Hermann Nicolai: Area-preserving diffeomorphisms and supermembrane Lorentz invariance, Commun. Math. Phys. 128 (1990) 39–62 [doi:10.1007/BF02097044, inSpire:277216]

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) [doi:10.1088/1126-6708/2002/05/056, arXiv:hep-th/0205185]

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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]

Covariant discussion via volume-preserving diffeomorphism symmetry:

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

• So Katagiri: Quantum Stability at One Loop for BPS Membranes in a Lorentz-Covariant RVPD Matrix Model [arXiv:2509.23853]

See also:

• Eric Bergshoeff, Ergin Sezgin, Y. Tanii, Paul K. Townsend: Super $p$-branes as gauge theories of volume preserving diffeomorphisms, Annals of Physics 199 2 (1990) 340-365 [doi:10.1016/0003-4916(90)90381-W]