nLab matrix model

Idea

Matrix models are physical models in which the dynamical quantities are square matrices (in certain class of matrices, e.g. hermitian), in other words, a Lagrangian/Hamiltonian depends on matrix quantities and is usually taken at the limit when the size of matrix tends to infinity.

Matrix models are studied mainly in the context of statistical mechanics (see random matrix theory) or in quantum field theory.

Related entries

• Fredholm determinant, random matrix theory, determinental process?, Kontsevich matrix model?, large N limit

• integrable system, topological recursion

Matrix models that have been argued to capture D-brane dynamics and nonperturbative effects in string theory include

• BFSS matrix model

• IKKT matrix model

• BMN matrix model

• membrane matrix model

• K-matrix model

• matrix model for the 3-brane in 6d

References

For QCD

First inkling of matrix models from the large N limit of QCD:

• Tohru Eguchi, Hikaru Kawai, Reduction of Dynamical Degrees of Freedom in the Large-$N$ Gauge Theory, Phys. Rev. Lett. 48, 1063 (1982) (spire:176459, doi:10.1103/PhysRevLett.48.1063)

• A. Gonzalez-Arroyo, M. Okawa, A twisted model for large $N$ lattice gauge theory, Physics Letters B Volume 120, Issues 1–3, 6 January 1983, Pages 174-178 (doi:10.1016/0370-2693(83)90647-0)

• A. Gonzalez-Arroyo, M. Okawa, Twisted-Eguchi-Kawai model: A reduced model for large- $N$ lattice gauge theory, Phys. Rev. D 27, 2397 (1983) (doi:10.1103/PhysRevD.27.2397)

For 2D quantum gravity

On matrix models for D=2 quantum gravity:

• Edouard Brézin, Jean Zinn-Justin: Renormalization Group Approach to Matrix Models, Phys. Lett. B 288 (1992) 54-58 [arXiv:hep-th/9206035, doi:10.1016/0370-2693(92)91953-7]

• Astrid Eichhorn, Tim Koslowski: Continuum limit in matrix models for quantum gravity from the Functional Renormalization Group, Phys. Rev. D 88 (2013) 084016 [arXiv:1309.1690, doi:10.1103/PhysRevD.88.084016]

In string/M-theory

For references on the BFSS-, IKKT- and BMN matrix models see there.

Quantum Hall effect via non-commutative geometry

Discussion of the integer quantum Hall effect via a Brillouin torus with noncommutative geometry and using the Connes-Chern character:

• J. Bellissard, A. van Elst, H. Schulz-Baldes: The noncommutative geometry of the quantum Hall effect, J. Math. Phys. 35 5373 (1994) [doi:10.1063/1.530758, cond-mat/9411052]

Generalization of BvESB94 to the fractional quantum Hall effect:

• Matilde Marcolli, Varghese Mathai: Towards the fractional quantum Hall effect: a noncommutative geometry perspective, in Noncommutative Geometry and Number Theory, Vieweg (2006) [doi:10.1007/978-3-8348-0352-8_12, arXiv:cond-mat/0502356]

See also exposition in:

• Vincent Pasquier: Quantum Hall Effect and Non-commutative Geometry, in: Quantum Spaces, Progress in Mathematical Physics 53, Birkhäuser (2007) [doi:10.1007/978-3-7643-8522-4_1, pdf]

Discussion of the fractional quantum Hall effect via abelian but noncommutative (matrix model-)Chern-Simons theory:

• Leonard Susskind: The Quantum Hall Fluid and Non-Commutative Chern Simons Theory [arXiv:hep-th/0101029]

• Simeon Hellerman, Leonard Susskind: Realizing the Quantum Hall System in String Theory [arXiv:hep-th/0107200]

  (relating this to M5-branes via the BFSS matrix model)

• Alexios P. Polychronakos: Quantum Hall states as matrix Chern-Simons theory, JHEP 0104:011 (2001) [doi:10.1088/1126-6708/2001/04/011, arXiv:hep-th/0103013]

• Simeon Hellerman, Mark Van Raamsdonk: Quantum Hall Physics = Noncommutative Field Theory, JHEP 0110:039 (2001) [doi:10.1088/1126-6708/2001/10/039, arXiv:hep-th/0103179]

• Eduardo Fradkin, Vishnu Jejjala, Robert G. Leigh: Non-commutative Chern-Simons for the Quantum Hall System and Duality, Nucl. Phys. B 642 (2002) 483-500 [doi:10.1016/S0550-3213(02)00616-8, arXiv:cond-mat/0205653]

• Andrea Cappelli, Ivan D. Rodriguez: Matrix Effective Theories of the Fractional Quantum Hall effect, J. Phys. A 42 (2009) 304006 [doi:10.1088/1751-8113/42/30/304006, arXiv:0902.0765]

• Zhihuan Dong, T. Senthil: Non-commutative field theory and composite Fermi Liquids in some quantum Hall systems, Phys. Rev. B 102 (2020) 205126 [doi:10.1103/PhysRevB.102.205126, arXiv:2006.01282]

See also

Discussion of topological recursion for matrix models originates with

• Leonid Chekhov, Bertrand Eynard, Nicolas Orantin, Free energy topological expansion for the 2-matrix model, JHEP 0612:053,2006 (arXiv:math-ph/0603003)

See also

• Raghav G. Jha, Introduction to Monte Carlo for Matrix Models (arXiv:2111.02410)

• Chong-Sun Chu, A Matrix Model Proposal for Quantum Gravity and the Quantum Mechanics of Black Holes [arXiv:2406.01466]

category: physics