nLab IKKT matrix model

Idea

The KK-compactification of D=10 super Yang-Mills theory all the way to the point yields a theory whose fields are simply elements of the gauge Lie algebra $\mathfrak{g}$, hence matrices for a matrix Lie algebra. This physics is called the IKKT matrix model.

Alternatively this model can be motivated from a certain regularization of the worldsheet action functional of the superstring. This is how it was originally obtained in (IKKT 96).

It has been argued that for $\mathfrak{g} = \mathfrak{su}(N)$ the special unitary Lie algebra and in the large N limit, this model captures aspects of non-perturbative type IIB string theory (which would be F-theory, see also at M-theory). Therefore this is also called the IIB matrix model (in contrast to the BFSS matrix model in type IIA string theory).

Several authors have explored the possibility to lift the derivation of the IKKT model from the superstring to the M2-brane. See at membrane matrix model for more on this.

In (Kim-Nishimura-Tsuchiya 12) it is claimed that computer simulation of the IKKT matrix model, regarded as non-perturbative type IIB string theory, shows a spontaneous emerging spacetime of macroscopic dimension 3+1, with 6 microscopic dimensions. (A similar claim results from a very different argument: the Brandenberger-Vafa mechanism.)

Related concepts

matrix models for brane dynamics:

D-brane	matrix model
D0-brane	BFSS matrix model, BMN matrix model
D(-1)-brane	IKKT matrix model
D4-brane	nuclear matrix model

M-brane	matrix model
D2-brane	membrane matrix model

See also:

• K-matrix model

• lattice gauge theory

References

General

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)

The original articles on the IKKT matrix model:

• Noboyuki Ishibashi, Hikaru Kawai, Y. Kitazawa, A. Tsuchiya, A Large-$N$ Reduced Model as Superstring, Nucl. Phys. B498 (1997) 467-491 (arXiv:hep-th/9612115, doi:10.1016/S0550-3213(97)00290-3)

• H. Aoki, S. Iso, Hikaru Kawai, Y. Kitazawa, T. Tada, A. Tsuchiya, IIB Matrix Model, Prog.Theor.Phys.Suppl.134:47-83,1999 (arXiv:hep-th/9908038)

See also

• Mikhail Movshev, Albert Schwarz, On maximally supersymmetric Yang-Mills theories, Nucl.Phys. B681 (2004) 324-350 (arXiv:hep-th/0311132)

Review of the grand perspective with emphasis on noncommutative geometry (fuzzy spheres):

• Harold Steinacker, Emergent Geometry and Gravity from Matrix Models: an Introduction, Class. Quant. Grav. 27 (2010) 133001 [arXiv:1003.4134]

• Harold Steinacker, Non-commutative geometry and matrix models, PoS QGQGS2011 (2011) 004 [arXiv:1109.5521]

• Badis Ydri, Review of M(atrix)-Theory, Type IIB Matrix Model and Matrix String Theory (arXiv:1708.00734), published as: Matrix Models of String Theory, IOP 2018 (ISBN:978-0-7503-1726-9)

See also at D-brane geometry.

Derivation from open string field theory is discussed in

• Taejin Lee, Covariant Open String Field Theory on Multiple D$p$-Branes (arXiv:1703.06402)

• Harold C. Steinacker, Tung Tran, Soft limit of higher-spin interactions in the IKKT model [arXiv:2311.14163]

• Harold C. Steinacker, Tung Tran, Spinorial description for Lorentzian $\mathfrak{hs}$-IKKT [arXiv:2312.16110]

Arguments that full Yang-Mills theory generalized to noncommutative geometry is recovered as the perturbation theory around classical solutions of the IKKT model are in

• H. Aoki, Noboyuki Ishibashi, S. Iso, Hikaru Kawai, Y. Kitazawa, T. Tada, Noncommutative Yang-Mills in IIB Matrix Model, Nucl.Phys. B565 (2000) 176-192 (arXiv:hep-th/9908141)

• Tatsuo Azeyanagi, Masanori Hanada, Tomoyoshi Hirata, On Matrix Model Formulations of Noncommutative Yang-Mills Theories, Phys.Rev.D78:105017,2008 (arXiv:0806.3252)

Emergence of higher spin gravity from the IKKT matrix model:

• Harold Steinacker, Higher-spin gravity and torsion on quantized space-time in matrix models, Journal of High Energy Physics 2020 111 (2020) [arXiv:2002.02742, doi:10.1007/JHEP04(2020)111]

• Harold Steinacker, Gravity as a Quantum Effect on Quantum Space-Time, Phys.Lett. B 827 (2022) 136946 [arXiv:2110.03936, doi:10.1016/j.physletb.2022.136946]

• Yuhma Asano, Harold Steinacker, Spherically symmetric solutions of higher-spin gravity in the IKKT matrix model, Nuclear Physics B 980 (2022) 115843 [arXiv:2112.08204, doi:10.1016/j.nuclphysb.2022.115843]

• Harold Steinacker, Tung Tran, Spinorial higher-spin gauge theory from IKKT in Euclidean and Minkowski signatures [arXiv:2305.19351]

• Harold C. Steinacker, Tung Tran, Quantum $\mathfrak{hs}$-Yang-Mills from the IKKT matrix model [arXiv:2405.09804]

Arguments that closed string field theory arises from the dynamics of Wilson loops in the IKKT model are in

• M. Fukuma, Hikaru Kawai, Y. Kitazawa, A. Tsuchiya, String Field Theory from IIB Matrix Model, Nucl.Phys.B510:158-174,1998 (arXiv:hep-th/9705128)

• Daiji Ennyu, Hiroshi Kawabe, Naohito Nakazawa, Note on a Closed String Field Theory from Bosonic IIB Matrix Model, JHEP 0301 (2003) 025 (arXiv:hep-th/0212044)

Possibilities of generalizing the IKKT model from Lie algebras to Lie 2-algebras in some membrane matrix model are explored in

• Patricia Ritter, Christian Saemann, Lie 2-algebra models (arXiv:1308.4892)

Discussion of standard model phenomenology within the IKKT model includes

• Harold Steinacker, Jochen Zahn, An extended standard model and its Higgs geometry from the matrix model, PTEP 2014 (2014) 8, 083B03 (arXiv:1401.2020)

See also

• A. Stern, Chuang Xu, Signature change in matrix model solutions (arXiv:1808.07963)

• Frans R. Klinkhamer, On the emergence of an expanding universe from a Lorentzian matrix model, Prog. Theor. Exp. Phys. 2020, 103B03 (2020) (arXiv:1912.12229)

• Frans R. Klinkhamer, IIB matrix model: Emergent spacetime from the master field (arXiv:2007.08485)

• Frans R. Klinkhamer, IIB matrix model: Extracting spacetime points (arXiv:2008.01058)

• Frans R. Klinkhamer, Regularized big bang and IIB matrix model (arXiv:2009.06525)

• Frans R. Klinkhamer, M-theory and the birth of the Universe (arXiv:2102.11202)

• Frans R. Klinkhamer, A first look at the master-field equation of the IIB matrix model (arXiv:2105.05831)

• Emmanuele Battista, Harold C. Steinacker, One-loop effective action of the IKKT model for cosmological backgrounds [arXiv:2310.11126]

• Samuel Laliberte, Effective mass and symmetry breaking in the IKKT matrix model from compactification [arXiv:2401.16401]

• Yuhma Asano: Quantisation of type IIB superstring theory and the matrix model [arXiv:2408.04000]

• Sean A. Hartnoll, Jun Liu: The Polarised IKKT Matrix Model [arXiv:2409.18706]

• Alessandro Manta, Harold C. Steinacker, Tung Tran: $\mathfrak{hs}$-extended gravity from the IKKT matrix model [arXiv:2411.02598]

In relation to the Brandenberger-Vafa mechanism:

• Robert Brandenberger, Julia Pasiecznik: On the Origin of the $SO(9) \to SO(3) \times SO(6)$ Symmetry Breaking in the IKKT Matrix Model [arXiv:2409.00254]

• Suddhasattwa Brahma, Robert Brandenberger, Samuel Laliberte: BFSS Matrix Model Cosmology: Progress and Challenges, Physics 5 1 (2023) [arXiv:2210.07288, doi:10.3390/physics5010001]

• Robert Brandenberger: Emergent Metric Space-Time from the BFSS Matrix Model, talk at Perimeter Institute (May 2023) [doi:10.48660/23050109, video:pirsa:23050109, slides:pdf]

See also:

• Shota Komatsu, Adrien Martina, João Penedones, Antoine Vuignier, Xiang Zhao: Einstein gravity from a matrix integral – Part I [arXiv:2410.18173]

• Shota Komatsu, Adrien Martina, Joao Penedones, Antoine Vuignier, Xiang Zhao: Einstein gravity from a matrix integral – Part II [arXiv:2411.18678]

Computer simulation

There are claims that numerical computer simulations (as in lattice gauge theory, see the references there) show that the IKKT matrix model predicts a spontanously generated spacetime where exactly 3+1 dimensions become macroscopic (hence effectively predicts moduli stabilization in spintaneous KK-compactification of M-theory to $D = 3+1$ macroscopic dimensions ):

• S.-W. Kim, J. Nishimura, and A. Tsuchiya, Expanding (3+1)-dimensional universe from a Lorentzian matrix model for superstring theory in (9+1)-dimensions, Phys. Rev. Lett. 108, 011601 (2012), (arXiv:1108.1540).

• S.-W. Kim, J. Nishimura, and A. Tsuchiya, Late time behaviors of the expanding universe in the IIB matrix model, JHEP 10, 147 (2012), (arXiv:1208.0711).

• Yuta Ito, Jun Nishimura, Asato Tsuchiya, Large-scale computation of the exponentially expanding universe in a simplified Lorentzian type IIB matrix model (arXiv:1512.01923)

• Toshihiro Aoki, Mitsuaki Hirasawa, Yuta Ito, Jun Nishimura, Asato Tsuchiya, On the structure of the emergent 3d expanding space in the Lorentzian type IIB matrix model (arXiv:1904.05914)

• Kohta Hatakeyama, Akira Matsumoto, Jun Nishimura, Asato Tsuchiya, Atis Yosprakob, The emergence of expanding space-time and intersecting D-branes from classical solutions in the Lorentzian type IIB matrix model (arxiv:1911.08132)

• Konstantinos N. Anagnostopoulos, Takehiro Azuma, Yuta Ito, Jun Nishimura, Toshiyuki Okubo, Stratos Kovalkov Papadoudis, Complex Langevin analysis of the spontaneous breaking of 10D rotational symmetry in the Euclidean IKKT matrix model (arXiv:2002.07410)

Status report:

• Progress in the numerical studies of the type IIB matrix model [arXiv:2210.17537]

Further:

• The emergence of expanding space-time in the Lorentzian type IIB matrix model with a novel regularization [arXiv:2307.01681]