Skip to main content
SpringerLink
Log in

Model of M-theory with eleven matrices

  • Published:
Journal of High Energy Physics Aims and scope Submit manuscript

Abstract

We show that an action of a supermembrane in an eleven-dimensional spacetime with a semi-light-cone gauge can be written only with Nambu-Poisson bracket and an invariant symmetric bilinear form under an approximation. Thus, the action under the conditions is manifestly covariant under volume preserving diffeomorphism even when the world-volume metric is flat. Next, we propose two 3-algebraic models of M-theory which are obtained as a second quantization of an action that is equivalent to the supermembrane action under the approximation. The second quantization is defined by replacing Nambu-Poisson bracket with finite-dimensional 3-algebras’ brackets. Our models include eleven matrices corresponding to all the eleven space-time coordinates in M-theory although they possess not SO(1, 10) but SO(1, 2) × SO(8) or SO(1, 2) × SU(4) × U(1) covariance. They possess \( \mathcal{N} = 1 \) space-time supersymmetry in eleven dimensions that consists of 16 kinematical and 16 dynamical ones. We also show that the SU(4) model with a certain algebra reduces to BFSS matrix theory if DLCQ limit is taken.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. T. Banks, W. Fischler, S.H. Shenker and L. Susskind, M theory as a matrix model: A conjecture, Phys. Rev. D 55 (1997) 5112 [hep-th/9610043] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  2. J. Bagger and N. Lambert, Modeling multiple M2’s, Phys. Rev. D 75 (2007) 045020 [hep-th/0611108] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  3. A. Gustavsson, Algebraic structures on parallel M2-branes, Nucl. Phys. B 811 (2009) 66 [arXiv:0709.1260] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  4. J. Bagger and N. Lambert, Gauge Symmetry and Supersymmetry of Multiple M2-Branes, Phys. Rev. D 77 (2008) 065008 [arXiv:0711.0955] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  5. O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  6. J. Bagger and N. Lambert, Three-Algebras and N = 6 Chern-Simons Gauge Theories, Phys. Rev. D 79 (2009) 025002 [arXiv:0807.0163] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  7. Y. Honma, S. Iso, Y. Sumitomo and S. Zhang, Janus field theories from multiple M2 branes, Phys. Rev. D 78 (2008) 025027 [arXiv:0805.1895] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  8. Y. Matsuo, Nambu bracket as a new gauge symmetry in M theory, talk given at JPS 2008 Autumn Meeting, Yamagata Japan, 20-23 Sep. 2008.

  9. M. Sato, Covariant Formulation of M-theory, Int. J. Mod. Phys. A 24 (2009) 5019 [arXiv:0902.1333] [SPIRES].

    ADS  Google Scholar 

  10. K. Lee and J.-H. Park, Partonic description of a supersymmetric p-brane, JHEP 04 (2010) 043 [1001.4532] [SPIRES].

    Article  Google Scholar 

  11. Y. Matsuo, New Formulation of M-theory and Nambu Bracket, BUTSURI 65 (2010) 156.

    Google Scholar 

  12. B. de Wit, J. Hoppe and H. Nicolai, On the Quantum Mechanics of Supermembranes, Nucl. Phys. B 305 1988 545 [SPIRES].

    Article  ADS  Google Scholar 

  13. N. Ishibashi, H. Kawai, Y. Kitazawa and A. Tsuchiya, A large-N reduced model as superstring, Nucl. Phys. B 498 (1997) 467 [hep-th/9612115] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  14. Y. Nambu, Generalized Hamiltonian dynamics, Phys. Rev. D 7 (1973) 2405 [SPIRES].

    MathSciNet  ADS  Google Scholar 

  15. H. Awata, M. Li, D. Minic and T. Yoneya, On the quantization of Nambu brackets, JHEP 02 (2001) 013 [hep-th/9906248] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  16. L. Smolin, M theory as a matrix extension of Chern-Simons theory, Nucl. Phys. B 591 (2000) 227 [hep-th/0002009] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  17. L. Smolin, The cubic matrix model and a duality between strings and loops, hep-th/0006137 [SPIRES].

  18. T. Azuma, S. Iso, H. Kawai and Y. Ohwashi, Supermatrix models, Nucl. Phys. B 610 (2001) 251 [hep-th/0102168] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  19. K. Lee and J.-H. Park, Three-algebra for supermembrane and two-algebra for superstring, JHEP 04 (2009) 012 [arXiv:0902.2417] [SPIRES].

    Article  ADS  Google Scholar 

  20. S. Carlip, Loop Calculations For The Green-Schwarz Superstring, Phys. Lett. B 186 (1987) 141 [SPIRES].

    MathSciNet  ADS  Google Scholar 

  21. R.E. Kallosh, Quantization of Green-Schwarz Superstring, Phys. Lett. B 195 (1987) 369 [SPIRES].

    ADS  Google Scholar 

  22. Y. Kazama and N. Yokoi, Superstring in the plane-wave background with RR flux as a conformal field theory, JHEP 03 (2008) 057 [arXiv:0801.1561] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  23. T. Banks, N. Seiberg and S.H. Shenker, Branes from matrices, Nucl. Phys. B 490 (1997) 91 [hep-th/9612157] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  24. P.-M. Ho and Y. Matsuo, M5 from M2, JHEP 06 (2008) 105 [arXiv:0804.3629] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  25. P.-M. Ho, Y. Imamura, Y. Matsuo and S. Shiba, M5-brane in three-form flux and multiple M2-branes, JHEP 08 (2008) 014 [arXiv:0805.2898] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  26. E. Bergshoeff, E. Sezgin and P.K. Townsend, Supermembranes and eleven-dimensional supergravity, Phys. Lett. B 189 (1987) 75 [SPIRES].

    MathSciNet  ADS  Google Scholar 

  27. P.-M. Ho, Y. Matsuo and S. Shiba, Lorentzian Lie (3)-algebra and toroidal compactification of M/string theory, JHEP 03 (2009) 045 [arXiv:0901.2003] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  28. P. de Medeiros, J. Figueroa-O’Farrill, E. Mendez-Escobar and P. Ritter, Metric 3-Lie algebras for unitary Bagger-Lambert theories, JHEP 04 (2009) 037 [arXiv:0902.4674] [SPIRES].

    Article  Google Scholar 

  29. J. Gomis, G. Milanesi and J.G. Russo, Bagger-Lambert Theory for General Lie Algebras, JHEP 06 (2008) 075 [arXiv:0805.1012] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  30. S. Benvenuti, D. Rodriguez-Gomez, E. Tonni and H. Verlinde, N = 8 superconformal gauge theories and M2 branes, JHEP 01 (2009) 078 [arXiv:0805.1087] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  31. P.-M. Ho, Y. Imamura and Y. Matsuo, M2 to D2 revisited, JHEP 07 (2008) 003 [arXiv:0805.1202] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  32. M.A. Bandres, A.E. Lipstein and J.H. Schwarz, Ghost-Free Superconformal Action for Multiple M2-Branes, JHEP 07 (2008) 117 [arXiv:0806.0054] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  33. H. Nishino and S. Rajpoot, Triality and Bagger-Lambert Theory, Phys. Lett. B 671 (2009) 415 [arXiv:0901.1173] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  34. A. Gustavsson and S.-J. Rey, Enhanced N = 8 Supersymmetry of ABJM Theory on R(8) and R(8)/Z(2), arXiv:0906.3568 [SPIRES].

  35. P. de Medeiros, J. Figueroa-O’Farrill, E. Mendez-Escobar and P. Ritter, On the Lie-algebraic origin of metric 3-algebras, Commun. Math. Phys. 290 (2009) 871 [arXiv:0809.1086] [SPIRES].

    Article  MATH  ADS  Google Scholar 

  36. S. Cherkis, V. Dotsenko and C. Sämann, On Superspace Actions for Multiple M2-Branes, Metric 3-Algebras and their Classification, Phys. Rev. D 79 (2009) 086002 [arXiv:0812.3127] [SPIRES].

    ADS  Google Scholar 

  37. L. Susskind, Another conjecture about M(atrix) theory, hep-th/9704080 [SPIRES].

  38. A. Sen, D0 branes on T n and matrix theory, Adv. Theor. Math. Phys. 2 (1998) 51 [hep-th/9709220] [SPIRES].

    MATH  MathSciNet  Google Scholar 

  39. N. Seiberg, Why is the matrix model correct?, Phys. Rev. Lett. 79 (1997) 3577 [hep-th/9710009] [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  40. J. Polchinski, M-theory and the light cone, Prog. Theor. Phys. Suppl. 134 (1999) 158 [hep-th/9903165] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  41. J. Polchinski, String Theory Vol. 2: Superstring Theory and Beyond, Cambridge University Press, Cambridge U.K. (1998).

    Google Scholar 

  42. K. Becker, M. Becker and J.H. Schwarz, String Theory and M-theory, Cambridge University Press, Cambridge U.K. (2007).

    MATH  Google Scholar 

  43. O. Aharony, O. Bergman and D.L. Jafferis, Fractional M2-branes, JHEP 11 (2008) 043 [arXiv:0807.4924] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  44. Y. Pang and T. Wang, From N M2’s to N D2’s, Phys. Rev. D 78 (2008) 125007 [arXiv:0807.1444] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  45. M. Hanada, L. Mannelli and Y. Matsuo, Large-N reduced models of supersymmetric quiver, Chern-Simons gauge theories and ABJM, JHEP 11 (2009) 087 [arXiv:0907.4937] [SPIRES].

    Article  ADS  Google Scholar 

  46. G. Ishiki, S. Shimasaki and A. Tsuchiya, Large-N reduction for Chern-Simons theory on S 3, Phys. Rev. D 80 (2009) 086004 [arXiv:0908.1711] [SPIRES].

    ADS  Google Scholar 

  47. H. Kawai, S. Shimasaki and A. Tsuchiya, Large-N reduction on group manifolds, arXiv:0912.1456 [SPIRES].

  48. G. Ishiki, S. Shimasaki and A. Tsuchiya, A Novel Large-N Reduction on S 3 : Demonstration in Chern-Simons Theory, Nucl. Phys. B 834 (2010) 423 [1001.4917] [SPIRES].

    ADS  Google Scholar 

  49. S. Mukhi and C. Papageorgakis, M2 to D2, JHEP 05 (2008) 085 [arXiv:0803.3218] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  50. W. Taylor, D-brane field theory on compact spaces, Phys. Lett. B 394 (1997) 283 [hep-th/9611042] [SPIRES].

    ADS  Google Scholar 

  51. L. Susskind, T duality in M(atrix) theory and S duality in field theory, hep-th/9611164 [SPIRES].

  52. S. Sethi and L. Susskind, Rotational Invariance in the M(atrix) Formulation of Type IIB Theory, Phys. Lett. B 400 (1997) 265 [hep-th/9702101] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  53. M. Rozali, Matrix theory and U-duality in seven dimensions, Phys. Lett. B 400 (1997) 260 [hep-th/9702136] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  54. M. Berkooz, M. Rozali and N. Seiberg, Matrix description of M-theory on T 4 and T 5, Phys. Lett. B 408 (1997) 105 [hep-th/9704089] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  55. A. Mohammed, J. Murugan and H. Nastase, Looking for a Matrix model of ABJM, 1003.2599 [SPIRES].

Download references

Author information

Authors and Affiliations

  1. Department of Natural Science, Faculty of Education, Hirosaki University, Bunkyo-cho 1, Hirosaki, Aomori, 036-8560, Japan

    Matsuo Sato

Authors
  1. Matsuo Sato
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matsuo Sato.

Additional information

ArXiv ePrint: 1003.4694

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sato, M. Model of M-theory with eleven matrices. J. High Energ. Phys. 2010, 26 (2010). https://doi.org/10.1007/JHEP07(2010)026

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP07(2010)026

Keywords

Search

Navigation