Skip to main content
SpringerLink
Log in

A toy model of open membrane field theory in constant 3-form flux

  • Research Article
  • Published:
General Relativity and Gravitation Aims and scope Submit manuscript

Abstract

Based on an explicit computation of the scattering amplitude of four open membranes in a constant 3-form background, we construct a toy model of the field theory for open membranes in the large C field limit. It is a generalization of the noncommutative field theories which describe open strings in a constant 2-form flux. The noncommutativity due to the B-field background is now replaced by a nonassociative triplet product. The triplet product satisfies the consistency conditions of lattice 3D gravity, which is inherent in the world-volume theory of open membranes. We show the UV/IR mixing of the toy model by computing some Feynman diagrams. Inclusion of the internal degree of freedom is also possible through the idea of the cubic matrix.

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. de Wit B., Hoppe J., Nicolai H.(1988). On the quantum mechanics of supermembranes. Nucl. Phys. B 305: 545

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  3. de Wit B., Luscher M., Nicolai H.(1989). The supermembrane is unstable. Nucl. Phys. B 320: 135

    Article  ADS  Google Scholar 

  4. Nicolai, H., Helling, R.: Supermembranes and M(atrix) theory [arXiv:hep-th/9809103]

  5. Seiberg N., Witten E. (1999). String theory and noncommutative geometry. JHEP 9909: 032 [arXiv:hep-th/9908142]

    Article  ADS  MathSciNet  Google Scholar 

  6. Aoki H., Ishibashi N., Iso S., Kawai H., Kitazawa Y., Tada T.(2000). Noncommutative Yang–Mills in IIB matrix model. Nucl. Phys. B 565: 176 [arXiv:hep-th/9908141]

    Article  MATH  ADS  MathSciNet  Google Scholar 

  7. Minwalla S., Van Raamsdonk M., Seiberg N. (2000). Noncommutative perturbative dynamics. JHEP 0002: 020 [arXiv:hep-th/9912072]

    Article  ADS  Google Scholar 

  8. Hayakawa M. (2000). Perturbative analysis on infrared aspects of noncommutative QED on R**4. Phys. Lett. B 478: 394 [arXiv:hep-th/9912094]

    Article  MATH  ADS  MathSciNet  Google Scholar 

  9. Hoppe J.(1997). On M-algebras, the quantisation of Nambu-mechanics, and volume preserving diffeomorphisms. Helv. Phys. Acta 70: 302 [arXiv:hep-th/9602020]

    MATH  MathSciNet  Google Scholar 

  10. Awata H., Li M., Minic D., Yoneya T.(2001). On the quantization of Nambu brackets. JHEP 0102: 013 [arXiv:hep-th/9906248]

    Article  ADS  MathSciNet  Google Scholar 

  11. Xiong C.S.(2000). A note on the quantum Nambu bracket. Phys. Lett. B 486: 228 [arXiv:hep-th/0003292]

    Article  MATH  ADS  MathSciNet  Google Scholar 

  12. Bergshoeff E., Berman D.S., van der Schaar J.P., Sundell P. (2000). Anoncommutative M-theory five-brane. Nucl. Phys. B 590: 173 [arXiv:hep-th/0005026]

    Article  MATH  ADS  Google Scholar 

  13. Kawamoto S., Sasakura N. (2000). Open membranes in a constant C-field background and noncommutative boundary strings. JHEP 0007: 014 [arXiv:hep-th/0005123]

    Article  ADS  MathSciNet  Google Scholar 

  14. Matsuo Y., Shibusa Y.(2001). Volume preserving diffeomorphism and noncommutative branes. JHEP 0102: 006 [arXiv:hep-th/0010040]

    Article  ADS  MathSciNet  Google Scholar 

  15. Sakakibara M. (2000). Remarks on a deformation quantization of the canonical Nambu bracket. Prog. Theor. Phys. 104: 1067

    Article  ADS  MathSciNet  Google Scholar 

  16. Ikeda N. (2001). Deformation of BF theories, topological open membrane and a generalization of the star deformation. JHEP 0107: 037 [arXiv:hep-th/0105286]

    Article  ADS  Google Scholar 

  17. Kawamura Y.(2003). Cubic matrix, Nambu mechanics and beyond. Prog. Theor. Phys. 109: 153 [arXiv:hep-th/0207054]

    Article  MATH  ADS  MathSciNet  Google Scholar 

  18. Pioline B. (2002). Comments on the topological open membrane. Phys. Rev. D 66: 025010 [arXiv:hep-th/0201257]

    Article  ADS  MathSciNet  Google Scholar 

  19. Curtright T., Zachos C.K. (2002). Deformation quantization of superintegrable systems and Nambu mechanics. New J. Phys. 4: 83 [arXiv:hep-th/0205063]

    Article  ADS  MathSciNet  Google Scholar 

  20. Curtright T., Zachos C.K.(2003). Classical and quantum Nambumechanics. Phys. Rev. D 68: 085001 [arXiv:hep-th/0212267]

    Article  ADS  Google Scholar 

  21. Berman D.S., Pioline B. (2004). Open membranes, ribbons and deformed Schild strings. Phys. Rev. D 70: 045007 [arXiv:hep-th/0404049]

    Article  ADS  MathSciNet  Google Scholar 

  22. Basu A., Harvey J.A.(2005). The M2–M5 brane system and a generalized Nahm’s equation. Nucl. Phys. B 713: 136 [arXiv:hep-th/0412310]

    Article  MATH  ADS  MathSciNet  Google Scholar 

  23. Sasakura N. (2006). An invariant approach to dynamical fuzzy spaces with a three-index variable. Mod. Phys. Lett. A 21: 1017 [arXiv:hep-th/0506192]

    Article  MATH  ADS  MathSciNet  Google Scholar 

  24. Sasakura N.(2006). Tensor model and dynamical generation of commutative nonassociative fuzzy spaces.Class. Quant. Grav. 23: 5397 [arXiv:hep-th/0606066]

    Article  MATH  ADS  MathSciNet  Google Scholar 

  25. Bagger, J., Lambert, N.: Modeling multiple M2’s [arXiv:hep-th/0611108]

  26. Lee, S.: Superconformal field theories from crystal lattices. [arXiv:hep-th/0610204]

  27. Nambu Y. (1973). Generalized Hamiltonian dynamics. Phys. Rev. D 7: 2405

    Article  ADS  MathSciNet  Google Scholar 

  28. Takhtajan L.(1994). On foundation of the generalized Nambu mechanics (Second Version). Commun. Math. Phys. 160: 295 [arXiv:hep-th/9301111]

    Article  MATH  ADS  MathSciNet  Google Scholar 

  29. Dito G., Flato M., Sternheimer D., Takhtajan L.(1997). Deformation quantization and Nambu mechanics. Commun. Math. Phys. 183: 1 [arXiv:hep-th/9602016]

    Article  MATH  ADS  MathSciNet  Google Scholar 

  30. Jackiw R.(1985) 3-Cocycle in mathematics and physics. Phys. Rev. Lett. 54: 159

    Article  ADS  MathSciNet  Google Scholar 

  31. Cornalba L., Schiappa R.(2002). Nonassociative star product deformations for D-brane worldvolumes in curved backgrounds. Commun. Math. Phys. 225: 33 [arXiv:hep-th/0101219]

    Article  MATH  ADS  MathSciNet  Google Scholar 

  32. Ho P.M.(2001). Making non-associative algebra associative. JHEP 0111: 026 [arXiv:hep-th/0103024]

    Article  ADS  Google Scholar 

  33. Bouwknegt P., Hannabuss K., Mathai V.(2006). Nonassociative tori and applications to T-duality. Commun. Math. Phys. 264: 41 [arXiv:hep-th/0412092]

    Article  MATH  ADS  MathSciNet  Google Scholar 

  34. Ellwood, I., Hashimoto, A.: Effective descriptions of branes on non-geometric tori [arXiv:hep-th/ 0607135]

  35. Sasai Y., Sasakura N.(2006). One-loop unitarity of scalar field theories on poincare invariant commutative nonassociative spacetimes. JHEP 0609: 046 [arXiv:hep-th/0604194]

    Article  ADS  MathSciNet  Google Scholar 

  36. Ambjorn J., Durhuus B., Jonsson T.(1997). Quantum Geometry. Cambridge University Press, Cambridge

    Google Scholar 

  37. De Pietri R., Petronio C. (2000). Feynman diagrams of generalized matrix models and the associated manifolds in dimension 4. J. Math. Phys. 41: 6671 [arXiv:gr-qc/0004045]

    Article  MATH  ADS  MathSciNet  Google Scholar 

  38. Klebanov I.R., Tseytlin A.A. (1996). Entropy of near-extremal black p-branes.Nucl. Phys. B 475: 164 [arXiv:hep-th/9604089]

    MATH  MathSciNet  Google Scholar 

  39. Harvey J.A., Minasian R., Moore G.W. (1998). Non-abelian tensor-multiplet anomalies. JHEP 9809: 004 [arXiv:hep-th/9808060]

    Article  ADS  MathSciNet  Google Scholar 

  40. Henningson M., Skenderis K. (1998). The holographicWeyl anomaly. JHEP 9807: 023 [arXiv:hep-th/9806087]

    Article  ADS  MathSciNet  Google Scholar 

  41. Berman D.S., Copland N.B. (2006). A note on the M2-M5 brane system and fuzzy spheres. Phys. Lett. B 639: 553 [arXiv:hep-th/0605086]

    Article  ADS  MathSciNet  Google Scholar 

  42. Ramgoolam S. (2002). Higher dimensional geometries related to fuzzy odd dimensional spheres. JHEP 0210: 064 [arXiv:hep-th/0207111]

    Article  ADS  MathSciNet  Google Scholar 

  43. Ramgoolam S. (2001). On spherical harmonics for fuzzy spheres in diverse dimensions. Nucl. Phys. B 610: 461 [arXiv:hep-th/0105006]

    Article  MATH  ADS  MathSciNet  Google Scholar 

  44. Zwiebach B. (1993). For a review on closed string field theory, see: closed string field theory: quantum action and the B-V master equation. Nucl. Phys. B 390: 33 [arXiv:hep-th/9206084]

    Article  ADS  MathSciNet  Google Scholar 

  45. Duff M.J., Howe P.S., Inami T., Stelle K.S.(1987). Superstrings in D = 10 from supermembranes in D = 11. Phys. Lett. B 191: 70

    Article  ADS  MathSciNet  Google Scholar 

  46. Ponzano G., Regge T., Bloch F. (eds) (1968). In Spectroscopic and Group Theoretical Methods in Physics. North-Holland, Amsterdam

    Google Scholar 

  47. Turaev, G., Viro, O.Y. State sum invariants of 3-manifolds and quantum 6 j Symbols (1990)

  48. Ooguri H., Sasakura N.(1991). Discrete and continuum approaches to three-dimensional quantum gravity. Mod. Phys. Lett. A 6: 3591–3600

    Article  MATH  ADS  MathSciNet  Google Scholar 

  49. Chu C.S., Ho P.M. (1999) Noncommutative open string and D-brane. Nucl. Phys. B 550: 151 [arXiv:hep-th/9812219]

    Article  MATH  ADS  MathSciNet  Google Scholar 

  50. Chu C.S., Ho P.M.(2000). Constrained quantization of open string in background B field and Nucl. Phys. B 568: 447 [arXiv:hep-th/9906192]

    MATH  MathSciNet  Google Scholar 

  51. Schomerus V.(1999). D-branes and deformation quantization. JHEP 9906: 030 [arXiv:hep-th/9903205]

    Article  ADS  MathSciNet  Google Scholar 

  52. Yin Z. (1999). A note on space noncommutativity. Phys. Lett. B 466: 234 [arXiv:hep-th/9908152]

    Article  MATH  ADS  MathSciNet  Google Scholar 

  53. Bigatti D., Susskind L. (2000). Magnetic fields, branes and noncommutative geometry. Phys. Rev. D 62: 066004 [arXiv:hep-th/9908056]

    Article  ADS  MathSciNet  Google Scholar 

  54. Sakaguchi M., Yoshida K. (2006). Noncommutative M-branes from covariant open supermembranes. Phys. Lett. B 642: 400 [arXiv:hep-th/0608099]

    Article  ADS  MathSciNet  Google Scholar 

  55. Chu C.S., Ho P.M., Li M. (2000). Matrix theory in a constant C field background. Nucl. Phys. B 574: 275 [arXiv:hep-th/9911153]

    Article  MATH  ADS  MathSciNet  Google Scholar 

  56. Kontsevich M. (2003). Deformation quantization of Poisson manifolds, I. Lett. Math. Phys. 66: 157 [arXiv:q-alg/9709040]

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics and Center for Theoretical Sciences, National Taiwan University, Taipei, 10617, Taiwan, ROC

    Pei-Ming Ho

  2. Department of Physics, Faculty of Science, University of Tokyo, Tokyo, Japan

    Yutaka Matsuo

Authors
  1. Pei-Ming Ho
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yutaka Matsuo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pei-Ming Ho.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ho, PM., Matsuo, Y. A toy model of open membrane field theory in constant 3-form flux. Gen Relativ Gravit 39, 913–944 (2007). https://doi.org/10.1007/s10714-007-0433-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10714-007-0433-3

Keywords

Search

Navigation