Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Letter
  • Open Access

Manakov-Zakharov-Ward model as an integrable decoupling limit of the membrane

David Osten*

  • *Contact author: david.osten@uwr.edu.pl

Phys. Rev. D 113, L061902 – Published 25 March, 2026

DOI: https://doi.org/10.1103/5gvf-rsdk

Abstract

This paper proposes a novel decoupling limit of the membrane, leading to the (1+2)-dimensional classically integrable model originally introduced by Manakov, Zakharov, and Ward. This limit is the large-wrapping regime of a membrane propagating toy background of the form Rt×T2×G subject to scaling limit, where G is a Lie group and the geometry is supported by a four-form flux. Such toy backgrounds can arise from consistent eleven-dimensional supergravity solutions, exemplified by the uplift of the pure Neveu-Schwarz-Neveu-Schwarz AdS3×S3×T4 background. The scaling limit can be interpreted as a simultaneous small tension and non- or hyperrelativistic limit.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (50)

  1. T. Banks, W. Fischler, S. H. Shenker, and L. Susskind, M theory as a matrix model: A conjecture, Phys. Rev. D 55, 5112 (1997).
  2. E. Bergshoeff, E. Sezgin, and P. Townsend, Supermembranes and eleven-dimensional supergravity, Phys. Lett. B 189, 75 (1987).
  3. M. Duff, P. S. Howe, T. Inami, and K. Stelle, Superstrings in D=10 from supermembranes in D=11, Phys. Lett. B 191, 70 (1987).
  4. O. Aharony, O. Bergman, D. L. Jafferis, and J. Maldacena, N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, J. High Energy Phys. 10 (2008) 091.
  5. S. Giombi and A. A. Tseytlin, Wilson loops at large N and the quantum M2-brane, Phys. Rev. Lett. 130, 201601 (2023).
  6. M. Beccaria, S. Giombi, and A. A. Tseytlin, Instanton contributions to the ABJM free energy from quantum M2 branes, J. High Energy Phys. 10 (2023) 029.
  7. S. V. Manakov and V. E. Zakharov, Three-dimensional model of relativistic-invariant field theory, integrable by the inverse scattering transform, Lett. Math. Phys. 5, 247 (1981).
  8. R. S. Ward, Soliton solutions in an integrable chiral model in 2+1 dimensions, J. Math. Phys. (N.Y.) 29, 386 (1988).
  9. R. Andringa, E. Bergshoeff, J. Gomis, and M. de Roo, “Stringy” Newton-Cartan gravity, Classical Quantum Gravity 29, 235020 (2012).
  10. T. Harmark, J. Hartong, L. Menculini, N. A. Obers, and Z. Yan, Strings with non-relativistic conformal symmetry and limits of the AdS/CFT correspondence, J. High Energy Phys. 11 (2018) 190.
  11. E. Bergshoeff, J. Gomis, and Z. Yan, Nonrelativistic string theory and T-duality, J. High Energy Phys. 11 (2018) 133.
  12. G. Oling and Z. Yan, Foundations of non-relativistic string theory, Front. Phys. 10, 832271 (2022).
  13. N. Lambert and J. Smith, Non-relativistic intersecting branes, Newton-Cartan geometry and AdS/CFT, J. High Energy Phys. 07 (2024) 224.
  14. N. Lambert and J. Smith, Reciprocal non-relativistic decoupling limits of string theory and M-theory, J. High Energy Phys. 12 (2024) 094.
  15. Chris D. A. Blair, J. Lahnsteiner, Niels A. Obers, and Z. Yan, Unification of decoupling limits in string and M theory, Phys. Rev. Lett. 132, 161603 (2024).
  16. C. D. A. Blair, J. Lahnsteiner, N. A. Obers, and Z. Yan, Matrix theory reloaded: A BPS road to holography, J. High Energy Phys. 02 (2025) 024.
  17. See Supplemental Material at http://link.aps.org/supplemental/10.1103/5gvf-rsdk for a very brief introduction to the non-Lorentzian limit of 11d supergravity, and a derivation of the double scaling limit for the membrane theory in the Polyakov formulation.
  18. B. de Wit, J. Hoppe, and H. Nicolai, On the quantum mechanics of supermembranes, Nucl. Phys. B305, 545 (1988).
  19. B. de Wit, M. Luscher, and H. Nicolai, The supermembrane is unstable, Nucl. Phys. B320, 135 (1989).
  20. F. K. Seibold and A. A. Tseytlin, Scattering on the supermembrane, J. High Energy Phys. 08 (2024) 102.
  21. B. Hoare, Integrable deformations of sigma models, J. Phys. A 55, 093001 (2022).
  22. S. Demulder, S. Driezen, B. Knighton, G. Oling, A. L. Retore, F. K. Seibold, A. Sfondrini, and Z. Yan, Exact approaches on the string worldsheet, J. Phys. A 57, 423001 (2024).
  23. A. Sergyeyev, Multidimensional integrable systems from contact geometry, Boletin de la Sociedad Matematica Mexicana 31, 26 (2025).
  24. T. A. Ioannidou and R. S. Ward, Conserved quantities for integrable chiral equations in (2+1)-dimensions, Phys. Lett. A 208, 209 (1995).
  25. L. Mason and N. Woodhouse, Integrability, Self-Duality, and Twistor Theory, London Mathematical Society Monographs, New Series Vol. 15 (Oxford University Press, New York, 1996).
  26. A. Dimakis and F. Muller-Hoissen, Dispersionless limit of the noncommutative potential KP hierarchy and solutions of the pseudodual chiral model in 2+1 dimensions, J. Phys. A 41, 265205 (2008).
  27. T. Ioannidou and N. Manton, Soliton scattering in the Ward model, Prog. Theor. Phys. 103, 1069 (2000).
  28. K. Costello, E. Witten, and M. Yamazaki, Gauge theory and integrability, I, ICCM Not. 06, 46 (2018).
  29. K. Costello, E. Witten, and M. Yamazaki, Gauge theory and integrability, II, ICCM Not. 06, 120 (2018).
  30. R. Bittleston and D. Skinner, Twistors, the ASD Yang-Mills equations and 4d Chern-Simons theory, J. High Energy Phys. 02 (2023) 227.
  31. A. Schenkel and B. Vicedo, 5d 2-Chern-Simons theory and 3D integrable field theories, Commun. Math. Phys. 405, 293 (2024).
  32. B. de Wit, J. Hoppe, and H. Nicolai, On the quantum mechanics of supermembranes, Nucl. Phys. B305, 545 (1988).
  33. J. Bjornsson and S. Hwang, The membrane as a perturbation around string like configurations, Nucl. Phys. B689, 37 (2004).
  34. J. Bjornsson and S. Hwang, Stretched quantum membranes, Nucl. Phys. B727, 77 (2005).
  35. C. D. A. Blair, D. Gallegos, and N. Zinnato, A non-relativistic limit of M-theory and 11-dimensional membrane Newton-Cartan geometry, J. High Energy Phys. 10 (2021) 015.
  36. D. Hansen, N. A. Obers, G. Oling, and B. T. Søgaard, Carroll expansion of general relativity, SciPost Phys. 13, 055 (2022).
  37. E. A. Bergshoeff, C. D. A. Blair, J. Lahnsteiner, and J. Rosseel, The surprising structure of non-relativistic 11-dimensional supergravity, J. High Energy Phys. 12 (2024) 010.
  38. A. Dimakis and F. Mueller-Hoissen, Bidifferential calculi and integrable models, J. Phys. A 33, 957 (2000).
  39. D. Wu, The Cauchy problem of the Ward equation, J. Funct. Anal. 256, 215 (2009).
  40. R. S. Ward, Classical solutions of the chiral model, unitons, and holomorphic vector bundles, Commun. Math. Phys. 128, 319 (1990).
  41. R. S. Ward, Nontrivial scattering of localized solitons in a (2+1)-dimensional integrable system, Phys. Lett. A 208, 203 (1995).
  42. D. Osten, Current algebras, generalised fluxes and non-geometry, J. Phys. A 53, 265402 (2020).
  43. R. Borsato and S. Driezen, Supergravity solution-generating techniques and canonical transformations of σ-models from O(D,D), J. High Energy Phys. 05 (2021) 180.
  44. R. Borsato, S. Driezen, and F. Hassler, An algebraic classification of solution generating techniques, Phys. Lett. B 823, 136771 (2021).
  45. M. Hatsuda and K. Kamimura, SL(5) duality from canonical M2-brane, J. High Energy Phys. 11 (2012) 001.
  46. Y. Sakatani and S. Uehara, Non-Abelian U-duality for membranes, Prog. Theor. Exp. Phys. 2020, 073B01 (2020).
  47. Y. Sakatani and S. Uehara, Born sigma model for branes in exceptional geometry, Prog. Theor. Exp. Phys. 2020, 073B05 (2020).
  48. D. Osten, Currents, charges and algebras in exceptional generalised geometry, J. High Energy Phys. 06 (2021) 070.
  49. M. Hatsuda, O. Hulík, W. D. Linch, W. D. Siegel, D. Wang, and Y.-P. Wang, A-theory—A brane world-volume theory with manifest U-duality, J. High Energy Phys. 10 (2023) 087.
  50. D. Osten, On the universal exceptional structure of world-volume theories in string and M-theory, Phys. Lett. B 855, 138814 (2024).

Outline

Information

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation