Skip to main content
SpringerLink
Log in

On the Lie-Algebraic Origin of Metric 3-Algebras

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

Since the pioneering work of Bagger–Lambert and Gustavsson, there has been a proliferation of three-dimensional superconformal Chern–Simons theories whose main ingredient is a metric 3-algebra. On the other hand, many of these theories have been shown to allow for a reformulation in terms of standard gauge theory coupled to matter, where the 3-algebra does not appear explicitly. In this paper we reconcile these two sets of results by pointing out the Lie-algebraic origin of some metric 3-algebras, including those which have already appeared in three-dimensional superconformal Chern–Simons theories. More precisely, we show that the real 3-algebras of Cherkis–Sämann, which include the metric Lie 3-algebras as a special case, and the hermitian 3-algebras of Bagger–Lambert can be constructed from pairs consisting of a metric real Lie algebra and a faithful (real or complex, respectively) unitary representation. This construction generalises and we will see how to construct many kinds of metric 3-algebras from pairs consisting of a real metric Lie algebra and a faithful (real, complex or quaternionic) unitary representation. In the real case, these 3-algebras are precisely the Cherkis–Sämann algebras, which are then completely characterised in terms of this data. In the complex and quaternionic cases, they constitute generalisations of the Bagger–Lambert hermitian 3-algebras and anti-Lie triple systems, respectively, which underlie N = 6 and N = 5 superconformal Chern–Simons theories, respectively. In the process we rederive the relation between certain types of complex 3-algebras and metric Lie superalgebras.

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. Bagger J., Lambert N.: Modeling multiple M2’s. Phys. Rev. D 75, 045020 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  2. Gustavsson, A.: Algebraic structures on parallel M2-branes. http://arxiv.org/abs/0709.1260v5[hep-th], 2008

  3. Bagger J., Lambert N.: Gauge symmetry and supersymmetry of multiple M2-branes. Phys. Rev. D 77, 065008 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  4. Gaiotto, D., Witten, E.: Janus Configurations, Chern-Simons Couplings, And The Theta-Angle in N = 4 Super Yang-Mills Theory. http://arxiv.org/abs/0804.2907v1[hep-th], 2008

  5. Hosomichi K., Lee K.-M., Lee S., Lee S., Park J.: N = 4 Superconformal Chern-Simons Theories with Hyper and Twisted Hyper Multiplets. JHEP 07, 091 (2008)

    Article  MathSciNet  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  7. Benna M., Klebanov I., Klose T., Smedbäck M.: Superconformal Chern–Simons theories and AdS4/CFT3 correspondence. JHEP 0809, 072 (2008)

    Article  ADS  Google Scholar 

  8. Mauri A., Petkou A.C.: An N = 1 Superfield Action for M2 branes. Phys. Lett. B 666, 527–532 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  9. Hosomichi K., Lee K.-M., Lee S., Lee S., Park J.: N = 5,6 Superconformal Chern-Simons Theories and M2-branes on Orbifolds. JHEP 09, 002 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  10. Bagger J., Lambert N.: Three-algebras and \(\mathcal{N}\,=\,6\) Chern–Simons gauge theories. Phys. Rev. D 79, 025002 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  11. Cherkis S., Sämann C.: Multiple M2-branes and generalized 3-Lie algebras. Phys. Rev. D 78, 066019 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  12. Schnabl, M., Tachikawa, Y.: Classification of N = 6 superconformal theories of ABJM type. http://arxiv.org/abs/0807.1102v1[hep-th], 2008

  13. Aharony O., Bergman O., Jafferis D.L.: Fractional M2-branes. JHEP 0811, 043 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  14. Ooguri H., Park C.-S.: Superconformal Chern-Simons Theories and the Squashed Seven Sphere. JHEP 0811, 082 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  15. Jafferis D.L., Tomasiello A.: A simple class of N = 3 gauge/gravity duals. JHEP 0810, 101 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  16. Bergshoeff E.A., de Roo M., Hohm O.: Multiple M2-branes and the embedding tensor. Class. Quant. Grav. 25, 142001 (2008)

    Article  ADS  Google Scholar 

  17. Bergshoeff E.A., de Roo M., Hohm O., Roest D.: Multiple Membranes from Gauged Supergravity. JHEP 0808, 091 (2008)

    Article  ADS  Google Scholar 

  18. Bergshoeff E.A., Hohm O., Roest D., Samtleben H., Sezgin E.: The Superconformal Gaugings in Three Dimensions. JHEP 07, 1111 (2008)

    Google Scholar 

  19. Schwarz J.H.: Superconformal Chern-Simons theories. JHEP 11, 078 (2004)

    Article  ADS  Google Scholar 

  20. Gaiotto D., Yin X.: Notes on superconformal Chern-Simons-matter theories. JHEP 08, 056 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  21. de Medeiros P., Figueroa-O’Farrill J., Méndez-Escobar E.: Lorentzian Lie 3-algebras and their Bagger–Lambert moduli space. JHEP 07, 111 (2008)

    Article  ADS  Google Scholar 

  22. Nagy, P.-A.: Prolongations of Lie algebras and applications. http://arxiv.org/abs/0712.1398v2[math.DG], 2008

  23. Papadopoulos G.: M2-branes, 3-Lie Algebras and Plucker relations. JHEP 05, 054 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  24. Gauntlett, J.P., Gutowski, J.B.: Constraining maximally supersymmetric membrane actions. http://arxiv.org/abs/0804.3078v3[hep-th], 2008; to appear JHEP

  25. Faulkner J.R.: On the geometry of inner ideals. J. Algebra 26, 1–9 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  26. Gustavsson A.: One-loop corrections to Bagger-Lambert theory. Nucl. Phys. B 807, 315–333 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  27. Van Raamsdonk M.: Comments on the Bagger-Lambert theory and multiple M2- branes. JHEP 0805, 105 (2008)

    Article  ADS  Google Scholar 

  28. Gomis J., Milanesi G., Russo J.G.: Bagger-Lambert Theory for General Lie Algebras. JHEP 06, 075 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  29. Benvenuti, S., Rodríguez-Gómez, D., Tonni, E., Verlinde, H.: N = 8 superconformal gauge theories and M2 branes. http://arxiv.org/abs/0805.1087v1[hep-th], 2008

  30. Ho P.-M., Imamura Y., Matsuo Y.: M2 to D2 revisited. JHEP 07, 003 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  31. Nambu Y.: Generalized Hamiltonian dynamics. Phys. Rev. D 7, 2405–2414 (1973)

    Article  MathSciNet  ADS  Google Scholar 

  32. Yamazaki M.: Octonions, G 2 and generalized Lie 3-algebras. Phys. Lett. B 670, 215–219 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  33. Figueroa-O’Farrill J.M., Meessen P., Philip S.: Supersymmetry and homogeneity of M-theory backgrounds. Class. Quant. Grav. 22, 207–226 (2005)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  34. Filippov V.: n-Lie algebras. Sibirsk. Mat. Zh. 26(6), 126–140, 191 (1985)

    MATH  MathSciNet  Google Scholar 

  35. Figueroa-O’Farrill J.M., Papadopoulos G.: Plücker-type relations for orthogonal planes. J. Geom. Phys. 49, 294–331 (2004)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  36. Jacobson N.: General representation theory of Jordan algebras. Trans. Amer. Math. Soc. 70, 509–530 (1951)

    Article  MATH  MathSciNet  Google Scholar 

  37. Lister W.G.: A structure theory of Lie triple systems. Trans. Am. Math. Soc. 72(2), 217–242 (1952)

    Article  MATH  MathSciNet  Google Scholar 

  38. Yamaguti K.: On algebras of totally geodesic spaces (Lie triple systems). J. Sci. Hiroshima Univ. Ser. A 21, 107–113 (1957/1958)

    MathSciNet  Google Scholar 

  39. Nilsson, B.E.W., Palmkvist, J.: Superconformal M2-branes and generalized Jordan triple systems. http://arxiv.org/abs/0807.5134v2[hep-th], 2008

  40. Faulkner J.R., Ferrar J.C.: Simple anti-Jordan pairs. Comm. Algebra 8(11), 993–1013 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  41. Okubo S.: Construction of Lie superalgebras from triple product systems. AIP Conf. Proc. 687, 33–40 (2003)

    Article  ADS  Google Scholar 

  42. Kamiya N., Okubo S.: Construction of Lie superalgebras D(2,1;α), G(3) and F(4) from some triple systems. Proc. Edinb. Math. Soc. (2) 46(1), 87–98 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  43. de Medeiros P., Figueroa-O’Farrill J., Méndez-Escobar E.: Metric Lie 3-algebras in Bagger–Lambert theory. JHEP 08, 045 (2008)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, James Clerk Maxwell Building, King’s Buildings, Edinburgh, EH9 3JZ, UK

    Paul de Medeiros, José Figueroa-O’Farrill, Elena Méndez-Escobar & Patricia Ritter

  2. Department de Física Teòrica& IFIC (CSIC-UVEG), Universitat de València, 46100, Burjassot, Spain

    José Figueroa-O’Farrill

Authors
  1. Paul de Medeiros
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. José Figueroa-O’Farrill
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Elena Méndez-Escobar
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Patricia Ritter
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to José Figueroa-O’Farrill.

Additional information

Communicated by G.W. Gibbons

Rights and permissions

Reprints and permissions

About this article

Cite this article

de Medeiros, P., Figueroa-O’Farrill, J., Méndez-Escobar, E. et al. On the Lie-Algebraic Origin of Metric 3-Algebras. Commun. Math. Phys. 290, 871–902 (2009). https://doi.org/10.1007/s00220-009-0760-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-009-0760-1

Keywords

Search

Navigation