Cartan-Weyl 3-algebras and the BLG theory. I: classification of Cartan-Weyl 3-algebras

Article
First Online:
Received:
Accepted:

Abstract

As Lie algebras of compact connected Lie groups, semisimple Lie algebras have wide applications in the description of continuous symmetries of physical systems. Mathematically, semisimple Lie algebra admits a Cartan-Weyl basis of generators which consists of a Cartan subalgebra of mutually commuting generators H I and a number of step generators E α that are characterized by a root space of non-degenerate one-forms α. This simple decomposition in terms of the root space allows for a complete classification of semisimple Lie algebras. In this paper, we introduce the analogous concept of a Cartan-Weyl Lie 3-algebra. We analyze their structure and obtain a complete classification of them. Many known examples of metric Lie 3-algebras (e.g. the Lorentzian 3-algebras) are special cases of the Cartan-Weyl 3-algebras. Due to their elegant and simple structure, we speculate that Cartan-Weyl 3-algebras may be useful for describing some kinds of generalized symmetries. As an application, we consider their use in the Bagger-Lambert-Gustavsson (BLG) theory.

Keywords

Gauge Symmetry M-Theory 
This is a preview of subscription content, log in to check access.

References

  1. [1]
    I.R. Klebanov and A.A. Tseytlin, Entropy of Near-Extremal Black p-branes, Nucl. Phys. B 475 (1996) 164 [hep-th/9604089] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  2. [2]
    J. Bagger and N. Lambert, Modeling multiple M2’s, Phys. Rev. D 75 (2007) 045020 [hep-th/0611108] [SPIRES].MathSciNetADSGoogle Scholar
  3. [3]
    J. Bagger and N. Lambert, Gauge Symmetry and Supersymmetry of Multiple M2-Branes, Phys. Rev. D 77 (2008) 065008 [arXiv:0711.0955] [SPIRES].MathSciNetADSGoogle Scholar
  4. [4]
    J. Bagger and N. Lambert, Comments On Multiple M2-branes, JHEP 02 (2008) 105 [arXiv:0712.3738] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  5. [5]
    A. Gustavsson, Algebraic structures on parallel M2-branes, Nucl. Phys. B 811 (2009) 66 [arXiv:0709.1260] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  6. [6]
    J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [hep-th/9711200] [SPIRES].MATHMathSciNetADSGoogle Scholar
  7. [7]
    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].CrossRefMathSciNetADSGoogle Scholar
  8. [8]
    I.R. Klebanov and G. Torri, M2-branes and AdS/CFT, Int. J. Mod. Phys. A 25 (2010) 332 [arXiv:0909.1580] [SPIRES].ADSGoogle Scholar
  9. [9]
    D. Berenstein and D. Trancanelli, Three-dimensional N =6 SCFT’s and their membrane dynamics, Phys. Rev. D 78 (2008) 106009 [arXiv:0808.2503] [SPIRES].MathSciNetADSGoogle Scholar
  10. [10]
    D. Gaiotto and D.L. Jafferis, Notes on adding D6 branes wrapping RP3 in AdS4 × CP3, arXiv:0903.2175 [SPIRES].
  11. [11]
    S. Kim, The complete superconformal index for N =6 Chern-Simons theory, Nucl. Phys. B 821 (2009) 241 [arXiv:0903.4172] [SPIRES].CrossRefADSGoogle Scholar
  12. [12]
    M.K. Benna, I.R. Klebanov and T. Klose, Charges of Monopole Operators in Chern-Simons Yang-Mills Theory, JHEP 01 (2010) 110 [arXiv:0906.3008] [SPIRES].CrossRefADSGoogle Scholar
  13. [13]
    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].
  14. [14]
    Y. Imamura, Monopole operators in N =4 Chern-Simons theories and wrapped M2-branes, Prog. Theor. Phys. 121 (2009) 1173 [arXiv:0902.4173] [SPIRES].MATHCrossRefADSGoogle Scholar
  15. [15]
    Y. Imamura and S. Yokoyama, A Monopole Index for N =4 Chern-Simons Theories, Nucl. Phys. B 827 (2010) 183 [arXiv:0908.0988] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  16. [16]
    C.-S. Chu and D.J. Smith, Towards the Quantum Geometry of the M5-brane in a Constant C-Field from Multiple Membranes, JHEP 04 (2009) 097 [arXiv:0901.1847] [SPIRES].CrossRefADSGoogle Scholar
  17. [17]
    C.-S. Chu and D.J. Smith, Multiple Self-Dual Strings on M5-Branes, JHEP 01 (2010) 001 [arXiv:0909.2333] [SPIRES].CrossRefADSGoogle Scholar
  18. [18]
    N. Lambert and C. Papageorgakis, Nonabelian (2,0) Tensor Multiplets and 3-algebras, JHEP 08 (2010) 083 [arXiv:1007.2982] [SPIRES].CrossRefADSGoogle Scholar
  19. [19]
    N. Jacobson, Lie Algebras, Dover Publications, New York (1979).Google Scholar
  20. [20]
    A.L. Onishchik and E.B. Vinberg, Lie Groups and Lie Algebras III: Structure of Lie Groups and Lie Algebras (Encyclopaedia of Mathematical Sciences), Volume III), Springer, (1994).Google Scholar
  21. [21]
    C.S. Chu, Cartan-Weyl 3-algebras and the BLG Theory II: Strong Semisimplicity and Generalized Cartan-Weyl 3-algebras, in preparation.Google Scholar
  22. [22]
    V. Filippov, n-Lie algebras, Sibirsk. Mat. Zh. 26 (1985) 126.MATHGoogle Scholar
  23. [23]
    S.M. Kasymov, Theory of n-Lie algebras, Algebra i Logika 26 (1987) 277.MATHMathSciNetGoogle Scholar
  24. [24]
    J. Gomis, G. Milanesi and J.G. Russo, Bagger-Lambert Theory for General Lie Algebras, JHEP 06 (2008) 075 [arXiv:0805.1012] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  25. [25]
    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].CrossRefMathSciNetADSGoogle Scholar
  26. [26]
    P.-M. Ho, Y. Imamura and Y. Matsuo, M2 to D2 revisited, JHEP 07 (2008) 003 [arXiv:0805.1202] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  27. [27]
    P. de Medeiros, J.M. Figueroa-O’Farrill and E. Mendez-Escobar, Metric Lie 3-algebras in Bagger-Lambert theory, JHEP 08 (2008) 045 [arXiv:0806.3242] [SPIRES].CrossRefGoogle Scholar
  28. [28]
    A. Basu and J .A. Harvey, The M2-M5 brane system and a generalized Nahm’s equation, Nucl. Phys. B 713 (2005) 136 [hep-th/0412310] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  29. [29]
    U. Gran, B.E.W. Nilsson and C. Petersson, On relating multiple M2 and D 2-branes, JHEP 10 (2008) 067 [arXiv:0804.1784] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  30. [30]
    G. Papadopoulos, M2-branes, 3-Lie Algebras and Plucker relations, JHEP 05 (2008) 054 [arXiv:0804.2662] [SPIRES].CrossRefADSGoogle Scholar
  31. [31]
    J.P. Gauntlett and J.B. Gutowski, Constraining Maximally Supersymmetric Membrane Actions, JHEP 06 (2008) 053 [arXiv:0804.3078] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  32. [32]
    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].CrossRefGoogle Scholar
  33. [33]
    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].CrossRefMathSciNetADSGoogle Scholar
  34. [34]
    J. Gomis, D. Rodriguez-Gomez, M. Van Raamsdonk and H. Verlinde, Supersymmetric Yang-Mills Theory From Lorentzian Three-Algebras, JHEP 08 (2008) 094 [arXiv:0806.0738] [SPIRES].CrossRefADSGoogle Scholar
  35. [35]
    B. Ezhuthachan, S. Mukhi and C. Papageorgakis, D2 to D2, JHEP 07 (2008) 041 [arXiv:0806.1639] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  36. [36]
    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].CrossRefMathSciNetADSGoogle Scholar
  37. [37]
    N.R. Constable, R.C. Myers and O. Tafjord, The noncommutative bion core, Phys. Rev. D 61 (2000) 106009 [hep-th/9911136] [SPIRES].MathSciNetADSGoogle Scholar
  38. [38]
    N.R. Constable, R.C. Myers and O. Tafjord, Fuzzy funnels: Non-abelian brane intersections, hep-th/0105035 [SPIRES].
  39. [39]
    H. Nastase, C. Papageorgakis and S. Ramgoolam, The fuzzy S 2 structure of M2-M5 systems in ABJM membrane theories, JHEP 05 (2009) 123 [arXiv:0903.3966] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  40. [40]
    N. Akerblom, C. Sämann and M. Wolf, Marginal Deformations and 3-Algebra Structures, Nucl. Phys. B 826 (2010) 456 [arXiv:0906.1705] [SPIRES].CrossRefADSGoogle Scholar
  41. [41]
    N. Lambert and D. Tong, Membranes on an Orbifold, Phys. Rev. Lett. 101 (2008) 041602 [arXiv:0804.1114] [SPIRES].CrossRefMathSciNetADSGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2010

Authors and Affiliations

  1. 1.Centre for Particle Theory and Department of MathematicsDurham UniversityDurhamU.K.

Personalised recommendations